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 the pseudolikelihood of is in discrete case and in continuous one. Here X is a vector of variables, x is a vector of values, is conditional density and is the vector of parameters we are to estimate. 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 and means that the coordinate x_i has been omitted. The expression is the probability that the vector of variables X has values equal to the vector x. This probability of course depends on the unknown parameter \theta. Because situations can often be described using state variables ranging over a set of possible values, the expression 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, namely (in discrete case) 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.