Noisy channel model
The noisy channel model is a framework used in spell checkers, question answering, speech recognition, and machine translation. In this model, the goal is to find the intended word given a word where the letters have been scrambled in some manner. Definition. Given an alphabet formula_1, let formula_2 be the set of all finite strings over formula_1. Let the dictionary formula_4 of valid words be some subset of formula_2, i.e., formula_6. The noisy channel is the matrix where formula_8 is the intended word and formula_9 is the scrambled word that was actually received. Example. Consider the English  alphabet formula_10. Some subset formula_6 makes up the dictionary of valid English words. There are several mistakes that may occur while typing, including: To construct the noisy channel matrix formula_12, we must consider the probability of each mistake, given the intended word (formula_13 for all formula_8 and formula_9). These probabilities may be gathered, for example, by considering the Damerau–Levenshtein distance between formula_16 and formula_17 or by comparing the draft of an essay with one that has been manually edited for spelling. Error-correction. The goal of the noisy channel model is to find the intended word given the scrambled  word that was received. The decision function formula_18 is a function that, given a scrambled word, returns the intended word. Methods of constructing a decision function include the maximum likelihood rule, the maximum a posteriori rule, and the minimum distance rule. In some cases, it may be better to accept the scrambled word as the intended word rather than attempt to find an intended word in the dictionary. For example, the word "schönfinkeling" may not be in the dictionary, but might in fact be the intended word.