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Inequality applying to random variables
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 discrete random variables and represent input and output messages with a joint probability . Let represent an occurrence of error; i.e., that , with being an approximate version of . Fano's inequality is
where denotes the support of , denotes the cardinality of (number of elements in) ,
is the conditional entropy,
is the probability of the communication error, and
is the corresponding binary entropy.
Proof
Define an indicator random variable , that indicates the event that our estimate is in error,
Consider . We can use the chain rule for entropies to expand this in two different ways
Equating the two
Expanding the right most term,
Since means ; being given the value of allows us to know the value of with certainty. This makes the term .
On the other hand, means that , hence given the value of , we can narrow down to one of different values, allowing us to upper bound the conditional entropy . Hence
The other term, , because conditioning reduces entropy. Because of the way is defined, , meaning that . Putting it all together,
Because is a Markov chain, we have by the data processing inequality, and hence , giving us
Intuition
Fano's inequality can be interpreted as a way of dividing the uncertainty of a conditional distribution into two questions given an arbitrary predictor. The first question, corresponding to the term , relates to the uncertainty of the predictor. If the prediction is correct, there is no more uncertainty remaining. If the prediction is incorrect, the uncertainty of any discrete distribution has an upper bound of the entropy of the uniform distribution over all choices besides the incorrect prediction. This has entropy . Looking at extreme cases, if the predictor is always correct the first and second terms of the inequality are 0, and the existence of a perfect predictor implies is totally determined by , and so . If the predictor is always wrong, then the first term is 0, and can only be upper bounded with a uniform distribution over the remaining choices.
Alternative formulation
Let 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
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
- P. Assouad, "Deux remarques sur l'estimation", Comptes Rendus de l'Académie 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 Économiques, Universite Paris X, Nanterre, France, 1983.
- T. Cover, J. Thomas (1991). Elements of Information Theory. pp. 38–42. ISBN 978-0-471-06259-2.
- 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.
- Fano, Robert (1968). Transmission of information: a statistical theory of communications. Cambridge, Mass: MIT Press. ISBN 978-0-262-56169-3. OCLC 804123877.
- 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
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