{
  "id": "enwiki-001100439604",
  "slug": "pseudolikelihood",
  "title": "Pseudolikelihood",
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  "lead": "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.",
  "body": "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 X = X_1, X_2, \\ldots, X_n the pseudolikelihood of X = x = (x_1,x_2, \\ldots, x_n) is L(\\theta) := \\prod_i \\mathrm{Pr}_\\theta(X_i = x_i\\mid X_j = x_j \\text{ for } j \\neq i)=\\prod_i \\mathrm {Pr}_\\theta (X_i = x_i \\mid X_{-i}=x_{-i}) in discrete case and L(\\theta) := \\prod_i p_\\theta(x_i \\mid x_j \\text{ for } j \\neq i)=\\prod_i p _\\theta (x_i \\mid x_{-i})=\\prod _i p_\\theta (x_i \\mid x_1,\\ldots, \\hat x_i, \\ldots, x_n) in continuous one. Here X is a vector of variables, x is a vector of values, p_\\theta(\\cdot \\mid \\cdot) is conditional density and \\theta =(\\theta_1, \\ldots, \\theta_p) is the vector of parameters we are to estimate. 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 and x_{-i}=(x_1, \\ldots,\\hat x_i, \\ldots, x_n) means that the coordinate x_i has been omitted. The expression \\mathrm {Pr}_\\theta(X = x) is the probability that the vector of variables X has values equal to the vector x . This probability of course depends on the unknown parameter \\theta . Because situations can often be described using state variables ranging over a set of possible values, the expression \\mathrm {Pr}_\\theta(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, namely (in discrete case) l(\\theta):=\\log L(\\theta) = \\sum_i \\log \\mathrm{Pr}_\\theta(X_i = x_i\\mid X_j = x_j \\text{ for } j \\neq i). 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== 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.",
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    "id": "enwiki-001100439604",
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    "title": "Pseudolikelihood",
    "author": [
      {
        "family": "Anticloud FZ LLE",
        "literal": "Anticloud FZ LLE"
      }
    ],
    "container-title": "encyclopegia.org",
    "publisher": "Anticloud FZ LLE",
    "publisher-place": "Dubai, United Arab Emirates",
    "URL": "https://encyclopegia.org/a/pseudolikelihood",
    "issued": {
      "date-parts": [
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          2026,
          11,
          15
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          2026,
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    "license": "CC BY-SA 4.0",
    "DOI": "eo1:enwiki-001100439604",
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