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 in the context of analysing data having spatial dependence.
Definition.
Given a set of random variables formula_1 the pseudolikelihood of formula_2 is
in discrete case and 
in continuous one. 
Here formula_5 is a vector of variables, formula_6 is a vector of values, formula_7 is conditional density and formula_8 is the vector of parameters we are to estimate. The expression formula_9 above means that each variable formula_10 in the vector formula_5 has a corresponding value formula_12 in the vector formula_6 and formula_14 means that the coordinate formula_15 has been omitted. The expression formula_16 is the probability that the vector of variables formula_5 has values equal to the vector formula_6. This probability of course depends on the unknown parameter formula_19. Because situations can often be described using state variables ranging over a set of possible values, the expression formula_16 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 formula_10 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.