Fano inequality

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 ${\displaystyle P(x,y)}$. Let e represent an occurrence of error; i.e., that ${\displaystyle X\neq {\tilde {X}}}$, with ${\displaystyle {\tilde {X}}=f(Y)}$ being an approximate version of ${\displaystyle X}$. Fano's inequality is

...
Wikipedia