Fano's inequality says the following:

Theorem: Let $X$ be a random variable with range $M$. Let $\hat{X} = g(Y)$ be the predicted value of $X$ given some transmitted value $Y$, where $g$ is a deterministic function. Let $p_e$ be the probability of making a mistake, i.e., $p_e = Pr[ \hat{X} \ne X ]$. Then $p_e \ge \frac{H(X\mid Y) - 1}{\log|M|}$.

It seems to me that Fano's inequality is useful when $X$ is the output of an algorithm computing some function, i.e., for any fixed input there is a unique correct output.

I'm interested in the following generalization: Suppose we are trying to reconstruct a valid output for a problem, where, for a given input $I$, there are multiple valid solutions. In other words, for each input $I$, there is a fixed subset $S_I$ of values that are correct, i.e., if $\hat{X} \in S_I$ we succeed (instead of just $\hat{X} = X$). For instance, $\hat{X}$ could be a spanning tree of a given input graph $I$, and $S_I$ would be the set of all spanning trees.

How would I bound the error probability of recovering any element in $S_I$?



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