The Kirkwood superposition approximation was introduced by Matsuda (2000) as a means of representing a discrete probability distribution. The name apparently refers to a 1942 paper by John G. Kirkwood

John Gamble Kirkwood

John "Jack" Gamble Kirkwood was a noted chemist and physicist, holding faculty positions at Cornell University, the University of Chicago, California Institute of Technology, and Yale University.-Early life and background:Kirkwood was born in Gotebo, Oklahoma, the oldest child of John Millard and...

. The Kirkwood approximation for a discrete probability density function

Probability density function

In probability theory, a probability density function , or density of a continuous random variable is a function that describes the relative likelihood for this random variable to occur at a given point. The probability for the random variable to fall within a particular region is given by the...

  is given by


where

is the product of probabilities over all subsets of variables of size i in variable set . This kind of formula has been considered by Watanabe (1960) and, according to Watanabe, also by Robert Fano. For the three-variable case, it reduces to simply


The Kirkwood approximation does not generally produce a valid probability distribution (the normalization condition is violated). Watanabe claims that for this reason informational expressions of this type are not meaningful, and indeed there has been very little written about the properties of this measure. The Kirkwood approximation is the probabilistic counterpart of the interaction information

Interaction information

The interaction information or co-information is one of several generalizations of the mutual information, and expresses the amount information bound up in a set of variables, beyond that which is present in any subset of those variables...

.

Judea Pearl

Judea Pearl

Judea Pearl is a computer scientist and philosopher, best known for developing the probabilistic approach to artificial intelligence and the development of Bayesian networks ....

 (1988 §3.2.4) indicates that an expression of this type can be exact in the case of a decomposable model, that is, a probability distribution that admits a graph

Graph (mathematics)

In mathematics, a graph is an abstract representation of a set of objects where some pairs of the objects are connected by links. The interconnected objects are represented by mathematical abstractions called vertices, and the links that connect some pairs of vertices are called edges...

 structure whose cliques

Clique (graph theory)

In the mathematical area of graph theory, a clique in an undirected graph is a subset of its vertices such that every two vertices in the subset are connected by an edge. Cliques are one of the basic concepts of graph theory and are used in many other mathematical problems and constructions on graphs...

 form a tree

Tree (graph theory)

In mathematics, more specifically graph theory, a tree is an undirected graph in which any two vertices are connected by exactly one simple path. In other words, any connected graph without cycles is a tree...

. In such cases, the numerator contains the product of the intra-clique joint distributions and the denominator contains the product of the clique intersection distributions.