# Pseudolikelihood

In statistical theory, a **pseudolikelihood** is an approximation to the joint probability distribution of a collection of random variables. The practical use of this is that it can provide an approximation to the likelihood function of a set of observed data which may either provide a computationally simpler problem for estimation, or may provide a way of obtaining explicit estimates of model parameters.

The pseudolikelihood approach was introduced by Julian Besag^{[1]} in the context of analysing data having spatial dependence.

## Definition

Given a set of random variables and a set of dependencies between these random variables, where implies is conditionally independent of given 's neighbors, the pseudolikelihood of is

Here is a vector of variables, is a vector of values. The expression above means that each variable in the vector has a corresponding value in the vector . The expression is the probability that the vector of variables has values equal to the vector . Because situations can often be described using state variables ranging over a set of possible values, the expression can therefore represent the probability of a certain state among all possible states allowed by the state variables.

The **pseudo-log-likelihood** is a similar measure derived from the above expression. Thus

One use of the pseudolikelihood measure is as an approximation for inference about a Markov or Bayesian network, as the pseudolikelihood of an assignment to may often be computed more efficiently than the likelihood, particularly when the latter may require marginalization over a large number of variables.

## Properties

Use of the pseudolikelihood in place of the true likelihood function in a maximum likelihood analysis can lead to good estimates, but a straightforward application of the usual likelihood techiques to derive information about estimation uncertainty, or for significance testing, would in general be incorrect.^{[2]}