Conditional mutual information

In probability theory, and in particular, information theory, the conditional mutual information is, in its most basic form, the expected value of the mutual information of two random variables given the value of a third.
Contents
Definition
For discrete random variables X, Y, and Z, we define
where the marginal, joint, and/or conditional probability mass functions are denoted by p with the appropriate subscript. This can be simplified as
Alternatively, we may write^{[1]}
 I(X;Y  Z) = H(X,Z) + H(Y,Z) − H(X,Y,Z) − H(Z)
Conditional mutual information can also be rewritten to show its relationship to mutual information
 I(X;Y  Z) = H(X  Z) − H(X  Y,Z) = I(X;Y) − H(Z) + H(Z  X) + H(Z  Y) − H(Z  X,Y)
Conditioning on a third random variable may either increase or decrease the mutual information: that is, the difference I(X;Y  Z) − I(X;Y), called the interaction information, may be positive, negative, or zero, but it is always true that
for discrete, jointly distributed random variables X, Y, Z. This result has been used as a basic building block for proving other inequalities in information theory, in particular, those known as Shannontype inequalities.
Like mutual information, conditional mutual information can be expressed as a KullbackLeibler divergence:
Or as an expected value of simpler KullbackLeibler divergences:
More general definition
A more general definition of conditional mutual information, applicable to random variables with continuous or other arbitrary distributions, will depend on the concept of regular conditional probability. (See also.^{[2]}^{[3]})
Let be a probability space, and let the random variables X, Y, and Z each be defined as a Borelmeasurable function from Ω to some state space endowed with a topological structure.
Consider the Borel measure (on the σalgebra generated by the open sets) in the state space of each random variable defined by assigning each Borel set the measure of its preimage in . This is called the pushforward measure The support of a random variable is defined to be the topological support of this measure, i.e.
Now we can formally define the conditional probability measure given the value of one (or, via the product topology, more) of the random variables. Let M be a measurable subset of Ω, (i.e. ) and let Then, using the disintegration theorem:
where the limit is taken over the open neighborhoods U of x, as they are allowed to become arbitrarily smaller with respect to set inclusion.
Finally we can define the conditional mutual information via Lebesgue integration:
where the integrand is the logarithm of a Radon–Nikodym derivative involving some of the conditional probability measures we have just defined.
Note on notation
In an expression such as I(A;B  C), A, B, and C need not necessarily be restricted to representing individual random variables, but could also represent the joint distribution of any collection of random variables defined on the same probability space. As is common in probability theory, we may use the comma to denote such a joint distribution, e.g. I(A_{0},A_{1};B_{1},B_{2},B_{3}  C_{0},C_{1}). Hence the use of the semicolon (or occasionally a colon or even a wedge ) to separate the principal arguments of the mutual information symbol. (No such distinction is necessary in the symbol for joint entropy, since the joint entropy of any number of random variables is the same as the entropy of their joint distribution.)
Multivariate mutual information
Main article: Multivariate mutual informationThe conditional mutual information can be used to inductively define a multivariate mutual information in a set or measuretheoretic sense in the context of information diagrams. In this sense we define the multivariate mutual information as follows:
where
This definition is identical to that of interaction information except for a change in sign in the case of an odd number of random variables. A complication is that this multivariate mutual information (as well as the interaction information) can be positive, negative, or zero, which makes this quantity difficult to interpret intuitively. In fact, for n random variables, there are 2^{n} − 1 degrees of freedom for how they might be correlated in an informationtheoretic sense, corresponding to each nonempty subset of these variables. These degrees of freedom are bounded by various Shannon and nonShannontype inequalities in information theory.
References
 ^ K. Makarychev et al. A new class of nonShannontype inequalities for entropies. Communications in Information and Systems, Vol. 2, No. 2, pp. 147–166, December 2002 PDF
 ^ Regular Conditional Probability on PlanetMath
 ^ D. Leao Jr. et al. Regular conditional probability, disintegration of probability and Radon spaces. Proyecciones. Vol. 23, No. 1, pp. 15–29, May 2004, Universidad Católica del Norte, Antofagasta, Chile PDF
Categories: Information theory
 Entropy and information
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