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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 $\lbrace X_{i},X_{j}\rbrace \notin E$ 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

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

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 $\Pr(X=x)$ 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 $\Pr(X=x)$ 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

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

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 $X_{i}$ 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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