Pseudolikelihood

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 } ∉ 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

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

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.