pseudolikelihoods

Pseudolikelihood is a computationally feasible surrogate for the likelihood of complex statistical models, especially when the full joint distribution is difficult to evaluate. It was introduced in the context of Markov random fields and spatial statistics to handle models with intractable normalizing constants. For a random vector X = (X1, ..., Xd) with parameter θ, the pseudolikelihood Lp(θ; x) is defined as the product of the local conditional densities or probabilities: Lp(θ; x) = ∏i fθ(xi | x−i). The maximum pseudolikelihood estimator (MPLE) then maximizes this product with respect to θ. In discrete settings, the conditionals are conditional probabilities; in continuous settings, they are conditional densities.

The idea rests on replacing the difficult joint distribution with a collection of simpler, local conditionals. This makes estimation feasible in models where the joint likelihood is expensive or intractable, such as many Markov random fields, Gibbs distributions, or lattice-based spin systems. Pseudolikelihood is often viewed as a special case of composite likelihood, and various extensions include conditional, pairwise, and other marginal likelihoods that combine lower-dimensional components to form an objective for estimation.

Properties and limitations: MPLE can be consistent and asymptotically normal under standard regularity conditions, but it is generally less statistically efficient than the full maximum likelihood estimator. Standard error estimation commonly uses robust (sandwich) variance estimators or resampling methods to account for the use of conditional components. The accuracy of pseudolikelihood depends on the adequacy of the local conditional models and the extent to which the conditionals capture the dependence structure of the full model.

Applications span spatial statistics, image analysis, network models, and temporal processes, where computational tractability is a priority and exact likelihoods are impractical.