Kirkwood approximation

Kirkwood approximation

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. The Kirkwood approximation for a discrete probability density function P(X_{1},X_{2},ldots ,X_{n}) is given by

:P^{prime }(X_{1},X_{2},ldots ,X_{n})=frac{frac{frac{prod_{mathcal{T}_{n-1}subseteq mathcal{Vp(mathcal{T}_{n-1})}{prod_{mathcal{T}_{n-2}subseteq mathcal{Vp(mathcal{T}_{n-2}){vdots {prod_{mathcal{T}_{1}subseteq mathcal{Vp(mathcal{T}_{1})}

where prod_{mathcal{T}_{i}subseteq mathcal{Vp(mathcal{T}_{i}) is the product of probabilities over all subsets of variables of size i in variable set mathcal{V}. 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

:P^{prime }(X_{1},X_{2},X_{3})=frac{p(X_{1},X_{2})p(X_{2},X_{3})p(X_{1},X_{3})}{p(X_{1})p(X_{2})p(X_{3})}

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.

Judea Pearl (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 which admits a graph structurewhose cliques form 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 intersectiondistributions.

References

* Jakulin, A. & Bratko, I. (2004), Quantifying and visualizing attribute interactions: An approach based on entropy, "Journal of Machine Learning Research", (submitted) pp. 38–43.

* Matsuda, H. (2000), Physical nature of higher-order mutual information: Intrinsic correlations and frustration, "Physical Review E" 62, 3096–3102.

* Pearl, J. (1988), "Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference", Morgan Kaufmann, San Mateo, CA.

* Watanabe, S. (1960), Information theoretical analysis of multivariate correlation, "IBM Journal of Research and Development" 4, 66–82.


Wikimedia Foundation. 2010.

Игры ⚽ Нужно решить контрольную?

Look at other dictionaries:

  • Kirkwood — is the name of some places in the United States of America: *Kirkwood, California **Kirkwood Mountain Resort *Kirkwood, Delaware *Kirkwood (Atlanta), Georgia *Kirkwood, Illinois *Kirkwood, Missouri *Kirkwood, New Jersey *Kirkwood, New York… …   Wikipedia

  • List of statistics topics — Please add any Wikipedia articles related to statistics that are not already on this list.The Related changes link in the margin of this page (below search) leads to a list of the most recent changes to the articles listed below. To see the most… …   Wikipedia

  • List of mathematics articles (K) — NOTOC K K approximation of k hitting set K ary tree K core K edge connected graph K equivalence K factor error K finite K function K homology K means algorithm K medoids K minimum spanning tree K Poincaré algebra K Poincaré group K set (geometry) …   Wikipedia

  • Interaction information — The interaction information (McGill 1954) or co information (Bell 2003) is one of several generalizations of the mutual information, and expresses the amount information (redundancy or synergy) bound up in a set of variables, beyond that which is …   Wikipedia

  • Théorie de la perturbation (mécanique quantique) — En mécanique quantique, la théorie de la perturbation (ou théorie des perturbations) est un ensemble de schémas d approximations liée à une perturbation mathématique utilisée pour décrire un système quantique complexe de façon simplifiée. L idée… …   Wikipédia en Français

  • Perturbation theory (quantum mechanics) — In quantum mechanics, perturbation theory is a set of approximation schemes directly related to mathematical perturbation for describing a complicated quantum system in terms of a simpler one. The idea is to start with a simple system for which a …   Wikipedia

  • celestial mechanics — the branch of astronomy that deals with the application of the laws of dynamics and Newton s law of gravitation to the motions of heavenly bodies. [1815 25] * * * Branch of astronomy that deals with the mathematical theory of the motions of… …   Universalium

  • Asteroid belt — For the album by Velvet Chain, see Asteroid Belt (album). The asteroid belt (shown in white) is located between the orbits of Mars and Jupiter. The asteroid belt is the region of the Solar System located roughly between the orbits of the planets… …   Wikipedia

  • CINÉTIQUE DES FLUIDES (THÉORIE) — La théorie cinétique des fluides appartient à une branche de la physique qui se propose d’expliquer les propriétés macroscopiques des fluides à partir d’une analyse statistique des mouvements des particules qui les constituent. On peut classer… …   Encyclopédie Universelle

  • Nikolay Bogolyubov — For the actor, see Nikolay Bogolyubov (actor). Nikolay Nikolaevich Bogolyubov Born 21 August 1909( …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”