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 $X = X_1, X_2, ..., X_n$ 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 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
2. ↑ Dodge, Y. (2003) The Oxford Dictionary of Statistical Terms, OUP. ISBN 0-19-920613-9