Here’s a conjecture: If you put any significant amount of text on the internet under different names, those identities can be linked using only the text itself. This is possible (I conject) because of the statistical “fingerprint” you leave in everything you write.

Imagine a website where you can paste in some brand-new text someone just wrote. In return, the website provides links to all the text that writer has ever published under any name. It’s not perfect, but it’s pretty good. As far as I know, no such website exists—at least not on the public internet. But I suspect it’s possible and will soon become easy. This will pose some difficulty for pseudonymous blogging.

Note: I wrote most of this essay in mid-2025, after which I idiotically sat on it for a year tinkering with theorem statements that none of you will read.¹ In the meantime, LLMs have gotten much better at guessing authors from text. (Given the first 1000 words of a draft of this post, Claude 4.8 knows it’s me.) Still, I think we’re just getting started. I expect to see increasingly obscure writers identified from increasingly small bits of text. I expect that this work even when people are writing in a different register or about unrelated subjects. And I expect that everything I’ve ever written under any pseudonym will soon be linked to my genuine-nym.²

A stronger conjecture is that we’re heading towards a sort of generalized pseudpocalypse. Perhaps, in the future, if you interact with the world through essentially any high-bandwidth channel, then you identify yourself. Say you wear a mask in public and only speak by sub-vocalizing into a voice changer. That’s fine, you’ll still be identified using your body shape, gait, or chemical signature. Or say you don’t like your car being tracked everywhere, so you stop carrying a phone and you somehow convince lawmakers to ban license plates. No problem, your car will still be tracked using tiny scratches or unique pinging sounds from the engine. Or say you don’t like being tracked on the internet, so you lock down your browser profile, buy stuff only with Monero, and connect through a chain of three VPNs. That’s OK. You’ll still be identified through how you wiggle your finger as you scroll down the page. We’re all just too unique, and the information theoretic limit is coming for us.

Starting bits

Let’s start from first principles. Imagine that at birth, everyone is assigned a random binary string. Whenever you post anything on the internet, you’re required to sign it with that string. If the strings are very short, like 0110, then lots of other people will have the same one as you. But if the strings are very long, then yours would almost certainly be unique and it would be trivial to link all your pseudonyms.

Where’s the transition point? If you only know that the author is currently alive and living somewhere in the Anglosphere, it’s around 29 bits. That’s because if there are K digits, then there are 2ᴷ possible binary strings, and if K = 28.86, then 2ᴷ ≈ 490,000,000 is the number of currently-alive Anglosphere-dwellers. If the strings have fewer than 29 bits, then someone else will probably share your string. If they have more than 29 bits, then your string is probably unique.

We don’t (yet?) have to sign the things we write with immutable government-issued strings. But the way you write still provides lots of clues about you by way of your tone, personality, word choice, and so on.

Theoretically speaking, I think it has to be possible to link the identities of anyone who writes enough. Imagine again that everyone is assigned a random binary string at birth, but instead of you needing to sign the stuff you write with your string, each time you write a word, there’s some chance that a random bit from your string is revealed and added as a signature to your message. For example, maybe a signature of bit[129]=1 is added, indicating that your string at position 129 has value 1.

Think of your string as representing all your writing style quirks, and a bit being revealed as representing when you write something that reveals a preference. For example, maybe bit 18 indicates if you prefer to write your em-dashes with hideous spaces — like this — or without spaces—like this. If you use an em-dash, that bit is revealed.

So imagine you’ve written a lot under Pseudonym A, enough that the full bit-string has been revealed. Maybe it’s this:

Pseudonym A: 
110000001111001101110000100001
010100100101011110111001101000
100111110010101001101010111010

Now say you start writing under Pseudonym B. Initially, none of the bits will be known:

Pseudonym B: 
??????????????????????????????
??????????????????????????????
??????????????????????????????

But slowly, you’ll start to leak a few bits:

Pseudonym B:
?????00???1????1????????1??0??
?????01??1?????????11????01???
???1??????1???10??????????????

And eventually you’ll leak a lot of bits:

Pseudonym B:
???0?00?1?11??110??1????10?0??
?10?001?0101?11???11100?101???
???11?1??010??10?1?01??011????

Now think about this from the perspective of an “attacker” who wants to know if A and B are the same person. Let’s assume they’ve only seen the above bits, and have no information about anyone else. Then here’s what the attacker knows:

1. A and B have revealed K overlapping bits, which all match.
2. Different people have a 50% chance of matching on any given revealed bit.
3. Non-different people have a 100% chance of matching on any given revealed bit.
4. There are 490,000,000 people.

Intuitively, if K was 5, then the fact that all bits match wouldn’t prove much, since with 490 million people, lots of people would match on those bits by chance. But if K was 70, it’s extremely unlikely that two different people would share all of them, even with such a gigantic pool to start with. It turns out that if there are N other people with random bits, and you pick K of your bits, the probability that someone exists who matches all of them is 1 - (1-2⁻ᴷ)ᴺ. When N is 490 million, that looks like this:

Look at that, 29 appears again. (Isn’t math wonderful?) In general, the transition happens around whatever number of bits K makes 2ᴷ ≈ N, namely K = log₂(N).

If you reveal significantly fewer than 29 bits under pseudonym B, then it’s almost guaranteed that there’s someone else out there who matches all of them. But if you reveal significantly more than 29 bits, then there’s almost no chance that anyone else exists who matches all of them. So the attacker essentially knows that A and B are the same person. And I stress again: They know that without needing to see anything from the other 490 million people.

Of course, we don’t literally leak bits of immutable feature strings as we write. But you can make the model more realistic, and the same issue persists. If you want to reflect that text only provides noisy information about the writer, then you can add noise to the bits before they’re revealed. If you want to reflect that some writing styles are more common than others, then you can make the distribution over bit strings non-uniform. If you want to reflect that certain quirks are more obvious than others, you can give different bits different probabilities of being revealed. All these make the math more complicated. But they don’t change the basic conclusion: If your writing style contains at least 29 bits of information, and you do enough writing, you’re done.

That’s my argument that pseudpocalypse is possible. But I don’t just want to claim that it could happen, eventually. I think it is likely to happen, soon, and that the amount of text you need to reveal isn’t very large. To make that argument, we need to get specific: What features do people have that are reflected in their writing? How many bits of information do those features contain? How accurately can those bits be guessed from written text?

Note: To avoid this turning into a giant information theory lecture, I’ll mostly use words like “bit” and “information” without being 100% fully precise about what they mean. I’m doing that because I expect that most people reading this aren’t definition-of-bit fetishists, and anyway being hyper-technical would obscure the big picture. If you’re an information theory enthusiast and/or skeptical that I know what I’m doing, I refer you to the Section For Skeptical Information Theory Enthusiasts, below. Until then, use your intuition and have faith.

Feature space

Say you knew nothing about me other than that I wrote the above words. And say you had to guess my age or religion or occupation. You could guess, right? It wouldn’t be perfect, but you’d do much better than you would without being able to read those words. Thus, somehow, those words contain information about my demographic characteristics. So I tried to make a list of similar things that you could plausibly guess from text at least somewhat better then chance. Here’s what I came up with:

• Age
• Education
• Ethnicity
• Family status
• Income
• Marital status
• Mental health
• Native language
• Occupation
• Physical health
• Political leanings
• Region
• Religious affiliation
• Sex

In the same spirit, if you only read the above words, could you guess how extroverted or conscientious I am? Again, not perfectly. (When I meet people who read this blog, they usually seem surprised I can survive direct sunlight.) But still, I’m sure you’d do OK. So, again, these words contain information about my personality.

What features does personality have? The HEXACO model lists six, namely honesty-humility, emotionality, extraversion, agreeableness, conscientiousness, and openness to experience. I suspect those can all be guessed with reasonable accuracy from a long-enough writing sample. But could you guess more? For each of those six factors, the HEXACO model lists four “facets”. In the abstract, trying to guess 6 × 4 = 24 different personality features from text sounds ludicrous, but just look at them:

• Honesty-humility
  • Sincerity
  • Fairness
  • Greed avoidance
  • Modesty
• Emotionality
  • Fearfulness
  • Anxiety
  • Dependence
  • Sentimentality
• Extraversion
  • Social self-esteem
  • Social boldness
  • Sociability
  • Liveliness
• Agreeableness
  • Forgivingness
  • Gentleness
  • Flexibility
  • Patience
• Conscientiousness
  • Organization
  • Diligence
  • Perfectionism
  • Prudence
• Openness to experience
  • Aesthetic appreciation
  • Inquisitiveness
  • Creativity
  • Unconventionality

If you think about specific people, I think you can convince yourself that these 24 represent real things, and that it’s plausible to guess them from text. (Your favorite existential angst + science blogger, for example, might score lower on “modesty” than the other honesty-humility facets.) The different sub-factors are surely correlated, but not perfectly correlated.

Of course, the biggest thing you learn from people’s writing is how they write. Do they tend to pointlessly split infinitives? Do they use hyphen-connected words? Do they, incorrectly, position their adverbial clauses?

The idea of attributing authorship using writing style features goes back to at least 1440, when Lorenzo Valla demonstrated that the Donation of Constantine—in which Emperor Constantine supposedly donated the Roman Empire to the Catholic Church—used a vernacular that came from 400 years after Constantine’s death and was therefore a forgery. In 1851, Augustus De Morgan observed that average word length tends to be stable for the same author. The first “modern” attempt seemingly came in 1964, when Mosteller and Wallace published Inference in an Authorship Problem:

This study [attempts] to solve the authorship question of The Federalist papers; […]

Word counts are the variables used for discrimination. Since the topic written about heavily influences the rate with which a word is used, care in selection of words is necessary. The filler words of the language such as an, of, and upon, and, more generally, articles, preposition, and conjunctions provide fairly stable rates, whereas more meaningful words like war, executive, and legislature do not.

After an investigation of the distribution of these counts, the authors execute an analysis […] based on Bayesian methods. The conclusions about the authorship problem are that Madison rather than Hamilton wrote all 12 of the disputed papers.

Get that? The idea is that your usage of the word war depends mostly on if you happen to be talking about war. But your usage of upon mostly depends mostly on how much you like the word upon. To demonstrate this, they took 48 papers written by Hamilton and 50 by Madison and made this table of how many times they used by, from, and to:

Madison liked by. Hamilton was more a to man. Using these kinds of statistics, they concluded that the disputed Federalist papers must have been written by Madison.

So I did some research looking for other writing style features that are believed to be stable when people write about different subjects. I found that there are a lot. There were so many that I struggle to even organize them into meaningful groups:

Low-level frequencies:

• Word lengths
• Sentence lengths
• Paragraph lengths
• Punctuation frequencies (commas, colons, dashes, parentheses)
• Function word frequencies (the, of, and, to)
• Adverb frequencies
  • Intensifiers (very, really, quite, pretty, so)
  • Evidential markers (apparently, evidently, obviously)
  • Downtoners (somewhat, fairly, rather)
• Pronoun usage
  • Overall preferences (I/we vs. you vs. he/she/they)
  • Third-person singular preferences (he, she, he or she, they, one)
• Modal verbs (can, could, might, must, should, will, would)
• Hedges (perhaps, maybe, possibly, probably)
• Conjunctions (and, but, yet, so)
• Known stable ratios (the/a, this/that, these/those, I/me/my)
• Character N-grams (3-grams and 4-grams)
• Word N-grams (often 3-grams)

Lexical features:

• Vocabulary size
• Lexical diversity / type-token ratio (Number of distinct words divided by number of words.)
• Frequencies of rare words
• Semantic density
• Discourse marker positions, combinations (So, anyway, so anyway)
• Use of abbreviations and acronyms
• Preference for latinate vs. germanic words (The majestic creature traversed the terrain vs. the mighty beast strode across the land.)

Syntactic features:

• Syntactic complexity
  • Subordination index
  • Average parse tree depth
• Use of passive voice.
• Nominalization (She was shocked I ate the pizza vs. My pizza consumption shocked her)
• Verb tense and aspect (I walk vs I walked vs I was walking vs I have walked)
• Sentence structure preferences:
  • Branching preferences (Cursed everyone had a good time when Alice taught some cool dogs I met and brought to dinner to juggle vs. clumsy-but-readable I met some dogs and they were cool and I took them to dinner and Alice taught them to juggle and and everyone had a good time.)
  • Adverbial clause positioning (Suddenly I was hungry vs. I was, suddenly, hungry vs. I was hungry, suddenly)
  • Sentence-final weight (Your plan won’t work because of the dyslexic bears vs. Dyslexic bears mean your plan won’t work.)
• Polysyndeton (I like dogs, cats, and ferrets vs. I like dogs and cats and ferrets.)
• Repetition / breaking of syntactic structures.

Style features:

• Register / formality.
• Patterns in sentence length (long/short/long/short vs. long/long/short/short)
• Stressed syllable interval preferences (e.g. iambic vs. trochaic)

Rule preference features:

• Minor punctuation (I laughed—you cried vs. I laughed — you cried, “…” (three periods) vs. “…” an actual ellipsis)
• Capitalization. (Job titles, seasons, after a colon, mistakes)
• Apostrophes (Steve Jobs’ car vs Steve Jobs’s car, 1990’s vs 1990s)
• Hyphenation (a highly-stable feature vs a highly stable feature)
• Oxford commas.
• Article omissions (Local dog was petted. vs. A local dog was petted.)
• Relative pronoun omissions (the dog you petted vs. the dog that you petted)
• Who vs. whom.
• Split infinitives (To obsessively blog vs. to blog obsessively)

Idiosyncratic features:

• Whitespace habits.
• Spelling errors (loose instead of lose)
• Grammar errors. (Between you and I)
• Consistent, unique typos
• Other consistent errors (repeated words, un-closed parentheses)

That’s a lot. There are surely more. And these are all “shallow” features that humans came up with using our tiny little brains. I strongly suspect that there are many more “deep” features that could be found by looking for statistical patterns in a sufficiently large dataset. Many of those features might not even have a coherent English-language description. But they’re still there, providing bits for those who seek them.

So we leak information about lots of different stuff when we write. But how much information? Is it possible to say how many words are needed to uniquely fingerprint someone?

No. To a first approximation, the answer is no. But to a second approximation, maybe? Within an order of magnitude? I’ll try, but it’s going to be hard.

Demographic bits

How many bits of identifying information does text provide by way of demographic features like age and sex and so on?

At first glance, this seems a perilous question, as it depends on the number of categories you consider those things to have. Take sex. For pseudpocalypse purposes, your opinion about how sex should be defined or how many sexes exist is irrelevant. Finer categorizations always provide more information, and our de-pseudonymizing attacker friends will use that information if they can. However, going beyond two categories for sex makes little difference, because the additional categories will be hard to guess and even if you could, categories with low prevalence don’t contribute much extra information.³ So, for us, two categories is the right answer.

And what about age? At first glance, converting age into a set of categories seems meaningless. If you code age by the millisecond, then there are 3.156 trillion categories for people born in the last 100 years. If you code age by the decade, there are only 10. Here, the thing to notice is that while you might be able to guess my decade of birth from how I write, you don’t have a snowball’s chance in hell of guessing the millisecond. (See what I did there? People born in certain decades are more likely to use expressions like snowball’s chance in hell?⁴) If we took age to have some crazy number of categories, we’d have to discount later to reflect the difficulty of guessing. My intuition is that it would be hard to guess age more accurately than around five years, so 20 categories seems reasonable.

Following this kind of logic, I chose a number of categories for each of the demographic variables, trying to hit the upper end of what could be guessed from text. (I’ll provide the actual categories below.)

If each of the age bins were equally likely, then knowing what bin someone fell into would provide 4.32 bits of information, because 2ᴷ ≈ 20 when K = 4.32. Doing that same calculation for each feature gives the maximum amount of information they could contain.

But there’s a problem. There are more people aged 30-35 than there are people aged 90-95. So, even if you could guess those age bins perfectly, they’d provide less than 4.32 bits of information on average. However, it turns out that categories need to get pretty damned uneven before information content drops very much. A perfectly balanced 50/50 distribution provides 1 bit of information, but if you switch to a 60/40 distribution, you still get 0.971 bits, and you need to go almost to 90/10 before information content drops to 0.5 bits.⁵ The same basic thing is true when there are more than two categories.⁶

So I went through all those features, rated them by how unevenly people are distributed, and tried to discount the bits accordingly. I’ve put the full details of what the original categories are and how I discounted them in a footnote.⁷

But there’s another problem. Female 65 to 70 year-old Asians living in Scotland tend to have different {occupations, family statuses, religious affiliations} than 15 to 20 year-old Latinos living in Southeast Australia. That is, the above features are correlated. So as you look at more of them, they gradually become less surprising and thus contribute less information.

How much less? Answering that the right way would require us to estimate how likely someone is to fall into each of the 20 × 6 × 6 × 2 × 11 × 3 × 3 × 2 × 23 × 3 × 3 × 23 × 3 × 2 = 8,144,737,920 joint categories. That seems hard. But a not-completely-ridiculous approximation is that if a group of variables are all pairwise correlated at a level of ρ>0, then the total information might be reduced by a fraction of ρ.⁸

So how correlated are those features? In the social sciences, a correlation of 0.5 is considered quite high. That’s plausible for some pairs of variables, e.g. age vs. health or political leaning vs. religious affiliation. But many of those correlations are are probably quite weak, e.g. age vs. native language or region vs. sex vs. marital status.⁹

Overall, my guess is that correlations reduce the total information by at least 10% but I doubt they reduce it by more than 60%. So I’d think the total information in the above features (if you could guess the categories perfectly) is somewhere between 10.6 and 23.9 bits. Let’s take the average and call it 17.2 bits.

Personality bits

What about personality features? Let’s use the same same recipe we used for demographic features, but faster: To start, let’s give each of the 24 personality features five bins, in deference to dynomight personality notation. That would correspond to 24 × 2.32 = 55.68 bits total, because 2ᴷ ≈ 5 when K = 2.32.

Then we need to discount for correlations. The six main HEXACO personality factors are designed to be uncorrelated, but the different “facets” inside each factor are correlated (usually with a coefficient between 0.3 and 0.6). It seems reasonable to use an overall discount factor of 0.3 to reflect strong intra-factor correlations but weak inter-factor correlations. That suggests 39.0 bits overall.

Style bits

And what about writing style features? How much information do they contain?

This seems hard. Some of the features, like character n-grams are actually themselves long lists of features. (Frequency of typing aaa, frequency of typing aab, etc.) However, many of those features contain little information, since almost everyone types zqx around 0% of the time. And, of course, writing style features are correlated, since people who write realise instead of realize are less likely to put spaces around their em-dashes.

In absence of a better idea, I’m going to give one bit for each leaf node in the above list of style features. I think of this as giving each feature two bins, and then assuming that uneven distributions of features and correlations (which reduce information) are canceled out by the fact that many features deserve more than one bin and that there are probably more “deep” features that aren’t listed (which increase information). This gives us the suspiciously round number of 50.0 bits.

Guessing bits

If you believe the above numbers, then we have at least 17.2 + 39.0 + 50.0 = 106.2 bits of identifying information that we leave clues about when we write. That’s a lot. If you could see all those features, it would be enough to identify people even on a planet with 93 million trillion trillion people.

But to argue that the pseudpocalypse is nigh, it’s not enough to argue that those bits exist. We need to argue that they can and will be guessed from a relatively small amount of text.

So obviously we need to talk about nuclear weapons. In a nuclear detonation, many unstable atoms are created. These spontaneously decay into more-stable atoms, in the process emitting radiation. Some types of atoms are very eager to decay, meaning they release a lot of radiation but stop existing within a few weeks (iodine-131). Others are reluctant to decay, meaning they don’t release as much radiation but they stick around for decades (strontium-90). Others stick around for millions of years, but they produce so little radiation that they’re not a big problem (cesium-135).¹⁰ So, the residual radiation produced after a nuclear detonation is the sum of many different exponential curves, one for each isotope created during the detonation.

I suspect that identifying bits in text are sort of like that. Your level of formality and your average sentence length are revealed almost immediately. Your preference for latinate vs. germanic words takes a while to come through. And your social boldness and the fact that you live in Queensland rather than Southeast Australia are revealed very slowly, perhaps so slowly that it’s effectively not revealed at all.

Right. So if you start with 106.2 bits, how many of those do you reveal after writing a given number of words?

I will answer that question through the noble method of making up numbers. But first, let’s calibrate. You just read 4500 words written by me. How well could you guess my demographic and personality features? As a sanity check, I gave the above words to an LLM and asked it to guess. It did unnervingly well. It wasn’t always right, but it usually was, and it did a great job of rating the confidence of the individual predictions.

I don’t think there’s any magical explanation for this. The fact is, if you look at the individual personality and demographic features, guessing them just isn’t that hard. So I’m sure you could do just as well. And given enough time, I’m pretty sure you’d do even better for writing style features.

Even so, you’re probably bad at it. Take the example of GeoGuessr, where people guess a location in the world from a random photo. Random people are sort of OK, but if you pick the top natural talents and have them practice obsessively, they’re really good. I don’t think LLMs are particularly good at guessing features from text, either. They weren’t trained for it. It’s just an emergent property of their general intelligence. The information-theoretic limit is surely much higher.

So here’s a very rough cut: After 4500 words, I’d think it’s possible to guess around:

• 60% of the demographic features
• 70% of the personality features
• 80% writing style features

If we model each of those with a separate exponential, and start them at 17.2 / 39.0 / 50.0 bits, then the total number of identifying bits that remain hidden after writing a given number of words is as plotted here:¹¹

Et voilà, pseudonymity is compromised when you leak 29 bits, which happens after 1071 words.

Seriously?

Of course not. The above figure stands on a creaking tower of tenuous assumptions. I’ve gone through the details of deriving that curve not because you should trust it, but because I think seeing the calculations makes the following points hard to argue with:

• You have far more than 29 bits of identifying information that you leak into your writing.
• Some of those bits take a long time to get revealed, but others are revealed pretty quickly.
• There are enough “fast leaking bits” that you can be identified from a writing sample that’s “pretty small”.

I’ve made lots of debatable choices in terms of choosing features, assigning numbers of categories, estimating distributions across those categories, discounting for correlations, and guessing how many bins can be guessed. Those choices are all individually suspect. But the above points are supported by a pretty wide margin of error. You can make different choices, but it seems very hard to avoid concluding that the above three points are true.¹²

How would this work?

You might be wondering why I’m using so many made-up numbers. After all, there’s a whole field devoted to identifying authors from text, usually called “stylometry” or “authorship attribution”. They have research papers and competitions and all that. However, as best I can tell, state of the art published results look something like this:

1. Take 50 people.
2. Get a few hundred writing samples from each author, each 1000-2000 words long.
3. Now, take a new writing sample from one of those authors.
4. Do some standard machine learning stuff.
5. Hey look, the author can be identified with ~95% accuracy!

That sounds OK, but that’s only identifying people against a pool of ~50 authors. For my claim to be true, similar accuracy would have to be possible with 490 million people. That’s seven orders of magnitude more.

The thing is, the methods those papers are using are extremely weak. All the above math assumes that you’re operating at the “information-theoretic limit”, making perfect use of all available information. If you want to get close to that, we now have some idea how to do it: You apply the “modern” machine learning recipe of gigantic dataset + gigantic neural network + gigantic fortune spent on GPUs. My guess is that for us, that would require something on the order of “all the words ever written” + “tens of billions of parameters” + “tens of millions of dollars”. I couldn’t find a single paper that came remotely close to attempting that.

So I don’t think those papers tell us much, for the same reason that a 3rd-order Markov model trained on a few books doesn’t tell us much about how good computers could be at writing text. LLMs have shown that if you use the above recipe, then computers can get close to the information-theoretic limit for generating text.¹³ So, I suspect that an LLM-level effort could achieve the same thing for identifying authors.

You might also wonder: Why am I talking about this as some possible future technology? Isn’t that technology just LLMs?

I suspect the technology will be quite LLM-like in how it models human language. But current general-purpose LLMs aren’t trained for this task. They’re good at it “by accident”. So, just like specialized chess AIs can crush LLMs at chess, I suspect specialized stylometry methods could crush general-purpose LLMs at stylometry. It’s just that those specialized stylometry methods don’t seem to exist yet, or at least aren’t public.¹⁴ So we shouldn’t imagine that current LLMs are anything close to what’s possible, even if you assume that generic LLM progress stopped today.¹⁵

Countermeasures

If this is all true, what could be done about it?

The most obvious “countermeasure” would be to get used to it. I mean, imagine that we did live in a world in which everyone literally had to sign everything they wrote with a unique immutable string. What would happen? I’d expect a mixture of:

1. People become more comfortable with their “full selves” being public, with less compartmentalization.
2. People pull back from communicating in public channels, relying more on group chats and the like.
3. People self-censor.

There are strong historical analogies here, since over the past 20 years many governments and tech companies have in fact decreed that people must sign the things they write with their real names.

The effects seem to vary quite a lot based on the ambient culture and political system. Overall, my impression is that people are already much more comfortable with the idea that their work colleagues might read their dating profile or learn that they go to furry conventions. I’m optimistic that culture will continue to adapt to respect the fact that we all encompass multitudes. This seems healthy.

Some effects seem clearly positive. Self-censoring is not necessarily bad. For example, on the margin, real-names surely stop some teenagers from engaging in cyber-bullying. On the other hand, were you ever a teenager? I’m pretty sure that for anyone who is “different”, having those differences broadcast to the world creates a much larger “bullying surface area”. So the effects are mixed. And adults aren’t as different from teenagers as we might like to think.

Twenty years ago, I might have predicted that real names would discourage people from expressing controversial political ideas online. Superficially, that seems completely wrong. At least in the West, lots of people are very happy to express minority political views, and if you disagree at all, then you can go to hell. But I also tend to think this hides a lot of self-censorship, where most people don’t want engage in political mortal combat and so are cowed by a feisty minority. And, obviously, people in certain countries know that it’s unwise to criticize the Party. So, getting used to it seems like an imperfect solution at best.

Another countermeasure would be to not build this technology, or not make it widely available. In the short term, this seems plausible. As far as I can tell, it’s been possible for years for a modestly-funded group to build a phone app that would identify most people on the street from a photo. And yet, almost no one reading this has access to such an app. If general-purpose LLMs continue to get better at stylometry, it seems entirely possible that AI companies might decide it’s a safety issue and train their AIs to refuse to do it.¹⁶ This could work for a while.

But if the technology is possible, it seems certain that governments will build it and use it. They might try to keep it out of the hands of normal people. Certain governments might restrict their own use. My privacy-minded allies always seem very jaded, but it wouldn’t surprise me at all if the Supreme Court declared that a warrant was needed before the FBI could de-pseudonymize a U.S. citizen. But when/if that technology becomes sufficiently cheap, it seems like it would be very difficult to keep it out of the hands of normal people and/or bad actors. My guess is that it’s possible to create a program that’s a few hundred gigabytes large and can run (slowly) on most modern laptops. If that program is made public, it would be hard to put the genie back in the bottle.

There are also technological countermeasures. Most obviously, you could run your writing through a “filter” to try to remove identifying bits, e.g. by asking an LLM to rewrite it. It’s hard to be sure how well this would work, since we don’t have accurate estimates of how many bits you’re starting with or how many bits this would remove. But I’d guess this would be pretty effective if done carefully. The reason is that the number of identifying bits you leave in writing probably isn’t that large, relative to the number needed to identify you. If you “homogenize” your writing to remove all style and personality, you should be able to remove most of those bits. Theoretically, you’ll still leak some information. But I’d think this would substantially increase the amount you could write while remaining pseudonymous.¹⁷

But after thinking about it, this makes me sad. Effectively, this countermeasure would preserve pseudonymity by taking writing and destroying all traces of humanity. It seems like this would work well for the “bad” uses of pseudonymity, like cyber-bullying or coordinated violence, but it wouldn’t work at all for the “good” uses, like for example someone who likes to write pseudonymously because they feel like it allows them to be more honest and vulnerable and more fully themselves, damn it.

Generalized pseudpocalypse

Maybe this isn’t just true for writing. Maybe it’s just a feature of our universe that if you interact with the world in any significant way, then you leave traces that make it possible to identify you.

• If you walk around in public, then you can likely be identified by your face, your gait, your voice, your DNA, your retinas, or your literal fingerprints.

• Or say you use the internet. Even if you lock down your browser fingerprint and hide your IP address using a VPN or Tor, a sufficiently powerful adversary could still identify you by analyzing global packet flow.

• Or say you use any phone or computer. You might be identified through keystroke dynamics or the way you jiggle your finger or mouse.

• Say you buy food at the grocery store, but you pay with cash and somehow shop at a grocery store with no cameras. If you buy more than a handful of items, I’d bet you can still be identified through the patterns in the stuff you buy.

• (Incidentally, did you ever notice that cash has serial numbers on it? And did you know that more and more ATMs are starting to track those numbers?)

• Or say you don’t like your car being tracked, so you stop carrying a phone and somehow get lawmakers to outlaw license plates. Still, your car surely has a few small unique scratches, and the engine probably doesn’t sound exactly the same as other cars, even from the same model and year. So if there’s any high-resolution video or audio, that’s still enough to track you.

• Say you plug your headphones into a charging station at the airport. Your headphones have eccentricities in their analog charging circuits. If someone really wanted to, they could track that.

• Or say you use electricity. Given high-resolution power-usage data, what can be said about how many people live with you? And what devices you’re using? Probably a lot?

• Or say you use a toilet. Many places already test sewage and know, at a population level, what drugs people are using and how prevalent various diseases are. Imagine this was upgraded to test many places in the system, with high temporal resolution, possibly correlated with flow measurements from individual houses. That would be exciting.

• Or say you are a country and you have submarines. Can they be detected by adversaries using distributed acoustic sensing? What about satellite-based synthetic aperture radar? Gravity Gradiometers? Quantum magnetometry?

As far as I can tell, the general trend is that without countermeasures, almost everything can be identified. Countermeasures can make it harder, but they’re costly, and on the whole, the arms race seems to favor the identifier, not the person who doesn’t want to be identified.

I stress: This is not all bad. The goodness / badness of a generalized pseudpocalypse depends on how society is structured. After all, the foundation of civilization is finding ways for people to make deals, and arguably less privacy makes that easier. The degree that we live in a vulnerable world where it’s easy to create civilization-destroying technologies, perhaps we’re very lucky to find ourselves in a non-private world. Still, I do worry that privacy has long provided a kind of “slack” from laws and norms. Historically, that slack has limited the power of institutions to enforce their rules. If privacy is going away, we need to think about how to preserve slack, particularly when institutions don’t want to.

Appendix: Section for skeptical information theory enthusiasts

Above, I tried to estimate the number of bits of identifying information in writing. But what is a “bit”? In general, if x is a discrete random variable, then the Shannon entropy of x in bits is H(x) = ∑ₓ p(x) log₂(1/p(x)), where the sum is over all the values x can take. This is always bounded between zero and the logarithm of the number of values x can take.

That’s fine, but “writing style” is not a discrete variable with a discrete number of categories. So how can I estimate the entropy of writing style? The short answer is that I can’t. What I’ve actually estimated above is the mutual information between writing and writing style.

Let s be a random variable representing writing style. Think of this as some sort of high dimensional continuous vector representing all the quirks of how different people write. And let x be a writing sample of some length. This is discrete because we can represent writing on digital computers. Then what I’ve estimated above is the mutual information I(x;s) = H(x) - H(x|s), where H(x|s) is the conditional entropy of x given s. This can be measured in bits because both H(x) and H(x|s) can be measured in bits. So that’s what my estimate above really says: I(x;s) ≈ 106.2 bits.

Now, you still might be skeptical. Above, I’ve implicitly assumed something like the following was true:

It’s possible to identify one person out of N possibilities with low accuracy if and only if the mutual information between identifying features and writing is at least log₂(N) bits.

That’s how I justified pseudonymity being compromised around 29 bits. But is it really true? Strictly speaking, no. Actually, even more strictly speaking, it’s “not even untrue” because it’s not precise enough to be true or false. But as far as I can tell, basically any precise version of that statement is false. However, it’s possible to find versions of that statement that are true, provided you add some extra not-too-crazy assumptions.

To start, let’s consider an extremely simple model of information leakage.

Theorem. Suppose the world consists of you plus N other people, and suppose each person has a binary identity string, drawn at uniform from the distribution over M-bit binary strings. All these strings are known to the attacker. Suppose you pick some subset of K bits and reveal them. Then the probability that this identifies you is

  (1-2⁻ᴷ)ᴺ.

Furthermore, in order to hold the probability of being identified below

  (1-1/N)ᴺ ≈ exp(-1) ≈ 36.7%,

it is necessary that K ≤ log₂(N).

Proof. The probability that all K observed features collide with any random person in the crowd is 2⁻ᴷ. Thus, the probability of no collisions after checking the crowd of N people (meaning you are the only one matching the observed features) is (1-2⁻ᴷ)ᴺ. □

That’s simple. But it’s not realistic at all, since it assumes that people have immutable binary strings that they leak into their writing. Can we make it more realistic?

Well, there is a simple lower bound. That is, we can say in general that if the mutual information is significantly less than log₂(N), then it’s not possible to reliably identify someone.

Theorem. Suppose N random people are selected and their full writing style features are made public. One person from that group is chosen and produces a writing sample. Then, the attacker must guess who produced it. The average success rate of the attacker (averaged over the random pool, the random choice of author, and the random writing sample) is at most (I(x;s)+1)/log₂(N).

Proof. Let S=(s₁, s₂, s₃, …) be the pool of N styles and let n be a random variable indicating which person was chosen. Fano’s inequality says that the highest possible success rate is bounded by the conditional mutual information between the writing sample x and the identity n, conditioning on the pool of writing styles, i.e. the probability of success is at most

  (I(x;n|S)+1)/log₂(N).

However, we can bound that conditional mutual information as

  I(x;n|S) ≤ I(x;n,S) = I(x;n,sₙ) = I(x;sₙ) = I(x;s).

The first inequality is standard. The second step uses the fact that given n, the writing x is conditionally independent of all styles except the chosen writer. The third step uses the fact that n is conditionally independent of x given sₙ. The last step uses that (x,sₙ) is distributed as (x,s). Substituting this bound gives the claimed result. □

So, if mutual information is much less than log₂(N), reliable identification is impossible, even if the attacker knows all the style vectors perfectly. So, provided you don’t leak that many bits, you’re definitely safe.

But is the converse true? Does leaking more than log₂(N) bits always identify you? The general answer is no. The basic problem is that I(x;s) is the average information that an average person leaks in an average writing sample. Without further assumptions, you can construct scenarios where some rare people and writing samples contain gigantic amounts of information, but most people usually leak nothing. That would mean that the attacker is very certain in some cases but usually learns nothing.

So, to get a guarantee that identification is actually possible, you need to make some kind of additional assumption that the information leakage rate doesn’t vary too much between different writers or between different things they write.

Suppose that p(x,s) is the joint distribution over writing styles s and writing samples x. Let’s suppose that the attacker knows the true style vector ŝ for some person. Then, they will be given a writing sample x that either came from that person or came from a randomly chosen person, and must decide which. Formally, the attacker’s goal is to guess if x was sampled from the writing distribution for that person, p(x|ŝ) or from the population marginal p(x). Intuition suggests that the attacker’s best strategy will be to look at the ratio

  p(x|ŝ)/p(x),

and “accept” x as coming from ŝ if above some threshold, and reject it otherwise. In fact, the Neyman-Pearson lemma guarantees that this is the optimal strategy, in a very strong sense: That ratio contains all the information that’s useful for making that decision.

Now here’s something interesting: Instead of looking at the ratio, the attacker could look at the logarithm of the ratio. It makes no difference since it’s monotonic. But if you take the logarithm of that ratio, and take the expectation over people and over texts, what do you get? Well:

  𝔼 ln (p(x|s)/p(x)) = 𝔼 ln (p(x,s)/(p(x) p(s))) = I(x;s)

It’s the mutual information! So, intuitively, the mutual information is how much an attacker learns about the style of the writer “on average”, where that average is over both writers and text.

The following theorem will look at the average information in text for a writer with a particular style. I’ll define this as

  D(s) = KL(p(X|s) || p(X)).

Intuitively, this is how different the writing of someone with style s is from the population average. That’s because if you take the average of this value over different styles, you get the mutual information. That is, I(x;s) = 𝔼[D(s)].¹⁸

Theorem (informal). Suppose that the attacker will observe some text and wishes to classify it as either coming from a writer with specific known style ŝ, or coming from someone with a random style. Suppose that the attacker is only willing to tolerate some small risk ε of a false positive. Provided that D(ŝ) is significantly larger than -ln(ε), the attacker can achieve that, while also keeping the risk of false negatives very low, provided that the variance of how much information is revealed in a random writing sample is bounded.

Theorem. Let D(ŝ) = KL(p(X|ŝ) || p(X)) to be the divergence between the target’s writing distribution and the marginal distribution. Also, define qₜ(x) ∝ p(x|ŝ)ᵗ p(x)¹⁻ᵗ to be the family that interpolates between those two distributions. To formalize the idea that “information leakage” for ŝ doesn’t vary that much, we assume that some constant V exists such that for 0 < t < 1, the variance of log(p(x|ŝ)/p(x)) under qₜ is bounded by V.

Then for any ε satisfying exp(-D) < ε < exp(-D + ½ V), it is possible for the attacker to simultaneously achieve a false positive rate of FPR ≤ ε and a false negative rate of FNR ≤ exp( - ½ (D+ ln ε)² / V). This false positive rate reflects the mistake rate provided the writing sample x came from a randomly chosen other person, while the false negative rate reflects the mistake rate provided the writing sample x actually came from the person with style ŝ.

Proof sketch. Let f be the distribution of l(x) = log(p(x|ŝ)/p(x)) with respect to p(x|ŝ) and let g be the distribution of l(x) with respect to p(x). The stated variance assumption implies a quadratic bound K(u) ≤ D u +½ V u^2 for -1 < u < 0, where K is the cumulant generating function of f. Observe that g is an exponential tilting of f. The attacker’s strategy must be to “accept” x as coming from ŝ if l is above some threshold c and “reject” it otherwise. Use K in a Chernoff bound on the probability l is less than c under f to upper-bound FNR ≤ exp( - ½ (D-c)²/V). Now, using that g(l) = exp(-l) f(l), again use K in a Chernoff bound on the probability l exceeds c under g to upper-bound FPR ≤ exp( -c - ½ (D-c)²/V). Both of these bounds are simultaneously valid when D-V < c < D. Setting c to make the false-positive bound equal to ε gives FPR ≤ ε and FNR ≤ exp( -½ (V - √(V² - 2V(D + ln ε)))²/V). The latter can be relaxed into the stated result using that √(1-x) ≤1-x/2 for 0 ≤ x ≤ 1. □

Now, if we suppose that the attacker wants to find a particular person, with a particular known style s. And suppose that the attacker has a pool of N people and will see one writing sample from each, but wants to limit the total probability of a false positive to δ after seeing one sample from each person. Then, they will need that

  (1-ε)ᴺ ≈ exp(-εN) = (1-δ),

which is satisfied by ε ≈ δ/N. Substituting this into the previous result says that the attacker can hold the total risk of a false positive to δ while achieving a false-negative risk of

  FNR ≤ exp( - ½ (D(s) + ln δ - ln N)² / V).

These results use natural logarithms because the math is easier if you measure information in nats. If you measure information in bits then you would get log₂ δ and log₂ N. (Rescaling D and V appropriately.)

So, again, as long as the average information for user s is significantly larger than log₂ N, the attacker can identify that user with minimal risk of false positives.

Some writers might leak more information (higher D(s)) and some writers might leak less information (lower D(s)). But remember, I(x;s)=𝔼 D(s). So as long as information leakage doesn’t vary too much between people, and assuming that I(x;s) is much larger than log₂ N (and assuming that variance condition), almost everyone can be identified.

For a while there, many people thought vitamin D was magical—that it could improve bones, the heart, infections, cancer, heart disease, longevity, even mental health. But among people I respect, opinion is now overwhelmingly that taking vitamin D does nothing unless you’re severely deficient. The central argument is that while vitamin D levels are correlated with ~all positive health outcomes, when you actually test vitamin D supplements against placebo in randomized trials, nothing ever happens.

That’s what I used to think, too. But I’ve come to think the skeptics have over-corrected. Yes, randomized trials have shown that the magical correlations are not causal. But if you start with non-insane expectations, the trials look like weak but positive evidence. And if you consider what we know about biology and evolution, I think the balance of evidence tips pretty clearly in the direction that people with low-ish levels would be wise to supplement.

Am I certain that vitamin D is beneficial for people with low-ish levels? Absolutely not! But I claim that’s the best bet given the limits of our knowledge.

The classical view: Boring bone vitamin

Most vitamins are “ingredients” that the body uses to do stuff. Vitamin D is more like a “signal” that the body uses to communicate with itself about what to do.¹ The classical “endocrine” story of vitamin D is that your body uses it to tell your guts to take in more calcium from food. If you don’t get enough vitamin D, then you have calcium problems.

That’s all you really need to know about the classical view. But if you enjoy gawking at biology’s complexity, I recommend this diagram and the following three paragraphs:

Ready for science? OK: Almost all the cells in your body make provitamin D.² Usually, this is all converted to cholesterol, but your skin cells leave some sitting around. When UVB light hits those skin cells, provitamin D is transformed (physically by the light itself) into previtamin D and then (by heat) into vitamin D. This diffuses from the skin cells into blood vessels. There it binds to a protein³ and starts circulating in the blood, where it is joined by vitamin D from food.⁴ Eventually, the liver converts it into more-stable storage vitamin D. It also soaks in and out of fat and muscle tissue, which acts as a slow-release reservoir.

Now, a fun fact: If calcium levels in your blood get too low, then your heart will stop working and you will die. To avoid this, you have parathyroid glands in your neck that sense when calcium is getting low, and release parathyroid hormone into the blood. This tells your bones to release some of their stored calcium. It also tells your kidneys to convert some of the storage vitamin D from your blood into active vitamin D. And when that gets to your guts, they try to absorb more calcium from food.

So what happens if you don’t get enough vitamin D? Well, your body is not going to let calcium levels drop too low, because your body is designed to avoid death. Parathyroid hormone will still get secreted, and calcium will still get scavenged from your bones. But without vitamin D, your guts never get the signal to gather extra calcium from food. So the body scavenges a lot of calcium from your bones, and you end up with weak bones, which is bad.

Now here’s the thing: In this story, only active vitamin D actually does anything. The kidneys make this on demand in response to calcium levels, not in response to storage vitamin D levels. General opinion is that as long as the blood has above ~25 nmol/L of storage vitamin D, then the kidneys have no trouble making active vitamin D.⁵ Furthermore, survey data suggests that only ~2% of the population has levels below that threshold. This suggests that for ~98% of people, supplementing vitamin D should do approximately nothing.

The correlation view: Magical mystery cure

Rickets is a terrible disease that involves soft bones, stunted growth, and skeletal deformities. It’s probably been with us since ancient times, but it became common in the West after the industrial revolution. In 1890, a Scottish missionary named Theobald Palm observed that rickets was common in smog-ridden UK cities but almost unheard of in sunny countries with poor sanitation, suggesting sunlight itself was the issue. This contributed to the discovery that rickets could be cured with UV light or cod-liver oil, and eventually the discovery of vitamin D.

In 1941, Apperly noticed that the amount of sunlight in different US states was positively correlated with skin cancer but inversely correlated with overall cancer mortality.⁶ He gave this charming graph:

Apperly never mentions vitamin D, presumably because he thought it was a boring bone vitamin.

Things took off in 1980, when Cedric and Frank Garland published, “Do Sunlight and Vitamin D Reduce the Likelihood of Colon Cancer?” Seemingly unaware of Apperly, they gave a similar, but uglier, graph:

They point out that regional diets (like meat and fiber) didn’t seem to explain this pattern. Instead, they propose a mechanistic story:

    Sunlight
        ↓
    Vitamin D
        ↓
    Adequate calcium in blood
        ↓
    Reduced inflammation of epithelial cells in the colon
        ↓
    Less colon cancer

(It’s always inflammation.) This paper was rejected many times before finally being published. I wish I could find an un-gated copy to link to, because it would have made a magnificent blog post.⁷

Following that paper, there was an explosion of work that found negative correlations between sunlight (or latitude) and other types of cancers as well as blood pressure, diabetes, and multiple sclerosis.

Then people started measuring vitamin D in blood. In 1989, the Garlands and collaborators found blood samples takin in 1974 from 25,000 people. They found that 34 of those people had since gotten colon cancer. They matched these with 67 demographically similar people and measured vitamin D levels in the stored blood samples for all 101 people. Among that group, people with vitamin D levels below 50 nmol/L got colon cancer more than three times as often as people with higher levels.

Again, many similar studies followed. These linked higher vitamin D levels to better outcomes in cardiovascular disease, diabetes, obesity, infectious disease, Parkinson’s, and mood disorders. While results were mixed for non-colorectal cancer incidence, higher vitamin D levels predicted better survival of many cancers. Amazingly, all-cause mortality was roughly 30% lower for those at the 75th percentile of vitamin D levels compared to the 25th.

Vitamin D was looking like a miracle. But how could it do all that stuff if it was just a boring bone vitamin?

Meanwhile in biology

While all these correlations were being discovered, we learned that the body doesn’t just use vitamin D for bone stuff.

In 1969, we discovered the vitamin D receptor that active vitamin D binds to in the gut and bones. And in the 1980s came a shock: Almost all cells in the body have vitamin D receptors. These seem to do different things in different tissues. In the pancreas, they support insulin secretion. In immune cells, they boost antimicrobial peptides and reduce inflammation. In neurons, they influence proliferation and differentiation.

So… What? When calcium drops and the kidneys put out active vitamin D, does every part of the body start doing different unrelated stuff?

In the late 1990s, we cloned the gene for the enzyme that the kidneys use to convert storage vitamin D to active vitamin D. Soon came another shock: This enzyme also exists in tons of other cells, including immune cells, the heart, the skin, the prostate, the breast, and colon. (Another win for the Garlands.)

So it’s not just the kidneys making active vitamin D to trigger the gut. Cells everywhere are making their own active vitamin D and using it to trigger vitamin D receptors in neighboring cells, or even inside the same cell.⁸ This often has little to do with calcium or bones.⁹

So:

1. The kidneys use vitamin D as a boring bone hormone.
2. As long as the blood contains at least ~25 nmol/L of storage vitamin D, the kidneys don’t care. They create the same amount of active vitamin D, in response to calcium levels.
3. But now cells everywhere are using storage vitamin D.
4. To do god-knows-what.
5. With god-knows-what sensitivity to circulating vitamin D levels.

And remember how only active vitamin D does anything? That’s wrong. In the mid-1970s, we learned that storage vitamin D also binds to the vitamin D receptor. The binding affinity is 100-1000× lower, but you have ~1000× more in your blood. So maybe circulating levels of storage vitamin D themselves matter, independently of how much active vitamin D gets made?

If that’s not confusing enough, people also noticed that while active vitamin D levels in the blood aren’t correlated with storage vitamin D (above ~25 nmol/L), levels of parathyroid hormone (the thing your parathyroid glands use to tell your kidneys to make active vitamin D) seem to decline as levels of storage vitamin D rise from ~25 to 50 or 75 nmol/L. Huh?¹⁰

On the one hand, all this makes the idea that vitamin D could be a miracle more plausible. On the other hand, this is getting complicated. And do we really believe that raising your vitamin D levels from the 25th to the 75th percentile could reduce your risk of death from any cause by thirty percent? Maybe we should try giving people vitamin D and see what happens.

Then came the RCTs

There have been many randomized trials. The “right” thing to do in such cases is to look at meta analyses that carefully combine all the data. We’ll get to those. But they conceal a lot of important nuance about what actually happens on the ground during these trials. So let’s start by going over the three main “megatrials”.

The Women’s Health Initiative (WHI) trial came out in 2006 and is still the largest vitamin D trial ever done. This took 36,000 postmenopausal American women and assigned half to take 400 IU daily with calcium and the other half to placebo.¹¹ After seven years, here’s what happened:¹²

(The hazard ratio is the ratio of the rate that something happens in the treatment vs. placebo groups. So, a number less than one suggests a benefit to taking vitamin D, while a number larger than one suggests a harm. The numbers in parentheses show a 95% confidence interval.)

The only statistically significant result was a bad one: Extra kidney stones, likely from the extra calcium.¹³ The other outcomes look vaguely good, but none were statistically significant despite the massive sample size.

This was disappointing. However, the WHI trial had limitations: Many subjects in both the vitamin D and placebo groups were already taking vitamin D, and continued taking it through the trial. The dose of 400 IU was fairly low, many subjects stopped taking their pills, and vitamin D levels didn’t actually change that much. They also measured vitamin D levels in only 6% of subjects, meaning we can’t compare the fates of subjects who started out with low versus high levels.

The next big hope was VITAL, which came out in 2018. They recruited 26,000 older people across the United States, half of them men and 20% Black (and thus far more likely to be vitamin-D deficient). They measured vitamin D levels in most people, and they gave the treatment group 2,000 IU per day.¹⁴ Here were the results after 5.3 years:

Some of the results look good-ish, but cardiovascular mortality was higher in the treatment group, leading to almost no effect on all-cause mortality.¹⁵ More disappointment.

The last megatrial was D-Health, which came out in 2022 based on 21,000 older Australians. Instead of daily supplements, it used a monthly “bolus” dose of 60,000 IU or placebo. Unlike in VITAL, there was no exclusion for people with a history of cardiovascular disease or cancer, and less restriction on how much vitamin D participants could take on their own during the trial.¹⁶ Here were the results after 6 years:

Now, the treatment group did better in terms of cardiovascular disease, but worse in cancer and worse in all-cause mortality. Even more disappointment.

Just from these three large trials, the main lesson should already be clear: Vitamin D is not a miracle. The correlations were wrong.¹⁷ There is essentially zero remaining hope that taking vitamin D could reduce all-cause mortality by a third.

In this sense, the vitamin D skeptics are definitely right. But what about the other trials? And is there a more subtle lesson?

I made some tables

I wanted a big table that summarized all the major vitamin D RCTs and what they found for different health outcomes. Annoyingly, no such overview appears to exist. So I made my own:¹⁸

Lots of the hazard ratios are less than one, suggesting a benefit to supplementation. But lots of them are also higher than one, suggesting a harm. The numbers that are far from one almost always come from smaller trials, which manifest as larger confidence intervals. If you’re interested in the details of how these trials were run, I refer you to more gigantic tables in a footnote.¹⁹

If big tables aren’t your thing, here are some formal meta-analyses, both some recent ones and an older but more comprehensive Cochrane review:

Squinting at the data

There are various ways you could try to squint at these RCT. In almost all of them, most people already had pretty high levels before they started. So why don’t we separate out people who started low? Usually we can’t, because most trials didn’t measure baseline vitamin D.²⁰ And among the trials that did, there are few people with low levels, so the results are noisy and confusing.²¹

Or, you might theorize that benefits would take time to show up, meaning the first couple years just add noise. In some cases—notably VITAL—excluding the first two years seems to help, but in other cases things get worse.²²

Finally, some people speculate that taking gigantic monthly or quarterly “bolus” doses of vitamin D might be dangerous. For example, here’s an enjoyable paragraph from Kunzia et al. in their meta-analysis of vitamin D and cancer mortality:

Our results showing efficacy of daily, but not bolus, vitamin D3 supplementation in reducing cancer mortality are consistent with previous meta-analyses on cancer mortality or all-cause mortality (Guo et al., 2022; Keum et al., 2022; Keum et al., 2019; Zhang et al., 2022; Zhang et al., 2019). However, by including more trials than these previous meta-analyses, we were able to detect statistically significant effect modification by treatment regimen for the first time with statistical significance (pinteraction=0.042). The pattern of intake could be important for a favourable steady state of the bioavailability of the active 1,25 (OH)₂D hormone. Daily administration counteracts the fast excretion of vitamin D from the circulation (Hollis and Wagner, 2013; Keum et al., 2022). Moreover, the enzymes CYP27B1 (converts 25(OH)D to 1,25 (OH)₂D) and CYP24A1 (inactivates 25(OH)D and 1,25(OH)₂D) follow first-order reaction kinetics (Vieth, 2009). This means that doubling the concentration of the precursor doubles the yield of the product, unlike other steroid hormones (e.g., cortisol, oestrogen, testosterone) that follow zero-order kinetics (Vieth, 2020). Intermittent, non-physiologically large vitamin D3 bolus doses may lead to unstable cycling of 25(OH)D and 1,25(OH)₂D levels in blood because the system needs time to adapt to the large doses (Hollis and Wagner, 2013; Keum et al., 2019; Vieth, 2020). In the long run, intermittent bolus regimens at weekly or larger intervals can lead to an up-regulation of countervailing factors (e.g., 24-hydroxylase (CYP24A1), 24,25(OH)2D and fibroblast growth factor 23), all of which ultimately leads to lower synthesis or higher degradation of 1,25(OH)₂D levels (Mazess et al., 2021). Bolus doses, unlike daily doses, failed to reduce C-reactive protein response and actually elevated anti-inflammatory cytokines and doubled the risk of hypercalcemia in previous studies (Krishnan et al., 2012; Martineau et al., 2017; Mazess et al., 2021).

Oh no, up-regulation of fibroblast growth factor 23!²³

I don’t feel like I understand this deeply enough to have any opinion beyond the surface level that the body seems to adapt to large doses of vitamin D in ways that could possibly be bad.²⁴ It seems intuitive that small daily doses would be safer than gigantic monthly doses, but I’m always suspicious of post-hoc mechanistic speculation. Also, if people get enough sun, they can apparently synthesize 10,000-25,000 IU per day, which isn’t that far from the 60,000 IU they got in the D-Health trial. But then again, I think Kunzia et al. are suggesting that the body is designed to adapt to regular exposure to large doses but not intermittent exposure?

Well, if you split up the trails by daily vs. bolus dosing, there’s a decent pattern of daily dosing leading to better results:

If those bolus dosing trials didn’t exist, I’d think this looked pretty good. So, maybe? Or maybe this is a story made up to hallucinate a positive trend. I would lean towards the latter theory, but there are papers like Mazess et al.’s “Vitamin D: Bolus is Bogus”, that suggested this pattern before D-Health’s dismal results came out. There are even some trials that suggest bolus doses don’t even work for treating rickets. So… I’m still not convinced. But maybe.

Aside: There are also many Mendelian randomization studies that look at correlations between health and genes that are related to vitamin D. But I don’t think these provide much information, because the assumptions are shaky and the genes don’t explain much of the variance.²⁵

Where are we?

Still with me? Here’s a summary of the above 5200 words:

• The body uses vitamin D in all sorts of weird and complicated ways. It’s biologically plausible that vitamin D could matter beyond bone stuff with severe deficiency, but there’s no convincing mechanistic evidence that it is.
• Vitamin D levels are strongly correlated with good health outcomes, but RCTs have conclusively shown that most of these correlations are non-causal.
• RCTs haven’t conclusively shown any benefit for anything beyond bone stuff. At best, they’ve given weak evidence for hazard ratios slightly below one.

So you might be wondering: Isn’t that quite weak? Wasn’t this post supposed to be a defense of vitamin D?

The case for supplementing anyway

It’s biologically plausible that vitamin D is good

Everyone agrees that severe vitamin D deficiency (below ~25 nmol/L) is bad. It leads to rickets, adult rickets, osteoporosis, muscle weakness or even—with profound deficiency—to seizures or cardiac arrhythmia. This makes sense, because below ~25 nmol/L, the kidneys have trouble converting storage vitamin D into active vitamin D, meaning you don’t absorb enough calcium from food.

The question is if taking supplement to further raise your levels (say to 50 or 90 nmol/L) is important. We have no mechanistic proof, but it might be true, because many parts of the body use vitamin D as a local signal and because cells are at least somewhat sensitive to circulating storage levels. There’s also this weird thing where parathyroid hormone continues to decline as vitamin D levels rise above ~25 nmol/L even while this seems to make little difference to how much active vitamin D the kidneys make.

Nothing in this world comes without trade-offs. Surely, supplementing vitamin D comes with some downsides. But it seems very unlikely that raising vitamin D levels to a “normal” level would cause more harm than benefit. Especially because…

Humans evolved to have a lot of vitamin D

According to Luxwolda et al.’s 2012 paper, “Traditionally living populations in East Africa have a mean serum 25-hydroxyvitamin D concentration of 115 nmol/L”, traditionally living populations in East Africa have a mean serum 25-hydroxyvitamin D concentration of 115 nmol/L.

Meanwhile, Wahl et al. 2012 try to estimate mean levels around the world today:

This map looks weird because of varying lifestyle, diet, supplementation, and needing to combine fragmented studies. But you get the idea. And remember, those are just averages. So there are lots of people with levels far lower than that in our evolutionary history.

Of course, just the fact that vitamin D levels have dropped doesn’t mean it’s important. Parasitic worm load, wood smoke inhalation, and cousin marriage have also dropped, but we aren’t rushing to restore those to ancestral levels.

But there’s another piece of evidence: After humans migrated out of East Africa, some of them evolved pale skin. Pale skin is bad, because it allows light to destroy folate, which is crucial for pregnancy.²⁶ Evolution doesn’t typically do things that harm fertility, because evolution wants to increase reproductive fitness. The most common explanation is that pale skin allows more UV light to penetrate, and thus allows people to synthesize more vitamin D. If evolution was willing to pay the high “price” of folate destruction for more vitamin D, that seems like good evidence that vitamin D is important.

Some even see contrasts like the Inuits versus Scandinavians as a kind of natural experiment: They lived at similar latitudes, but Inuits ate a diet with vitamin D (fatty fish and whale blubber) and Scandinavians didn’t. The result is that Inuits have darker skin than Scandinavians.²⁷

This is all speculative, and even if true, might be driven by severe deficiency and rickets. Or perhaps prehistoric benefits don’t translate to your lifestyle. But all the people in Luxwolda’s sample in East Africa had levels above ~60 nmol/L. I just don’t see how you can look at this and not see it as providing some suggestive evidence in favor of the idea that raising levels above severe deficiency is unlikely to be harmful, and could be important. So I think the prior is favorable.

What do you expect from vitamin D?

A hazard ratio like HR = 0.96 doesn’t look very impressive. But hold on. Suppose that life expectancy is 80 years and that taking vitamin D every day reduces your risk of all-cause mortality by a factor of HR. A reasonable approximation in rich countries is that this would increase your life expectancy by

    80 × 0.15 × (1-HR) years = 12 × (1-HR) years,

where 0.15 is derived from the entropy of lifespan in rich countries.²⁸ For example, if all-cause mortality had a true hazard ratio of HR = 0.96, then taking vitamin D every day of your life would increase life expectancy by around

    0.48 years.

I claim that this would be a lot. Certainly, if I were about to face my destiny, I would pay a lot of money for an extra 0.48 years. Or, you can calculate that this corresponds to an increase of life expectancy per-vitamin-D-pill of 8.6 minutes.²⁹ A common rule-of-thumb is that smoking a cigarette costs around 11 minutes of life in expectation. If you think HR = 0.96 is trivial, do you also think that smoking one cigarette each day is fine?³⁰

The correlational studies suggested that vitamin D might drop your risk of all-cause mortality by a third. It’s disappointing that the RCTs refuted this. But those correlational studies were crazy. They imply³¹ an increase of life expectancy of around 4 years or around 6.5 cigarettes per day. Could we really believe that you could smoke 6.5 cigarettes, then take a vitamin D pill, and you’re even?

Personally, I think hazard ratios just slightly less than one are the best we can reasonably hope for. But I also think that they would be an excellent return on investment. Arguably, modern human life expectancy comes from stacking lots of modest hazard ratios on top of each other.

What do you expect from vitamin D trials?

Let’s play a game. Let’s hallucinate some numbers for what vitamin D might do, and then simulate what trials would show. Here are the strongest effects I consider plausible for different baseline levels, along with how common those levels are in the United States.

Suppose that were real. Now, say we pick 26,000 people at random, and give half of them vitamin D for five years. Here are the results of a million simulated trials, assuming a baseline mortality risk of 0.7%:³²

Overall, 9% of trials would find a significant benefit, 63% would find a non-significant benefit, 27% would find a non-significant harm, and 1% would find a significant harm.

If you wanted to have an 80% chance of finding a significant decrease, you’d need to run a trial with something like 570,000 people, almost five times more than in all the above trials combined.³³ If you don’t like my numbers, I’ve put up a page where you can run your own simulations with different ones.

My point is, the results we see in vitamin D RCTs are what we should expect to see if vitamin D had plausible benefits. That’s not proof, of course—just that if you start with realistic expectations, the trials don’t provide much evidence in either direction.

The trials do find slightly helpful numbers

Recent meta-analyses have not consistently found a statistically significant benefit to vitamin D supplementation. But they do suggest a small benefit for cancer mortality and all-cause mortality, and they’re close to being statistically significant. That’s something.

And if you buy the argument that bolus dosing is bad, the results get even better. Kunzia et al. did a meta-analysis of cancer mortality using only trials with daily dosing, and found a hazard ratio of 0.88 (confidence interval 0.78 to 0.98). I’d keep this at arm’s length. The bolus dosing trials might have done worse by random chance, meaning this is a kind of p-hacking. But there’s a reasonable chance (maybe 25-50%) that bolus dosing really is bad, in which case those trials would be convincing evidence.

I actually think it’s surprising that the meta-analyses look as good as they do, because there just aren’t that many people who started out with low vitamin D levels. Only a handful of trials had mean levels below 60 nmol/L, and they all give semi-promising results:³⁴

Again, it’s dangerous to dig too deeply looking for these kinds of patterns. If you dig enough, you can always find a way to confirm whatever theory you want. But also again, maybe?

You’re probably already taking vitamin D

You might not personally supplement vitamin D. But for most people reading this, someone else is supplementing it for you.³⁵

Fortified food is common across the Anglosphere and Scandinavian peninsula. However, it’s rare in the rest of Europe (exceptions: Belgium, Poland) and even-more rare in the rest of the world (exceptions: Chile, Ethiopia, Pakistan).

I think this is important for two reasons. First, vitamin D is oddly self-defeating. There are some places in the world where people care about vitamin D. These are the places that run large trials. But these places also fortify their food and tend to be full of people that already supplement vitamin D. These places also tend to believe it’s unethical to tell the control group not to take vitamin D.

And here’s another question: If you think vitamin D is worthless, are you comfortable recommending removing vitamin D from food? If not, then why is the particular amount of fortification in food now the right one?

Some might argue that the purpose of fortification is to reach the severely deficient, or children, the elderly or pregnant mothers. Maybe! But again, if you could press a button and remove fortification from everyone else, would you feel comfortable pushing that button? Remember, trials don’t test going down from current levels, only going up.

So that’s my story

• Biology and evolution suggest a prior that moderate levels of vitamin D (say 80 nmol/L) are quite possibly better than low levels (like 40 nmol/L) and unlikely to be worse.
• Observational studies say that vitamin D is magical, but those studies are bad and we should ignore them.
• The RCTs show that vitamin D is non-miraculous. But beyond that they don’t provide much information, because they mostly enrolled people with moderate vitamin D levels, meaning plausible effects would require colossal sample sizes to reliably detect.
• What evidence the RCTs do provide points weakly towards a modest benefit.
• If real, that benefit would far exceed the cost of taking vitamin D.
• Therefore, if you have low vitamin D, it seems wise to supplement.

This is all very weak, I know! But sometimes weak evidence is all we’ve got.

I wish we had at least one large trial done in a population with low starting levels. But as far as I can tell, none are underway. In fact, it’s unlikely that there will be any more large trials anytime soon. So weak evidence is how it’s going to be.

The heritability wars have been a-raging. Watching these, I couldn’t help but notice that there’s near-universal confusion about what “heritable” means. Partly, that’s because it’s a subtle concept. But it also seems relevant that almost all explanations of heritability are very, very confusing. For example, here’s Wikipedia’s definition:

Any particular phenotype can be modeled as the sum of genetic and environmental effects:

   Phenotype (P) = Genotype (G) + Environment (E).

Likewise the phenotypic variance in the trait – Var (P) – is the sum of effects as follows:

   Var(P) = Var(G) + Var(E) + 2 Cov(G,E).

In a planned experiment Cov(G,E) can be controlled and held at 0. In this case, heritability, H², is defined as

   H² = Var(G) / Var(P)

H² is the broad-sense heritability.

Do you find that helpful? I hope not, because it’s a mishmash of undefined terminology, unnecessary equations, and borderline-false statements. If you’re in the mood for a mini-polemic:

1. Phenotype (P) is never defined. This is a minor issue, since it just means “trait”.
2. Genotype (G) is never defined. This is a huge issue, since it’s very tricky and heritability makes no sense without it.
3. Environment (E) is never defined. This is worse than it seems, since in heritability, different people use “environment” and E to refer to different things.
4. When we write P = G + E, are we assuming some kind of linear interaction? The text implies not, but why? What does this equation mean? If this equation is always true, then why do people often add other stuff like G × E on the right?
5. The text states that if you do a planned experiment (how?) and make Cov(G, E) = 0, then heritability is Var(G) / Var(P). But in fact, heritability is always defined that way. You don’t need a planned experiment and it’s fine if Cov(G, E) ≠ 0.
6. And—wait a second—that definition doesn’t refer to environmental effects at all. So what was the point of introducing them? What was the point of writing P = G + E? What are we doing?

Reading this almost does more harm than good. While the final definition is correct, it never even attempts to explain what G and P are, it gives an incorrect condition for when the definition applies, and instead mostly devotes itself to an unnecessary digression about environmental effects. The rest of the page doesn’t get much better. Despite being 6700 words long, I think it would be impossible to understand heritability simply by reading it.

Meanwhile, some people argue that heritability is meaningless for human traits like intelligence or income or personality. They claim that those traits are the product of complex interactions between genes and the environment and it’s impossible to disentangle the two. These arguments have always struck me as “suspiciously convenient”. I figured that the people making them couldn’t cope with the hard reality that genes are very important and have an enormous influence on what we are.

But I increasingly feel that the skeptics have a point. While I think it’s a fact that most human traits are substantially heritable, it’s also true the technical definition of heritability is really weird, and simply does not mean what most people think it means.

In this post, I will explain exactly what heritability is, while assuming no background. I will skip everything that can be skipped but—unlike most explanations—I will not skip things that can’t be skipped. Then I’ll go through a series of puzzles demonstrating just how strange heritability is.

What is heritability?

How tall you are depends on your genes, but also on what you eat, what diseases you got as a child, and how much gravity there is on your home planet. And all those things interact. How do you take all that complexity and reduce it to a single number, like “80% heritable”?

The short answer is: Statistical brute force. The long answer is: Read the rest of this post.

It turns out that the hard part of heritability isn’t heritability. Lurking in the background is a slippery concept known as a genotypic value. Discussions of heritability often skim past these. Quite possibly, just looking at the words “genotypic value”, you are thinking about skimming ahead right now. Resist that urge! Genotypic values are the core concept, and without them you cannot possibly understand heritability.

For any trait, your genotypic value is the “typical” outcome if someone with your DNA were raised in many different random environments. In principle, if you wanted to know your genotypic height, you’d need to do this:

1. Create a million embryonic clones of yourself.
2. Implant them in the wombs of randomly chosen women around the world who were about to get pregnant on their own.
3. Convince them to raise those babies exactly like a baby of their own.
4. Wait 25 years, find all your clones and take their average height.

Since you can’t / shouldn’t do that, you’ll never know your genotypic height. But that’s how it’s defined in principle—the average height someone with your DNA would grow to in a random environment. If you got lots of food and medical care as a child, your actual height is probably above your genotypic height. If you suffered from rickets, your actual height is probably lower than your genotypic height.

Comfortable with genotypic values? OK. Then (broad-sense) heritability is easy. It’s the ratio

   heritability = var[genotype] / var[height].

Here, var is the variance, basically just how much things vary in the population. Among all adults worldwide, var[height] is around 50 cm². (Incidentally, did you know that variance was invented for the purpose of defining heritability?)

Meanwhile, var[genotype] is how much genotypic height varies in the population. That might seem hopeless to estimate, given that we don’t know anyone’s genotypic height. But it turns out that we can still estimate the variance using, e.g., pairs of adopted twins, and it’s thought to be around 40 cm². If we use those numbers, the heritability of height would be

   heritability ≈ (40 cm²) / (50 cm²) ≈ 0.8.

People often convert this to a percentage and say “height is 80% heritable”. I’m not sure I like that, since it masks heritability’s true nature as a ratio. But everyone does it, so I’ll do it too. People who really want to be intimidating might also say, “genes explain 80% of the variance in height”.

Of course, basically the same definition works for any trait, like weight or income or fondness for pseudonymous existential angst science blogs. But instead of replacing “height” with “trait”, biologists have invented the ultra-fancy word “phenotype” and write

   heritability = var[genotype] / var[phenotype].

The word “phenotype” suggests some magical concept that would take years of study to understand. But don’t be intimidated. It just means the actual observed value of some trait(s). You can measure your phenotypic height with a tape measure.

On meaning

Let me make two points before moving on.

First, this definition of heritability assumes nothing. We are not assuming that genes are independent of the environment or that “genotypic effects” combine linearly with “environmental effects”. We are not assuming that genes are in Hardy-Weinberg equilibrium, whatever that is. No. I didn’t talk about that stuff because I don’t need to. There are no hidden assumptions. The above definition always works.

Second, many normal English words have parallel technical meanings, such as “field”, “insulator”, “phase”, “measure”, “tree”, or “stack”. Those are all nice, because they’re evocative and it’s almost always clear from context which meaning is intended. But sometimes, scientists redefine existing words to mean something technical that overlaps but also contradicts the normal meaning, as in “salt”, “glass”, “normal”, “berry”, or “nut”. These all cause confusion, but “heritability” must be the most egregious case in all of science.

Before you ever heard the technical definition of heritability, you surely had some fuzzy concept in your mind. Personally, I thought of heritability as meaning how many “points” you get from genes versus the environment. If charisma was 60% heritable, I pictured each person has having 10 total “charisma points”, 6 of which come from genes, and 4 from the environment:

  Genes       ★★★☆☆☆
  Environment ★☆☆☆
  Total       ★★★★☆☆☆☆☆☆

If you take nothing else from this post, please remember that the technical definition of heritability does not work like that. You might hope that if we add some plausible assumptions, the above ratio-based definition would simplify into something nice and natural, that aligns with what “heritability” means in normal English. But that does not happen. If that’s confusing, well, it’s not my fault.

Intermission

Not sure what’s happening here, but it seems relevant.

Heritability puzzles

So “heritability” is just the ratio of genotypic and phenotypic variance. Is that so bad?

I think… maybe?

How heritable is eye color?

Close to 100%.

This seems obvious, but let’s justify it using our definition that heritability = var[genotype] / var[phenotype].

Well, people have the same eye color, no matter what environment they are raised in. That means that genotypic eye color and phenotypic eye color are the same thing. So they have the same variance, and the ratio is 1. Nothing tricky here.

How heritable is speaking Turkish?

Close to 0%.

Your native language is determined by your environment. If you grow up in a family that speaks Turkish, you speak Turkish. Genes don’t matter.

Of course, there are lots of genes that are correlated with speaking Turkish, since Turks are not, genetically speaking, a random sample of the global population. But that doesn’t matter, because if you put Turkish babies in Korean households, they speak Korean. Genotypic values are defined by what happens in a random environment, which breaks the correlation between speaking Turkish and having Turkish genes.

Since 1.1% of humans speak Turkish, the genotypic value for speaking Turkish is around 0.011 for everyone, no matter their DNA. Since that’s basically constant, the genotypic variance is near zero, and heritability is near zero.

How heritable is speaking English?

Perhaps 30%. Probably somewhere between 10% and 50%. Definitely more than zero.

That’s right. Turkish isn’t heritable but English is. Yes it is. If you ask an LLM, it will tell you that the heritability of English is zero. But the LLM is wrong and I am right.

Why? Let me first acknowledge that Turkish is a little bit heritable. For one thing, some people have genes that make them non-verbal. And there’s surely some genetic basis for being a crazy polyglot that learns many languages for fun. But speaking Turkish as a second language is quite rare, meaning that the genotypic value of speaking Turkish is close to 0.011 for almost everyone.

English is different. While only 1 in 20 people in the world speak English as a first language, 1 in 7 learn it as a second language. And who does that? Educated people.

Say there’s an island where genes have no impact on height. How heritable is height among people on this island?

0%.

There’s nothing tricky here.

Say there’s an island where genes entirely determine height. How heritable is height?

100%.

Again, nothing tricky.

Say there’s an island where neither genes nor the environment influence height and everyone is exactly 165 cm tall. How heritable is height?

It’s undefined.

In this case, everyone has exactly the same phenotypic and genotypic height, namely 165 cm. Since those are both constant, their variance is zero and heritability is zero divided by zero. That’s meaningless.

Say there’s an island where some people have genes that predispose them to be taller than others. But the island is ruled by a cruel despot who denies food to children with taller genes, so that on average, everyone is 165 ± 5 cm tall. How heritable is height?

0%.

On this island, everyone has a genotypic height of 165 cm. So genotypic variance is zero, but phenotypic variance is positive, due to the ± 5 cm random variation. So heritability is zero divided by some positive number.

Say there’s an island where some people have genes that predispose them to be tall and some have genes that predispose them to be short. But, the same genes that make you tall also make you semi-starve your children, so in practice everyone is exactly 165 cm tall. How heritable is height?

∞%. Not 100%, mind you, infinitely heritable.

To see why, note that if babies with short/tall genes are adopted by parents with short/tall genes, there are four possible cases.

If a baby with short genes is adopted into random families, they will be shorter on average than if a baby with tall genes. So genotypic height varies. However, in reality, everyone is the same height, so phenotypic height is constant. So genotypic variance is positive while phenotypic variance is zero. Thus, heritability is some positive number divided by zero, i.e. infinity.

(Are you worried that humans are “diploid”, with two genes (alleles) at each locus, one from each biological parent? Or that when there are multiple parents, they all tend to have thoughts on the merits of semi-starvation? If so, please pretend people on this island reproduce asexually. Or, if you like, pretend that there’s strong assortative mating so that everyone either has all-short or all-tall genes and only breeds with similar people. Also, don’t fight the hypothetical.)

Say there are two islands. They all live the same way and have the same gene pool, except people on island A have some gene that makes them grow to be 150 ± 5 cm tall, while on island B they have a gene that makes them grow to be 160 ± 5 cm tall. How heritable is height?

It’s 0% for island A and 0% for island B, and 50% for the two islands together.

Why? Well on island A, everyone has the same genotypic height, namely 150 cm. Since that’s constant, genotypic variance is zero. Meanwhile, phenotypic height varies a bit, so phenotypic variance is positive. Thus, heritability is zero.

For similar reasons, heritability is zero on island B.

But if you put the two islands together, half of people have a genotypic height of 150 cm and half have a genotypic height of 160 cm, so suddenly (via math) genotypic variance is 25 cm². There’s some extra random variation so (via more math) phenotypic variance turns out to be 50 cm². So heritability is 25 / 50 = 50%.

Var[150 cm + 10 cm × Bernoulli(0.5)]
 = (10 cm)² × Var[Bernoulli(0.5)]
 = (10 cm)² × 0.25
 = 25 cm².

Var[150 cm + 10 cm × Bernoulli(0.5) + 5 cm × Normal(0,1)]
 = (10 cm)² × Var[Bernoulli(0.5)] + (5 cm)² × Var[Normal(0,1)]
 = (10 cm)² × 0.25 + (5 cm)² × 1
 = 50 cm².

Say there’s an island where neither genes nor the environment influence height. Except, some people have a gene that makes them inject their babies with human growth hormone, which makes them 5 cm taller. How heritable is height?

0%.

True, people with that gene will tend be taller. And the gene is causing them to be taller. But if babies are adopted into random families, it’s the genes of the parents that determine if they get injected or not. So everyone has the same genotypic height, genotypic variance is zero, and heritability is zero.

Suppose there’s an island where neither genes nor the environment influence height. Except, some people have a gene that makes them, as babies, talk their parents into injecting them with human growth hormone. The babies are very persuasive. How heritable is height?

We’re back to 100%.

The difference with the previous scenario is that now babies with that gene get injected with human growth hormone no matter who their parents are. Since nothing else influences height, genotype and phenotype are the same, have the same variance, and heritability is 100%.

Suppose there’s an island where neither genes nor the environment influence height. Except, there are crabs that seek out blue-eyed babies and inject them with human growth hormone. The crabs, they are unstoppable. How heritable is height?

Again, 100%.

Babies with DNA for blue eyes get injected. Babies without DNA for blue eyes don’t. Since nothing else influences height, genotype and phenotype are the same and heritability is 100%.

Note that if the crabs were seeking out parents with blue eyes and then injecting their babies, then height would be 0% heritable.

It doesn’t matter that human growth hormone is weird thing that’s coming from outside the baby. It doesn’t matter if we think crabs should be semantically classified as part of “the environment”. It doesn’t matter that heritability would drop to zero if you killed all the crabs, or that the direct causal effect of the relevant genes has nothing to do with height. Heritability is a ratio and doesn’t care.

What good is heritability?

So heritability can be high even when genes have no direct causal effect on the trait in question. It can be low even when there is a strong direct effect. It changes when the environment changes. It even changes based on how you group people together. It can be larger than 100% or even undefined.

Even so, I’m worried people might interpret this post as a long way of saying heritability is dumb and bad, trolololol. So I thought I’d mention that this is not my view.

Say a bunch of companies create different LLMs and train them on different datasets. Some of the resulting LLMs are better at writing fiction than others. Now I ask you, “What percentage of the difference in fiction writing performance is due to the base model code, rather than the datasets or the GPUs or the learning rate schedules?”

That’s a natural question. But if you put it to an AI expert, I bet you’ll get a funny look. You need code and data and GPUs to make an LLM. None of those things can write fiction by themselves. Experts would prefer to think about one change at a time: Given this model, changing the dataset in this way changes fiction writing performance this much.

Similarly, for humans, I think what we really care about is interventions. If we changed this gene, could we eliminate a disease? If we educate children differently, can we make them healthier and happier? No single number can possibly contain all that information.

But heritability is something. I think of it as saying how much hope we have to find an intervention by looking at changes in current genes or current environments.

1. If heritability is high, then given current typical genes, you can’t influence the trait much through current typical environmental changes. If you only knew that eye color was 100% heritable, that means you won’t change your kid’s eye color by reading to them, or putting them on a vegetarian diet, or moving to higher altitude. But it’s conceivable you could do it by putting electromagnets under their bed or forcing them to communicate in interpretive dance.

2. If heritability is high, that also means that given current typical environments you can influence the trait through current typical genes. If the world was ruled by an evil despot who forced red-haired people to take pancreatic cancer pills, then pancreatic cancer would be highly heritable. And you could change the odds someone gets pancreatic cancer by swapping in existing genes for black hair.

3. If heritability is low, that means that given current typical environments, you can’t cause much difference through current typical genetic changes. If we only knew that speaking Turkish was ~0% heritable, that means that doing embryo selection won’t much change the odds that your kid speaks Turkish.

4. If heritability is low, that also means that given current typical genes, you might be able change the trait through current typical environmental changes. If we only know that speaking Turkish was 0% heritable, then that means there might be something you could do to change the odds your kid speaks Turkish, e.g. moving to Turkey. Or, it’s conceivable that it’s just random and moving to Turkey wouldn’t do anything.

But be careful. Just because heritability is high doesn’t mean that changing genes is easy. And just because heritability is low doesn’t mean that changing the environment is easy.

And heritability doesn’t say anything about non-typical environments or non-typical genes.

If an evil despot is giving all the red-haired people cancer pills, perhaps we could solve that by intervening on the despot. And if you want your kid to speak Turkish, it’s possible that there’s some crazy genetic modifications that would turn them into unstoppable Turkish learning machine.

Heritability has no idea about any of that, because it’s just an observational statistic based on the world as it exists today.

Recommended reading

• Heritability: Five Battles by Steven Byrnes. Covers similar issues in way that’s more connected to the world and less shy about making empirical claims.

• A molecular genetics perspective on the heritability of human behavior and group differences by Alexander Gusev. I find the quantitative genetics literature to be incredibly sloppy about notation and definitions and math. (Is this why LLMs are so bad at it?) This is the only source I’ve found that didn’t drive me completely insane.

Appendix

This is an article that just appeared in Asimov Press, who kindly agreed that I could publish it here and also humored my deep emotional need to use words like “Sparklepuff”.

Do you like information theory? Do you like molecular biology? Do you like the idea of smashing them together and seeing what happens? If so, then here’s a question: How much information is in your DNA?

When I first looked into this question, I thought it was simple:

1. Human DNA has about 3.1 billion base pairs.
2. Each base pair can take one of four values (A, T, C, or G)
3. It takes 2 bits to encode one of four possible values (00, 01, 10, or 11)
4. Thus, human DNA contains 6.2 billion bits.

Easy, right? Sure, except:

1. You have two versions of each base pair, one from each of your parents. Should you count both?
2. All humans have almost identical DNA. Does that matter?
3. DNA can be compressed. Should you look at the compressed representation?
4. It’s not clear how much of our DNA actually does something useful. The insides of your cells are a convulsing pandemonium of interacting “hacks”, designed to keep working even as mutations constantly screw around with the DNA itself. Should we only count the “useful” parts?

Such questions quickly run into the limits of knowledge for both biology and computer science. To answer them, we need to figure out what exactly we mean by “information” and how that’s related to what’s happening inside cells. In attempting that, I will lead you through a frantic tour of information theory and molecular biology. We’ll meet some strange characters, including genomic compression algorithms based on deep learning, retrotransposons, and Kolmogorov complexity.

Ultimately, I’ll argue that the intuitive idea of information in a genome is best captured by a new definition of a “bit”—one that’s unknowable with our current level of scientific knowledge.

On counting

What is “information”? This isn’t just a pedantic question, as there are actually several different mathematical definitions of a “bit”. Often, the differences don’t matter, but for DNA, they turn out to matter a lot, so let’s start with the simplest.

In the storage space definition, a bit is a “slot” in which you can store one of two possible values. If some object can represent 2ⁿ possible patterns, then it contains n bits, regardless of which pattern actually happens to be stored.

So here’s a question we can answer precisely: How much information could your DNA store?

A few reminders: DNA is a polymer. It’s a long chain of chunks of ~40 atoms called “nucleotides”. There are four different chunks, commonly labeled A, T, C, and G. In humans, DNA comes in 23 pieces of different lengths, called “chromosomes”. Humans are “diploid”, meaning we have two versions of each chromosome. We get one from each of our parents, made by randomly weaving together sections from the two chromosomes they got from their parents.

Chromosomes 1-22 have a total of 2.875 billion nucleotides; the X chromosome has 156 million, and the Y chromosome has 62 million. From here, we can calculate the total storage space in your DNA. Remember, each nucleotide has 4 options, corresponding to 2 bits. So if you’re female, your total storage space is:

  (2×2875 + 2×156) million nucleotides
     × 2 bits / nucleotide
     = 12.12 billion bits
     = 1.51 GB.

If you’re male, the total storage space is:

  (2×2875 + 156 + 62) million nucleotides
     × 2 bits / nucleotide
     = 11.94 billion bits
     = 1.49 GB.

For comparison, a standard single-layer DVD can store 37.6 billion bits or 4.7 GB. The code for your body, magnificent as it is, takes up as much space as around 40 minutes of standard definition video.

So in principle, your DNA could represent around 2^{12,000,000,000} different patterns. But hold on. Given human common ancestry, the chromosome pair you got from your mother is almost identical to the one you got from your father. And even ignoring that, there are long sequences of nucleotides that are repeated over and over in your DNA, enough to make up a significant fraction of the total. It seems weird to count all this repeated stuff. So perhaps we want a more nuanced definition of “information.”

On compression

A string of 12 billion zeros is much longer than this article. But most people would (I hope) agree that this article contains more information than a string of 12 billion zeros. Why?

One of the fundamental ideas from information theory is to define information in terms of compression. Roughly speaking, the “information” in some string is the length of the shortest possible compressed representation of that string.

So how much can you compress DNA? Answers to this question are all over the place. Some people claim it can be compressed by more than 99 percent, while others claim the state of the art is only around 25 percent. This discrepancy is explained by different definitions of “compression”, which turn out to correspond to different notions of “information”.

The fact that we share so much DNA is key to how some algorithms can compress DNA by more than 99 percent. They do this by first storing a reference genome, which includes all the DNA that’s shared by all people and perhaps the most common variants for regions of DNA where people differ. Then, for each individual person, these algorithms only store the differences from the reference genome. Because that reference only has to be stored once, it isn’t counted in the compressed representation.

That’s great if you want to cram as many of your friends’ genomes on a hard drive as possible. But it’s a strange definition to use if you want to measure the “information content of DNA”. It implies that any genomic content that doesn’t change between individuals isn’t important enough to count as “information”. However, we know from evolutionary biology that it’s often the most crucial DNA that changes the least precisely because it’s so important. Heritability tends to be lower for genes more closely related to reproduction.

The best compression without a reference seems to be around 25 percent. (I expect this number to rise a bit over time, as the newest methods use deep learning and research is ongoing.) That’s not a lot of compression. However, these algorithms are benchmarked in terms of how well they compress a genome that includes only one copy of each chromosome. Since your two chromosomes are almost identical (at least, ignoring the Y chromosome), I’d guess that you could represent the other half almost for free, meaning a compression rate of around 50 percent + ½ × 25 percent ≈ 62 percent.

On information

So if you compress DNA using an algorithm with a reference genome, it can be compressed by more than 99 percent, down to less than 120 million bits. But if you compress it without a reference genome, the best you can do is 62 percent, meaning 4.6 billion bits.

Which of these is right? The answer is that either could be right. There are two different definitions of a “bit” in information theory that correspond to different types of compression.

In the Kolmogorov complexity definition, named after the remarkable Soviet mathematician Andrey Kolmogorov, a bit is a property of a particular string of 1s and 0s. The number of bits of information in the string is the length of the shortest computer program that would output that string.

In the Shannon information definition, named after the also-remarkable American polymath Claude Shannon, a bit is again a property of a particular sequence of 1s and 0s, but it’s only defined relative to some large pool of possible sequences. In this definition, if a given sequence has a probability p of occurring, then it contains n bits for whatever value of n satisfies 2ⁿ=1/p. Or, equivalently, n=-log₂ p.

The Kolmogorov complexity definition is clearly related to compression. But what about Shannon’s?

Well, say you have three beloved pet rabbits, Fluffles, Marmalade, and Sparklepuff. And say you have one picture of each of them, each 1 MB large when compressed. To keep me updated on how you’re feeling, you like to send me these same pictures over and over again, with different pets for different moods. You send a picture of Fluffles ½ the time, Marmalade ¼ of the time, and Sparklepuff ¼ of the time. (You only communicate in rabbit pictures, never with text or images.)

But then you decide to take off in a spacecraft, and your data rates go way up. Continuing the flow of pictures is crucial, so what’s the cheapest way to do that? The best thing would be that we agree that if you send me a 0, I should pull up the picture of Fluffles, while if you send 10 I should pull up Marmalade, and if you send 11, I should pull up Sparklepuff. This is unambiguous: If you send 0011100, that means Fluffles, then Fluffles again, then Sparklepuff, then Marmalade, then Fluffles one more time.

It all works out. The “code length” for Fluffles is the number n so that 2ⁿ=1/p:

Intuitively, the idea is that if you want to send as few bits as possible over time, then you should give short codes to high-probability patterns and long codes to low-probability patterns. If you do this optimally (in the sense that you’ll send the fewest bits over time), it turns out that the best thing is to code a pattern with probability p with about n bits, where 2ⁿ=p. (In general, things don’t work out quite this nicely, but you get the idea.)

In the Fluffles scenario, the Kolmogorov complexity definition would say that each of the images contains 1 MB of information since that’s the smallest each image can be compressed. But under the Shannon information definition, the Fluffles image contains 1 bit of information, and the Marmalade and Sparklepuff images contain 2 bits. This is quite a difference!

Now, let’s return to DNA. There, the Kolmogorov complexity definition basically corresponds to the best possible compression algorithm without a reference. As we saw above, the best-known current algorithm can compress by 62 percent. So, under the Kolmogorov complexity definition, DNA contains at most 12 billion × (1-0.62) ≈ 4.6 billion bits of information.

Meanwhile, under the Shannon information definition, you can assume that the distribution of all human genomes is known. The information in your DNA only includes the bits needed to reconstruct your genome. That’s essentially the same as compressing with a reference. So, under the Shannon information definition, your DNA contains less than 12 billion × (1-0.01) ≈ 120 million bits of information.

While neither of these is “wrong” for DNA, I prefer the Kolmogorov complexity definition for its ability to best capture DNA that codes for features and functions shared by all humans. After all, if you’re trying to measure how much “information” our DNA carries from our evolutionary history, surely you want to include that which has been universally preserved.

On biology

At some point, your high-school biology teacher probably told you (or will tell you) this story about how life works:

1. First, your DNA gets transcribed into matching RNA.

2. Next, that RNA gets translated into protein.

3. Then the protein does Protein Stuff.

If things were that simple, we could easily calculate the information density of DNA just by looking at what fraction of your DNA ever becomes a protein (only around 1 percent). But it’s not that simple. The rest of your DNA does other important things, like regulating what proteins get made. Some of it seems to exist only for the purpose of copying itself. Some of it might do nothing, or it might do important things we don’t even know about yet.

So let me tell you that story again with slightly more detail:

1. In the beginning, your DNA is relaxing in the nucleus.

2. Some parts of your DNA, called promoters, are designed so that if certain proteins are nearby, they’ll stick to the DNA.

3. If that happens, then a hefty little enzyme called “RNA polymerase” will show up, crack open the two strands of DNA, and start transcribing the nucleotides on one side into “pre-messenger RNA” (pre-mRNA).

4. Eventually, for one of several reasons—none of which make any sense to me—the enzyme will decide it’s time to stop transcribing, and the pre-mRNA will detach and float off into the nucleus. At this point, it’s a few thousand or a few tens of thousands of nucleotides long.

5. Then, my personal favorite macromolecular complex, the “spliceosome”, grabs the pre-mRNA, cuts away most of it, and throws those parts away. The sections of DNA that code for the parts that are kept are called exons, while the sections that code for parts that are thrown away are called introns.

6. Next, another enzyme called “RNA guanylyltransferase” (we can’t all be beautiful) adds a “cap” to one end, and an enzyme called “poly(A) polymerase” adds a “tail” to the other end.

7. The pre-mRNA is now all grown up and has graduated to being regular mRNA. At this point, it is a few hundred or a few thousand nucleotides long.

8. Then, some proteins notice that the mRNA has a tail, grab it, and throw it out of the nucleus into the cytoplasm, where the noble ribosome lurks.

9. The ribosome grabs the mRNA and turns it into a protein. It does this by starting at one end and looking at chunks of three nucleotides at a time, called "codons". When it sees a certain "start" pattern, it starts translating each chunk into one of 20 amino acids and continues until it sees a chunk with a "stop" pattern.

   Since there are 4 kinds of nucleotides, there are 4³=64 possible chunks, while your body only uses 20 amino acids. So the ribosome, logically, gives some amino acids (like leucine) six different codons, and others (like tryptophan) only one codon. Also there are three different stop codons, but only one start codon, and that start codon is also the codon for methionine. So all proteins have methionine at one end unless something else comes and removes it later. Biology is layer after layer of this kind of exasperating complexity, totally indifferent to your desire to understand it.
10. The resulting protein lives happily ever after.

It’s thought that ~1 percent of your DNA is exons and ~24 percent is introns. What’s the rest of it doing?

Well, while the above dance is happening, other sections of DNA are “regulating” it. Enhancers are regions of DNA where a certain protein can bind and cause the DNA to physically bend so that some promoter somewhere else (typically within a million nucleotides) is more likely to get activated. Silencers do the opposite. Insulators block enhancers and silencers from influencing regions they shouldn’t influence.

While that might sound complicated, we’re just warming up. The same region of DNA can be both an intron and an enhancer and/or a silencer. That’s right, in the middle of the DNA that codes for some protein, evolution likes to put DNA that regulates some other, distant protein. When it’s not regulating, it gets transcribed into (probably useless) pre-RNA and then cut away and recycled by the spliceosome.

Further complicating this picture are many regions of DNA that code for RNA that’s never translated into a protein but still has some function. Some regions make tRNA, whose job is to bring amino acids to the ribosome. Other regions make rRNA, which bundle together with some proteins to become the ribosome. There’s siRNA, microRNA, and piRNA that screw around with mRNA produced. And there’s scaRNA, snoRNA, rRNA, lncRNA, and mrRNA. Many more types are sure to be defined in the future, both because it’s hard to know for sure if DNA gets transcribed, it’s hard to know what functions RNA might have, and because academics have strong incentives to invent ever-finer subcategories.

On messiness

Why? Why is this all such a mess? Why is it so hard to say if a given section of DNA does anything useful?

Biologists hate “why” questions. We can’t re-run evolution, so how can we say “why” evolution did things the way it did? Better to focus on how biological systems actually work. This is probably wise. But since I’m not a biologist (or wise), I’ll give my theory: Cells work like this because DNA is under constant attack from mutations.

Mutations most commonly arise during cell replication. Your DNA is composed of around 250 billion atoms. Making a perfect copy of all those atoms is hard. Your body has amazing nanomachines with many redundant mechanisms to try to correct errors, and it’s estimated that the error rate is less than one per billion nucleotides. But with several billion nucleotides, mutations happen.

There are also environmental sources of mutations. Ultraviolet light has more energy than visible light. If it hits your skin, that energy can sort of knock atoms out of place. The same thing happens if you’re exposed to radiation. Certain chemicals, like formaldehyde, benzene, or asbestos, can also do this or can interfere with your body’s error correction tricks.

Mutations in your regular cells will just affect you, but mutations in your sperm/eggs could affect all future generations. Evolution helps manage this through selection. Say you have 10 bad mutations, and I have 10 bad mutations, but those mutations are in different spots. If we have some babies together, some of them might get 13 bad mutations, but some might only get 7, and the latter babies are more likely to pass on their genes.

But as well as selection, cells seem designed to be extremely robust to these kinds of errors. Instead of just relying on selection, there are many redundant mechanisms to tolerate them without much issue.

And remember, evolution is a madman. If it decides to tolerate some mutation, everything else will be optimized against it. So even if a mutation is harmful at first, evolution may later find a way to make use of it.

On information again

So, in theory, how should we define the “information content” of DNA? I propose a definition I call the “phenotypic Kolmogorov complexity”. (This has surely been proposed by someone before, but I can’t find a reference, try as I might.) Roughly speaking, this is how short you could make DNA and still get a “human”.

The “phenotype” of an animal is just a fancy way of referring to its “observable physical characteristics and behaviors”. So this definition says, like Kolmogorov complexity, to try and find the shortest compressed representation of the DNA. But instead of needing to lead to the same DNA you have, it just needs to lead to an embryo that would look and behave like you do.

So what would this number be? My guess is that you could reduce the amount of DNA by at least 75 percent, but not by more than 98 percent, meaning the information content is:

  12 billion bits
     × 2 bits / nucleotide
     × (2 to 25 percent)
     = 480 million to 6 billion bits
     = 60 MB to 750 MB

But in reality, nobody knows. We still have no idea what (if anything) lots of DNA is doing, and we’re a long way from fully understanding how much it can be reduced. Probably, no one will know for a long time.