# Bayes Nets - Understanding Inference

I was wondering if anyone could explain the following to me (from https://lips.cs.princeton.edu/complexity-of-inference-in-bayes-nets/):

"Briefly, recall that a Bayesian network consists of a directed acyclic graph with a random variable X_i at each vertex. Let $$\pi_i$$ be the parents of $$X_i$$. Then the Bayes net defines a distribution over $$X = (X_1,\dots,X_n)$$ of the form

$$\begin{equation*} \Pr[X] = \prod_{i=1}^n \Pr[X_i | \pi_i]\end{equation*}$$

Inference in a Bayes net corresponds to calculating the conditional probability $$\Pr[Y | Z = z]$$, where $$Y,Z \subset \{ X_1,\dots,X_n \}$$ are sets of latent and observed variables, respectively."

I understand the $$\Pr[X]$$ sectopm well enough, but am confused about the inference section; my guess is that $$\Pr[Y]$$ is the probability of the outcome variable/node but I'm not sure what $$Z = z$$ means and how this would be represented as a graph.

Thank you! :)

• by the looks of things, Z is just the set of X which is the parent of Y, since you can estimate Y only once you have a distribution over its parents Jul 11, 2022 at 11:30

One of the standard inference tasks in Bayesian networks (BNs) is to compute the conditional probability distribution function (cpdf) of some nodes ($$Y$$) in the BN, given the value of some other nodes ($$Z$$). In this case, one observes (i.e. "fixes") the values $$z$$ of a set of nodes $$Z$$ and infers the probability $$P(Y|Z=z)$$ of other nodes $$Y$$. Because the values of $$Y$$ are not observed, they are called "latent".
E.g., you could have a BN of binary ($$\{0, 1\}$$) nodes $$(X_1, X_2, X_3,X_4)$$:
and you are interested in the probabilities of both values of $$Y = X_2$$ under the condition that $$Z=X_4=0$$, i.e. you are interested in the cpdf $$P(X_2|X_4=0)$$, which consists of the two values $$P(X_2=0|X_4=0)$$ and $$P(X_2=1|X_4 = 0)$$.