P(A,B) to P(H,D)??

written by Eric J. Ma on 2020-06-14 | tags: bayesian data science statistical inference

Bayes' rule looks like this:

$$P(A|B) = \frac{P(B|A)P(A)} {P(B)} $$

It is a natural result that comes straight from the rules of probability, being that the joint distribution of two random variables can be written in two equivalent ways:

$$P(A, B) = P(A|B)P(B) = P(B|A)P(A)$$

Now, I have encountered in many books write, regarding the application of Bayes' rule to statistical modelling, something along the lines of the following:

Now, there is an alternative interpretation of Bayes' rule, one that replaces the symbol "A" with "Hypothesis", and "B" with the "Data", such that we get:

$$P(H|D) = \frac{P(D|H)P(H)}{P(D)}$$

At first glance, nothing seems wrong about this statement, but I did remember having a lingering nagging feeling that there was a logical jump unexplained here.

More specifically, that logical jump yielded the following question: Why are we allowed to take this interpretation?

It took asking the question to a mathematician friend, Colin Carroll, to finally "grok" the idea. Let me try to explain.

Spaces of models and data

We have to think back to the fundamental idea of possible spaces.

If we set up a Bernoulli probability distribution with parameter $p$, then the space of possible probability distributions that we could instantiate is infinite! This result should not surprise you: $p$ can take on any one of an infinite set of values between 0 and 1, each one giving a different instantiated Bernoulli. As such, a Bernoulli(p) hypothesis is drawn from a (very large) space of possible Bernoulli(p)s, or more abstractly, hypotheses, thereby giving us a $P(H)$.

Moreover, consider our data. The Bernoulli data that came to us, which for example might be 0, 1, 1, 1, 0, were drawn from a near-infinite space of possible configurations of data. First off, there's no reason why we always have to have three 1s and two 0s in five draws; it could have been five 1s or five 0s. Secondly, the order of data (though it doesn't really matter in this case) for three 1s and two 0s might well have been different. As such, we have the $P(D)$ interpretation.

As a modelling decision, we choose to say that our data and model are jointly distributed, thus we have the joint distribution between model and data, $P(H, D)$.

And once we have the joint distribution between model and data, we can begin to reason about Bayes' rule on it.