Probabilistic finite automata (PFAs) are com- mon statistical language model in natural lan- guage and speech processing. A typical task for PFAs is to compute the probability of all strings that match a query pattern. An impor- tant special case of this problem is computing the probability of a string appearing as a pre- fix, suffix, or infix. These problems find use in many natural language processing tasks such word prediction and text error correction. Recently, we gave the first incremental algorithm to efficiently compute the infix probabilities of each prefix of a string (Cognetta et al., 2018). We develop an asymptotic improvement of that algorithm and solve the open problem of computing the infix probabilities of PFAs from streaming data, which is crucial when process- ing queries online and is the ultimate goal of the incremental approach.

In natural language processing, a common task is to compute the probability of a phrase appearing in a document or to calculate the probability of all phrases matching a given pattern. For instance, one computes affix (prefix, suffix, infix, etc.) probabilities of a string or a set of strings with respect to a probability distribution of patterns. The problem of computing infix probabilities of strings when the pattern distribution is given by a probabilistic context-free grammar or by a probabilistic finite automaton is already solved, yet it was open to compute the infix probabilities in an incremental manner. The incremental computation is crucial when a new query is built from a previous query. We tackle this problem and suggest a method that computes infix probabilities incrementally for probabilistic finite automata by representing all the probabilities of matching strings as a series of transition matrix calculations. We show that the proposed approach is theoretically faster than the previous method and, using real world data, demonstrate that our approach has vastly better performance in practice.