Pseudolikelihood

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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[edit]

Given a set of random variables X = X_1, X_2, ... X_n and a set E of dependencies between these random variables, where  \lbrace X_i,X_j \rbrace \notin 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

\Pr(X = x) = \prod_i \Pr(X_i = x_i|X_j = x_j\ \mathrm{for\ all\ } j\ \mathrm{for\ which}\ \lbrace X_i,X_j \rbrace \in E).

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

\log \Pr(X = x) = \sum_i \log \Pr(X_i = x_i|X_j = x_j\ \mathrm{for\ all}\ \lbrace X_i,X_j \rbrace \in E).

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[edit]

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 techiques to derive information about estimation uncertainty, or for significance testing, would in general be incorrect.[2]

References[edit]

  1. ^ Besag, J. (1975) "Statistical Analysis of Non-Lattice Data." The Statistician, 24(3), 179–195
  2. ^ Dodge, Y. (2003) The Oxford Dictionary of Statistical Terms, OUP. ISBN 0-19-920613-9