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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 in the context of analysing data having spatial dependence.

Definition

Given a set of random variables $X=X_1,X_2,...X_n$ and a set $E$ of dependencies between these random variables, where $\{X_i,X_j\}\notin E$ implies $X_i$ is conditionally independent of $X_j$ given $X_i$'s neighbors, the pseudolikelihood of $X=x=(x_1,x_2,...x_n)$ is

$\mathrm{Pr}(X=x)=\prod _i\mathrm{Pr}(X_i=x_i|X_j=x_j\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}j\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{w}\mathrm{h}\mathrm{i}\mathrm{c}\mathrm{h}\{X_i,X_j\}\in E).$

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

$\log \mathrm{Pr}(X=x)=\sum _i\log \mathrm{Pr}(X_i=x_i|X_j=x_j\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}\{X_i,X_j\}\in E).$

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 $X_i$ 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 techiques to derive information about estimation uncertainty, or for significance testing, would in general be incorrect.

References

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