# 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 ${\displaystyle X=X_{1},X_{2},\ldots ,X_{n}}$ and a set ${\displaystyle E}$ of dependencies between these random variables, where ${\displaystyle \lbrace X_{i},X_{j}\rbrace \notin E}$ implies ${\displaystyle X_{i}}$ is conditionally independent of ${\displaystyle X_{j}}$ given ${\displaystyle X_{i}}$'s neighbors, the pseudolikelihood of ${\displaystyle X=x=(x_{1},x_{2},\ldots ,x_{n})}$ is

${\displaystyle \Pr(X=x)=\prod _{i}\Pr(X_{i}=x_{i}\mid X_{j}=x_{j}{\text{ for all }}j{\text{ for which }}\lbrace X_{i},X_{j}\rbrace \in E).}$

Here ${\displaystyle X}$ is a vector of variables, ${\displaystyle x}$ is a vector of values. The expression ${\displaystyle X=x}$ above means that each variable ${\displaystyle X_{i}}$ in the vector ${\displaystyle X}$ has a corresponding value ${\displaystyle x_{i}}$ in the vector ${\displaystyle x}$. The expression ${\displaystyle \Pr(X=x)}$ is the probability that the vector of variables ${\displaystyle X}$ has values equal to the vector ${\displaystyle x}$. Because situations can often be described using state variables ranging over a set of possible values, the expression ${\displaystyle \Pr(X=x)}$ 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

${\displaystyle \log \Pr(X=x)=\sum _{i}\log \Pr(X_{i}=x_{i}\mid X_{j}=x_{j}{\text{ for all }}\lbrace X_{i},X_{j}\rbrace \in E).}$

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 ${\displaystyle X_{i}}$ 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 techniques to derive information about estimation uncertainty, or for significance testing, would in general be incorrect.[2]

## References

1. ^ Besag, J. (1975), "Statistical Analysis of Non-Lattice Data", The Statistician, 24 (3): 179–195, JSTOR 2987782
2. ^ Dodge, Y. (2003) The Oxford Dictionary of Statistical Terms, Oxford University Press. ISBN 0-19-920613-9[full citation needed]