7.2. Inference by Enumeration

Inference by Enumeration

Inference by Enumeration is the most direct method for computing posterior probabilities in any probabilistic model. It applies to all models that define a full joint distribution over variables, including Bayes Nets, Markov Models, and Hidden Markov Models, but is computationally expensive for large or structured systems.

The goal is to compute the posterior
$P(Q\mid e)=\alpha P(Q,e),$
where $Q$ is the query variable, $e$ is observed evidence, and $\alpha =\frac{1}{P(e)}$ is a normalization constant.

Using marginalization, the joint term expands as a sum over all hidden variables h:
$P(Q,e)={\sum }_{h}P(Q,h,e).$
This operation (i.e., enumerating over all hidden variables and summing out their contributions) is what gives the method its name.