Herzog–Schönheim conjecture

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.

Fano's inequality

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 $X\neq Y$. Fano's inequality is

$H(X|Y)\leq H(P(e))+P(e)\log(|{\mathcal {X}}|-1),$

where ${\mathcal {X}}$ denotes the support of X,

$H\left(X|Y\right)=-\sum _{i,j}P(x_{i},y_{j})\log P\left(x_{i}|y_{j}\right)$

is the conditional entropy,

$P(e)=P(X\neq Y)$

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 $f_{1},\ldots ,f_{r+1}$. Furthermore, the Kullback-Leibler divergence between any pair of densities cannot be too large,

$D_{KL}(f_{i}\|f_{j})\leq \beta$ for all $i\not =j.$

Let $\psi (X)\in \{1,\ldots ,r+1\}$ be an estimate of the index. Then

$\sup _{i}P_{i}(\psi (X)\not =i)\geq 1-{\frac {\beta +\log 2}{\log r}}$

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 ƒ_θ such that for any θ ≠ θ′

$\|f_{\theta }-f_{\theta '}\|_{L_{1}}\geq \alpha ,\,$

$D_{KL}(f_{\theta }\|f_{\theta '})\leq \beta .\,$

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

$\sup _{f\in \mathbf {F} }E\|f_{n}-f\|_{L_{1}}\geq {\frac {\alpha }{2}}\left(1-{\frac {n\beta +\log 2}{\log r}}\right)$

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

References

• P. Assouad, "Deux remarques sur l'estimation," Comptes Rendus de L'Academie des Sciences de Paris, Vol. 296, pp. 1021–1024, 1983.
• L. Birge, "Estimating a density under order restrictions: nonasymptotic minimax risk," Technical report, UER de Sciences Economiques, Universite Paris X, Nanterre, France, 1983.
• T. Cover, J. Thomas, Elements of Information Theory. pp. 43.
• L. Devroye, A Course in Density Estimation. Progress in probability and statistics, Vol 14. Boston, Birkhauser, 1987. ISBN 0-8176-3365-0, ISBN 3-7643-3365-0.
• R. Fano, Transmission of information; a statistical theory of communications. Cambridge, Mass., M.I.T. Press, 1961. ISBN 0-262-06001-9
• R. Fano, Fano inequality Scholarpedia, 2008.
• I. A. Ibragimov, R. Z. Has′minskii, Statistical estimation, asymptotic theory. Applications of Mathematics, vol. 16, Springer-Verlag, New York, 1981. ISBN 0-387-90523-5