Pseudolikelihood

Pseudolikelihood is a measure in statistics that serves as an approximation of the distribution of a random variable. Given a set of random variables $X\; =\; X\_1,\; X\_2,\; ...\; X\_n$ and a set $E$ of dependencies between these random variables, where $lbrace\; X\_i,X\_j\; brace\; otin\; E$ 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

$Pr(X\; =\; x)\; =\; prod\_i\; Pr(X\_i\; =\; x\_i|X\_j\; =\; x\_j\; mathrm\{for\; all\}\; lbrace\; X\_i,X\_j\; brace\; in\; E)$

$X$ 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 $P(X\; =\; x)$ 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 $P(X\; =\; x)$ can therefore represent the probability of a certain state among all possible states allowed by the state variables.

Pseudo-log-likelihood is a similar measure derived from the above expression.

$log\; Pr(X\; =\; x)\; =\; sum\_i\; log\; Pr(X\_i\; =\; x\_i|X\_j\; =\; x\_j\; mathrm\{for\; all\}\; lbrace\; X\_i,X\_j\; brace\; in\; E)$

One use of the pseudolikelihood measure is as an approximation of inference over a Markov network 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.

Citations

* Besag, J. (1975). "Statistical Analysis of Non-Lattice Data." The Statistician, 24(3):179--195.

See also

* Likelihood