Mutual Information (transinformation)
It turns out that one of the most useful and important measures of information is the mutual information, or transinformation. This is a measure of how much information can be obtained about one random variable by observing another. The mutual information of relative to (which represents conceptually the average amount of information about that can be gained by observing ) is given by:
A basic property of the mutual information is that:
That is, knowing Y, we can save an average of bits in encoding X compared to not knowing Y. Mutual information is symmetric:
Mutual information can be expressed as the average Kullback–Leibler divergence (information gain) of the posterior probability distribution of X given the value of Y to the prior distribution on X:
In other words, this is a measure of how much, on the average, the probability distribution on X will change if we are given the value of Y. This is often recalculated as the divergence from the product of the marginal distributions to the actual joint distribution:
Mutual information is closely related to the log-likelihood ratio test in the context of contingency tables and the multinomial distribution and to Pearson's χ2 test: mutual information can be considered a statistic for assessing independence between a pair of variables, and has a well-specified asymptotic distribution.
Read more about this topic: Quantities Of Information
Famous quotes containing the words mutual and/or information:
“Ties of blood are not always ties of friendship; but friendship founded on merit, on esteem, and on mutual trust, becomes more vital and more tender when strengthened by the ties of blood.”
—Philip Dormer Stanhope, 4th Earl Chesterfield (1694–1773)
“The real, then, is that which, sooner or later, information and reasoning would finally result in, and which is therefore independent of the vagaries of me and you. Thus, the very origin of the conception of reality shows that this conception essentially involves the notion of a COMMUNITY, without definite limits, and capable of a definite increase of knowledge.”
—Charles Sanders Peirce (1839–1914)