Fano's inequality

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In information theory, Fano's inequality (also known as the Fano converse and the Fano lemma) relates the average information lost in a noisy channel to the probability of the categorization error. It was derived by Robert Fano in the early 1950s while teaching a Ph.D. seminar in information theory at MIT, and later recorded in his 1961 textbook.

It is used to find a lower bound on the error probability of any decoder as well as the lower bounds for minimax risks in density estimation.

Let the random variables X and Y represent input and output messages with a joint probability . Let e represent an occurrence of error; i.e., that , with being an approximate version of . Fano's inequality is

where denotes the support of X,

is the conditional entropy,

is the probability of the communication error, and

is the corresponding binary entropy.

Alternative formulation[edit]

Let X be a random variable with density equal to one of possible densities . Furthermore, the Kullback–Leibler divergence between any pair of densities cannot be too large,

for all

Let be an estimate of the index. Then

where is the probability induced by

Generalization[edit]

The following generalization is due to Ibragimov and Khasminskii (1979), Assouad and Birge (1983).

Let F be a class of densities with a subclass of r + 1 densities ƒθ such that for any θ ≠ θ

Then in the worst case the expected value of error of estimation is bound from below,

where ƒn is any density estimator based on a sample of size n.

References[edit]