# 18 Confronting the Partition Function¶

Review:

An undirected graphical model is a structured probabilitic model defined on an undirected graph G. For each clique C in the graph, a factor $$\phi(C)$$ (also called a clique protential) measures the affinity of the variables in that clique for being in each of their possible joint states. $$\phi(C) > 0$$. Unnormalized probability distribution:

$\hat{p}(x) = \prod_{C\in G}\phi(C)$

A clique of the graph is a subset of nodes that are all connected to each other by an edge of the graph.

To obtain a valid probability distribution, we must use the corresponding normalized probability distribution.

$p(x) = \frac{1}{Z}\hat{p}(x)$

where Z is the value that results in the probability distribution summing or integrting to 1

$Z = \int \hat{p}(x)dx$

You can think of Z as constant when $$\phi$$ functions are held constant. Note that if the $$\phi$$ functions have parameters, then Z is a function of those parameters. Normalizing function Z is known as partition function.

Computing Z is intractable for many interesting models. How to confront the challenge

• models designed to have a tractable normalizing constant
• designed to be used in ways that do not involve comupting p(x) at all.
• directly confront the challenge of intractable partition function, as described in this chapter