Pseudolikelihood

From Wikipedia, the free encyclopedia

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 $\lbrace X_i,X_j \rbrace \notin E$ 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

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

$X$ 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.

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

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.