Given a set of random variables the pseudolikelihood of is
in discrete case and
in continuous one.
Here is a vector of variables, is a vector of values, is conditional density and is the vector of parameters we are to estimate. The expression above means that each variable in the vector has a corresponding value in the vector and means that the coordinate has been omitted. The expression is the probability that the vector of variables has values equal to the vector . This probability of course depends on the unknown parameter . 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 may often be computed more efficiently than the likelihood, particularly when the latter may require marginalization over a large number of variables.