criticalpoints

E.T. Jaynes introduces this really interesting concept called the Apdistribution in his book Probability Theory. Despite the book enjoying a cult following, the Ap 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 Ap distribution is a way to give a probability about a probability. It concerns a very special set of propositions {Ap} that serve as basis functions for our belief state about a proposition A—somewhat similarly to how the Dirac delta functions can serve as basis elements for measures over the real line. The proposition Ap says that "regardless of whatever additional evidence that I've observed, the probability that I will assign to proposition A is p". It's defined by the rule: P(A|ApE)=p 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 1/2 probability of some proposition A being true, that can represent very different epistemic states. We can be more or less certain that the probability is "really" 1/2. 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 A be the proposition "the next time I flip this coin, it will come up heads." Then your best guess for A is 1/2. 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 1/2 to the next coin flip landi