Distribution of Mutual Information

Author: Marcus Hutter (2001)

Comments: 8 LaTeX pages

Subj-class: Artificial Intelligence

ACM-class:

I.2

Reference: Advances in Neural Information Processing Systems, 14 (NIPS-2001) 399-406

Report-no: IDSIA-13-01 and cs.AI/0112019

Paper: PostScript (167kb) - PDF (172kb) - Html/Gif

Slides: PostScript - PDF

Keywords: Mutual Information, Cross Entropy, Dirichlet distribution, Second order distribution, expectation and variance of mutual information.

Abstract: The mutual information of two random variables i and j with joint probabilities t_ij is commonly used in learning Bayesian nets as well as in many other fields. The chances t_ij are usually estimated by the empirical sampling frequency n_ij/n leading to a point estimate I(n_ij/n) for the mutual information. To answer questions like "is I(n_ij/n) consistent with zero?" or "what is the probability that the true mutual information is much larger than the point estimate?" one has to go beyond the point estimate. In the Bayesian framework one can answer these questions by utilizing a (second order) prior distribution p(t) comprising prior information about t. From the prior p(t) one can compute the posterior p(t|n), from which the distribution p(I|n) of the mutual information can be calculated. We derive reliable and quickly computable approximations for p(I|n). We concentrate on the mean, variance, skewness, and kurtosis, and non-informative priors. For the mean we also give an exact expression. Numerical issues and the range of validity are discussed.

Table of Contents

• Introduction
• Mutual Information Distribution
• Results for I under the Dirichlet P(oste)rior
• Approximation of Expectation and Variance of I
• The Second Order Dirichlet Distribution
• Exact Value for E[I]
• Generalizations
• Numerics

BibTeX Entry

@InProceedings{Hutter:01xentropy,
 author = "Marcus Hutter",
 title = "Distribution of Mutual Information",
 _number = "IDSIA-13-01",
 booktitle = "Advances in Neural Information Processing Systems 14",
 editor = "T. G. Dietterich and S. Becker and Z. Ghahramani",
 publisher = "MIT Press",
 address = "Cambridge, MA",
 pages = "399--406",
 year = "2002",
 url = "http://www.hutter1.net/ai/xentropy.htm",
 url2 = "http://arxiv.org/abs/cs.AI/0112019",
 ftp = "ftp://ftp.idsia.ch/pub/techrep/IDSIA-13-01.ps.gz",
 categories = "I.2. [Artificial Intelligence]",
 keywords = "Mutual Information, Cross Entropy, Dirichlet distribution, Second
 order distribution, expectation and variance of mutual
 information.",
 abstract = "The mutual information of two random variables i and j with joint
 probabilities t_ij is commonly used in learning Bayesian nets as
 well as in many other fields. The chances t_ij are usually
 estimated by the empirical sampling frequency n_ij/n leading to a
 point estimate I(n_ij/n) for the mutual information. To answer
 questions like ``is I(n_ij/n) consistent with zero?'' or ``what is
 the probability that the true mutual information is much larger
 than the point estimate?'' one has to go beyond the point estimate.
 In the Bayesian framework one can answer these questions by
 utilizing a (second order) prior distribution p(t) comprising
 prior information about t. From the prior p(t) one can compute the
 posterior p(t|n), from which the distribution p(I|n) of the mutual
 information can be calculated. We derive reliable and quickly
 computable approximations for p(I|n). We concentrate on the mean,
 variance, skewness, and kurtosis, and non-informative priors. For
 the mean we also give an exact expression. Numerical issues and
 the range of validity are discussed.",
}