This probability is defined as the possibly infinite sum of the probabilities of all strings of the form xwy, for any pair of strings x and y over the alphabet of G. Besides applications in computation of the probability distribution for the next word token and in word error correction, infix probabilities can also be exploited in speech understanding systems to score partial hypotheses in algorithms based on beam search, as discussed by Corazza et al.
the algorithm in (Fred, 2000) does not apply to cases with multiple occurrences of the given infix within a string in the language, which is what was pointed out to be the problematic case.