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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.

Contents

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 P ( x , y ) . Let e represent an occurrence of error; i.e., that X ≠ X ~ , with X ~ = f ( Y ) being an approximate version of X . Fano's inequality is

H ( X | Y ) ≤ H ( e ) + P ( e ) log ⁡ ( | X | − 1 ) ,

where X denotes the support of X,

H ( X | Y ) = − ∑ i , j P ( x i , y j ) log ⁡ P ( x i | y j )

is the conditional entropy,

P ( e ) = P ( X ≠ X ~ )

is the probability of the communication error, and

H ( e ) = − P ( e ) log ⁡ P ( e ) − ( 1 − P ( e ) ) log ⁡ ( 1 − P ( e ) )

is the corresponding binary entropy.

Alternative formulation

Let X be a random variable with density equal to one of r + 1 possible densities f 1 , … , f r + 1 . Furthermore, the Kullback–Leibler divergence between any pair of densities cannot be too large,

D K L ( f i ∥ f j ) ≤ β for all i ≠ j .

Let ψ ( X ) ∈ { 1 , … , r + 1 } be an estimate of the index. Then

sup i P i ( ψ ( X ) ≠ i ) ≥ 1 − β + log ⁡ 2 log ⁡ r

where P i is the probability induced by f i

Generalization

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′

∥ f θ − f θ ′ ∥ L 1 ≥ α , D K L ( f θ ∥ f θ ′ ) ≤ β .

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

sup f ∈ F E ∥ f n − f ∥ L 1 ≥ α 2 ( 1 − n β + log ⁡ 2 log ⁡ r )

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

References

Fano's inequality Wikipedia
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