Pseudolikelihood in Graphical Models
Introduction
Because I did not use pseudolikelihood quite frequently, I often forget what it is and how to calculate it. The introductory materials from Wikipedia and other resources are always vague and confusing to beginners. So I decided to write a blog post on pseudolikelihood in detail in case in the future I forget it again.
Problem Definition
We usually have to calculate, given a graphical model (directed or undirected), what the probability of each possible state is. Concretely, we have the following undirected graphical model. We would like to know

Undirected Graphical Model
Therefore,
More generally, if there are
Without applying softmax function globally, the probability of the state
However, to calculate the conditional probabilities on the right side, it is still required to calculate the probabilies of all the states, which still takes
Pseudolikelihood Approximation
In our case,
We first calculate
Given
Apply softmax function to calculate
Similarly, we can calculate
Once the conditional probabilities have all been calculated, we could calculate the pseudolikelihood of state
More generally, if there are
Conclusion
Pseudolikelihood makes the calculation of the probability of the state from “intractable” to “tractable”, which accelerates our computation of the graphical models.
Pseudolikelihood in Graphical Models