Mutual information for a deletion channel

We study the binary deletion channel where each input bit is independently deleted according to a fixed probability. We relate the conditional probability distribution of the output of the deletion channel given the input to the hidden pattern matching problem. This yields a new characterization of the mutual information between the input and output of the deletion channel. Through this characterization we are able to comment on the the deletion channel capacity, in particular for deletion probabilities approaching 0 and 1.