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In [[statistical theory]], a '''pseudolikelihood''' is an [[approximation]] to the [[joint probability distribution]] of a collection of [[random variable]]s. 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 theory|estimation]], or may provide a way of obtaining explicit estimates of model parameters. | |||
The pseudolikelihood approach was introduced by [[Julian Besag]]<ref>Besag, J. (1975) "Statistical Analysis of Non-Lattice Data." ''The Statistician'', 24(3), 179–195</ref> in the context of analysing data having [[spatial dependence]]. | |||
==Definition== | |||
Given a set of random variables <math>X = X_1, X_2, ... X_n</math> and a set <math>E</math> of dependencies between these random variables, where <math> \lbrace X_i,X_j \rbrace \notin E </math> implies <math>X_i</math> is [[Conditional independence|conditionally independent]] of <math>X_j</math> given <math>X_i</math>'s neighbors, the pseudolikelihood of <math>X = x = (x_1,x_2, ... x_n)</math> is | |||
:<math>\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).</math> | |||
Here <math>X</math> is a vector of variables, <math>x</math> is a vector of values. The expression <math>X = x</math> above means that each variable <math>X_i</math> in the vector <math>X</math> has a corresponding value <math>x_i</math> in the vector <math>x</math>. The expression <math>\Pr(X = x)</math> is the probability that the vector of variables <math>X</math> has values equal to the vector <math>x</math>. Because situations can often be described using state variables ranging over a set of possible values, the expression <math>\Pr(X = x)</math> 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 | |||
:<math>\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).</math> | |||
One use of the pseudolikelihood measure is as an approximation for inference about a [[Markov random field|Markov]] or [[Bayesian network]], as the pseudolikelihood of an assignment to <math>X_i</math> 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 techiques to derive information about estimation uncertainty, or for [[Statistical significance|significance testing]], would in general be incorrect.<ref>Dodge, Y. (2003) ''The Oxford Dictionary of Statistical Terms'', OUP. ISBN 0-19-920613-9</ref> | |||
== References == | |||
{{reflist}} | |||
[[Category:Statistical inference]] | |||
Revision as of 20:26, 27 November 2012
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
Given a set of random variables and a set of dependencies between these random variables, where implies is conditionally independent of given 's neighbors, the pseudolikelihood of is
Here is a vector of variables, is a vector of values. The expression above means that each variable in the vector has a corresponding value in the vector . The expression is the probability that the vector of variables has values equal to the vector . 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. Thus
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
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
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