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
represent input and output messages with a joint probability
represent an occurrence of error; i.e., that
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.
Define an indicator random variable
, that indicates the event that our estimate
We can use the chain rule for entropies to expand this in two different ways Equating the two Expanding the right most term,
different values, allowing us to upper bound the conditional entropy
, because conditioning reduces entropy.
is a Markov chain, we have
by the data processing inequality, and hence
, giving us 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.
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
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.
be a random variable with density equal to one of
Furthermore, the Kullback–Leibler divergence between any pair of densities cannot be too large, Let
is the probability induced by
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.