Pseudolikelihood

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

:$L(\theta) := \prod_i \mathrm_\theta(X_i = x_i\mid X_j = x_j \text j \neq i)=\prod_i \mathrm _\theta (X_i = x_i \mid X_=x_)$

in discrete case and

:$L(\theta) := \prod_i p_\theta(x_i \mid x_j \text j \neq i)=\prod_i p _\theta (x_i \mid x_)=\prod _i p_\theta (x_i \mid x_1,\ldots, \hat x_i, \ldots, x_n)$

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 $x_=(x_1, \ldots,\hat x_i, \ldots, x_n)$ means that the coordinate $x_i$ has been omitted. The expression $\mathrm _\theta(X = x)$ 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 $\mathrm _\theta(X = x)$ 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)

:$l(\theta):=\log L(\theta) = \sum_i \log \mathrm_\theta(X_i = x_i\mid X_j = x_j \text j \neq i).$

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.Dodge, Y. (2003) ''The Oxford Dictionary of Statistical Terms'', Oxford University Press.

References

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Statistical inference