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[1] in the context of analysing data having spatial dependence.

Definition

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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.

Properties

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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.[2]

References

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  1. ^ Besag, J. (1975), "Statistical Analysis of Non-Lattice Data", The Statistician, 24 (3): 179–195, doi:10.2307/2987782, JSTOR 2987782
  2. ^ Dodge, Y. (2003) The Oxford Dictionary of Statistical Terms, Oxford University Press. ISBN 0-19-920613-9 [full citation needed]