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 $X=X_{1},X_{2},\ldots ,X_{n}$ 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 $X=x=(x_{1},x_{2},\ldots ,x_{n})$ is

\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 $X$ 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}\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 $X_{i}$ 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

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

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