marginal likelihood

1. marginal likelihood

Say that we have some model \(\alpha\) with parameters \(\theta \sim p(\theta\mid \alpha)\)
How should we compare between models? Recall that a model is a family of distributions parameterized by some \(\theta\). Let's take a look at the probability of the evidence given the assumption that it was generated from a distribution of a particular family
This is called the marginal likelihood of the observed variables \(X\), or "evidence"
It is given by \(P(X\mid \alpha) = \int p(X\mid\theta)p(\theta \mid \alpha)\; d\theta\)