Let $X$ be a random variable, and let $Y = g(X)$ where $g(\cdot)$ is a random process. Assume the existence of a procedure $f$ that given $y = g(x)$ can reconstruct $x$ with probability $p$. Then $$ I(X; Y) \ge pH(X) − H(p)$$
In other words, if I can reconstruct, there's a lot of mutual information in the system.
Is there a "converse" to Fano's inequality: something of the form
"Given a channel with sufficient mutual information, there is a procedure to reconstruct input from output with error that depends on the mutual information"
It would be too much to expect that this procedure would also be efficient, but it would also be interesting to see (natural) examples where the reconstruction exists but must be inefficient.