Generalized Fano-Type Inequality for Countably Infinite Systems with List-Decoding

This study investigates generalized Fano-type inequalities in the following senses: (i) the alphabet X of a random variable X is countably infinite; (ii) instead of a fixed finite cardinality of X, a fixed X-marginal distribution PX is given; (iii) information measures are generalized from the conditional Shannon entropy H(X | Y ) to a general type of conditional information measures without explicit form; and (iv) the average probability of decoding error is defined on list-decoding settings. As a result, we give tight upper bounds on such generalized conditional information measures for a fixed X-marginal, a fixed list size, and a fixed tolerated probability of error. Then, we also clarify a sufficient condition, which the Fano-type inequality is sharp, on the cardinality of the alphabet Y of a side information Y . Resulting Fano-type inequalities can apply to not only the conditional Shannon entropy but also the Arimoto’s and Hayashi’s conditional Rényi entropies, α-mutual information, and so on. Index Terms Fano’s inequality; list-decoding settings; conditional Rényi entropy; α-mutual information; majorization theory