. We propose and analyze a distribution learning algorithm for variable memory length Markov processes. These processes can be described by a subclass of probabilistic finite automata which we name Probabilistic Suffix Automata (PSA). Though hardness results are known for learning distributions generated by general probabilistic automata, we prove that the algorithm we present can efficiently learn distributions generated by PSAs. In particular, we show that for any target PSA, the KL-divergence between the distribution generated by the target and the distribution generated by the hypothesis the learning algorithm outputs, can be made small with high confidence in polynomial time and sample complexity. The learning algorithm is motivated by applications in human-machine interaction. Here we present two applications of the algorithm. In the first one we apply the algorithm in order to construct a model of the English language, and use this model to correct corrupted text. In the second ...

We study efficient algorithms for solving the following problem, which we call the switching distributions learning problem. A sequence S = oe 1 oe 2 : : : oe n , over a finite alphabet \Sigma is generated in the following way. The sequence is a concatenation of K runs, each of which is a consecutive subsequence. Each run is generated by independent random draws from a distribution ~ p i over \Sigma, where ~p i is an element in a set of distributions f~p 1 ; : : : ; ~p N g. The learning algorithm is given this sequence and its goal is to find approximations of the distributions ~p 1 ; : : : ; ~p N , and give an approximate segmentation of the sequence into its constituting runs. We give an efficient algorithm for solving this problem and show conditions under which the algorithm is guaranteed to work with high probability. 1 Introduction Our work is motivated by the Hidden Markov Model (HMM). The HMM is a model for the distribution of sequences over a finite alphabet \Sigma. An HMM ...

1 1 Introduction 3 1.1 The PAC Model and Some of its Extensions : : : : : : : : : : : : : : : : : : : : : : : 5 1.2 Background on Automata Learning : : : : : : : : : : : : : : : : : : : : : : : : : : : : 8 1.2.1 Learning Deterministic Automata : : : : : : : : : : : : : : : : : : : : : : : : 8 1.2.2 Learning Probabilistic Automata : : : : : : : : : : : : : : : : : : : : : : : : : 11 1.3 Overview of Results Presented in this Thesis : : : : : : : : : : : : : : : : : : : : : : 12 1.3.1 Results on DFA Learning : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 13 1.3.2 Results on PFA Learning : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 14 1.4 Suggestions for Further Research : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 16 2 Preliminaries 19 2.1 Strings and Sets of Strings : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 19 2.2 Deterministic Finite Automata : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 20 2.3 Probabilistic F...