Suppose we want to smooth the likelihood term of a noisy channel model of spelling. We are given two words, x and w, where x is the same as w, except...

Suppose we want to smooth the likelihood term of a noisy channel model of spelling. We are given two words, x and w, where x is the same as w, except the letter wi−1 in w has been miss typed as wi−1xi in x. Specifically, we want to apply add-one smoothing to P(x|w), the probability of typing wi−1xi instead of wi−1, where xi and wi−1 are single letters. For insertions, P(x|w)=ins[wi−1,xi]c(wi−1), where ins[wi−1,xi] is the number of times that xi is inserted after wi−1 in the corpus, and c(wi−1) is the number of times letter wi−1 appears in our corpus. Again, please note that here xi and wi−1 are individual letters, not words. What is the formula for P(x|w) if we use add-one smoothing to the insertion edit model? Assume the only characters we use are lowercase a-z, that there are V word types in our corpus, and n total characters, not counting spaces.