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From the output produced by a memoryless deletion channel from a uniformlyrandom input of known length $n$, one obtains a posterior distribution on thechannel input. The difference between the Shannon entropy of this distributionand that of the uniform prior measures the amount of information about thechannel input which is conveyed by the output of length $m$, and it is naturalto ask for which outputs this is extremized. This question was posed in aprevious work, where it was conjectured on the basis of experimental data thatthe entropy of the posterior is minimized and maximized by the constant strings$\texttt{000}\ldots$ and $\texttt{111}\ldots$ and the alternating strings$\texttt{0101}\ldots$ and $\texttt{1010}\ldots$ respectively. In the presentwork we confirm the minimization conjecture in the asymptotic limit usingresults from hidden word statistics. We show how the analytic-combinatorialmethods of Flajolet, Szpankowski and Vall\'ee for dealing with the hiddenpattern matching problem can be applied to resolve the case of fixed outputlength and $n\rightarrow\infty$, by obtaining estimates for the entropy interms of the moments of the posterior distribution and establishing itsminimization via a measure of autocorrelation.

A Proof of Entropy Minimization for Outputs in Deletion Channels via Hidden Word Statistics