E.T. Jaynes introduces this really interesting concept called the distribution in his book Probability Theory. Despite the book enjoying a cult following, the distribution has failed to become widely-known among aspiring probability theorists. After finishing the relevant chapter in the book, I googled the phrase “Ap distribution” in an attempt to learn more, but I didn’t get many search results. So here’s an attempt to explain it for a wider audience.
The distribution is a way to give a probability about a probability. It concerns a very special set of propositions that serve as basis functions for our belief state about a proposition —somewhat similarly to how the Dirac delta functions can serve as basis elements for measures over the real line. The proposition says that “regardless of whatever additional evidence that I’ve observed, the probability that I will assign to proposition is ”. It’s defined by the rule:
This is a fairly strange proposition. But it’s useful as it allows us to encode a belief about a belief. Here’s why it’s a useful conceptual tool:
If we say that there is a probability of some proposition being true, that can represent very different epistemic states. We can be more or less certain that the probability is “really” .
For example, imagine that you know for sure that a coin is fair: that in the long run, the number of heads flipped will equal the number of tails flipped. Let be the proposition “the next time I flip this coin, it will come up heads.” Then your best guess for is . And importantly, no matter what sequence of coin flips you observe, you will always guess that there is a 50% chance that the next coin flip lands on heads.
But imagine a different scenario where you haven’t seen a coin get flipped yet, but you are told that it’s perfectly biased towards either heads or tails. Because of the principle of indifference, we would assign a probability of to the next coin flip landing on heads—but it’s a very different from the case when we had a fair coin. Just one single flip of the coin is enough to collapse the probability of to either or for all time.
There is a useful law in probability called the law of total expectation. It says that your expectation for your future expectation should match your current expectation. So given a static coin, the probability that you give for the next coin flip being heads should be the same as the probability that you give for 100th or 1000000th coin flip being heads.
Mathematically, the law of total expectation can be expressed as:
The law of total expectation is sometimes bastardized as saying “you shouldn’t expect your beliefs to change.” But that’s not what it’s saying at all! It’s saying that you shouldn’t expect your expectation—the center of mass of your state of belief—to change. But all higher moments in belief space can and do change as you acquire more evidence.
And this makes sense. Information is the opposite of entropy; if entropy measures our uncertainty, than information is a measure of our certainty: how constrained the possible worlds are. If I have a biased coin, I give a 50% chance that the 100th flip lands on heads. But I know now that right before the 100th flip, I will be in an epistemic state of 100% certainty in the outcome—it’s just that current me doesn’t know what outcome future me will be 100% certain in.
A way to think about the proposition is as a kind of limit. When we have little evidence, each bit of evidence has a potentially big impact on our overall probability of a given proposition. But each incremental bit of evidence shifts our beliefs less and less. The proposition can be thought of a shorthand for an infinite collection of evidences where the collection leads to an overall probability of given to . This would perhaps explains why the proposition is so strange: we have well-developed intuitions for how “finite” propositions interact, but the characteristic absorbing property of the distribution is more reminiscent of how an infinite object interacts with finite objects.
For any proposition , the probability of can be found by integrating over our probabilities of
where can be said to represent “the probability that, after encountering an infinite amount of evidence, I will give a probability of to the proposition .”
For the fair coin, our belief in space would be represented by a delta function at .
For the biased coin (where we don’t know the bias), our belief would be represented by two deltas of weight 1⁄2 at and .
Both distributions assign a probability of to . But the two distributions greatly differ in their variance in space.
I don’t think this concept is useful.
What you are showing with the coin is a hierarchical model over multiple coin flips, and doesn’t need new probability concepts. Let Fi be the flips. All you need in life is the distribution P(F1,F2,…). You can decide to restrict yourself to distributions of the form ∫10dpcoinP(F,G|pcoin)p(pcoin). In practice, you start out thinking about pcoin as a variable atop all the Fi in a graph, and then think in terms of P(F,G|pcoin) and p(pcoin) separately, because that’s more intuitive. This is the standard way of doing things. All you do with Ap is the same, there’s no point at which you do something different in practice, even if you ascribed additional properties to Ap in words.
A concept like “the probability of me assigning a certain probability” makes sense but I don’t think Jaynes actually did anything like that for real. Here on lesswrong I guess @abramdemski knows about stuff like that.
--PS: I think Jaynes was great in his way of approaching the meaning and intuition of statistics, but the book is bad as a statistics textbook. It’s literally the half-complete posthumous publication of a rambling contrarian physicist, and it shows. So I would not trust any specific statistical thing he does. Taking the general vibe and ideas is good, but when you ask about a specific thing “why is nobody doing this?” it’s most likely because it’s outdated or wrong.
Thanks for the feedback.
This isn’t emphasized by Jaynes (though I believe it’s mentioned at the very end of the chapter), but the Ap distribution isn’t new as a formal idea in probability theory. It’s based on De Finetti’s representation theorem. The theorem concerns exchangeable sequences of random variables.
A sequence of random variables {Xi} is exchangeable if the joint distribution of any finite subsequence is invariant under permutations. A sequence of coin flips is the canonical example. Note that exchangeability does not imply independence! If I have a perfectly biased coin where I don’t know the bias, then all the random variables are perfectly dependent on each other (they all must obtain the same value).
De Finetti’s representation theorem says that any exchangeable sequence of random variables can be represented as an integral over identical and independent distributions (i.e binomial distributions). Or in other words, the extent to which random variables in the sequence are dependent on each other is solely due to their mutual relationship to the latent variable (the hidden bias of the coin).
P(X1=x1,…,Xn=xn)=∫10(nk)θk(1−θ)n−kdF(θ)
You are correct that all relevant information is contained in the joint distribution P(F1,F2,...). And while I have no deep familiarity with Bayesian hierarchical modeling, I believe your claim that the decomposition ∫10 dpcoinP(F,G|pcoin)p(pcoin) is standard in Bayesian modeling.
But I think the point is that the Ap distribution is a useful conceptual tool when considering distributions governed by a time-invariant generating process. A lot of real-world processes don’t fit that description, but many do fit that description.
Yes, this is correct. The part about “the probability of assigning a probability” and the part about interpreting the proposition Ap as a shorthand for an infinite collection evidences are my own interpretations of what the Ap distribution “really” means. Specifically, the part about the “probability that you will assign the probability in the infinite future” is loosely inspired by the idea of Cauchy surfaces from e.g general relativity (or any physical theory that has a causal structure built in). In general relativity, the idea is that if you have boundary conditions specified on a Cauchy surface, then you can time-evolve to solve for the distribution of matter and energy for all time. In something like quantum field theory, a principled choice for the Cauchy surface would be the infinite past (this conceptual idea shows up when understanding the vacuum in QFT). But I think in probability theory, it’s more useful conceptually to take your Cauchy surface of probabilities to be what you expect them to be in the “infinite future”. This is how I make sense of the Ap distribution.
And now that you mention it, this blog post was totally inspired by reading the first couple chapters of “Logical Inductors” (though the inspiration wasn’t conscious on my part).
Not a statistician, so I will defer to your expertise that the book is bad as a statistics book (never thought of it as a statistics book to be honest). I think the strongest parts of this book are when he derives statistical mechanics from the maximum entropy principle and when he generalizes the principle of indifference to consider more general group invariances/symmetries. As far as I’m aware, my opinion on which of Jaynes’ ideas are his best ideas matches the consensus.
I suspect the reason why I like the Ap distribution is that I come from a physics background, so his reformulation of standard ideas in Bayesian modeling makes some amount of sense to me even if comes across as weird and crankish to statisticians.
I still don’t understand your “infinite limit” idea. If in your post I drop the following paragraph:
the rest is standard hierarchical modeling. So even if your words here are suggestive, I don’t understand how to actually connect the idea to calculations/concrete things, even at a vague indicative level. So I guess I’m not actually understanding it.
For example, you could show me a conceptual example where you do something with this which is not standard probabilistic modeling. Or maybe it’s all standard but you get to a solution faster. Or anything where applying the idea produces something different, then I would see how it works.
(Note: I don’t know if you noticed, but De Finetti applies to proper infinite sequences only, not finite ones, people forget this. It is not relevant to the discussion though)