Since deterministic inference is often intractable with probabilistic models as we saw just now, we now turn to approximation methods based on numerical sampling, which are known as Monte Carlo techniques. The key question we will look at with these methods is computing the expectation of a target function f(z) given a probability distribution p(z). Recall that the simple definition of expectation is given as an integral:
Source: PRML¹ Eq. 11.1
As we will see, these integrals are too computationally complex, so we will turn to sampling methods in this article.
In this article, we will look at 3 core sampling methods: inverse transformation, Markov chain Monte Carlo (MCMC), and Gibbs Sampling. By understanding the underlying statistical properties and computational requirements of these methods, we will learn that: