A recursive learning algorithm for model reduction of Hidden Markov Models

This paper is concerned with a recursive learning algorithm for model reduction of Hidden Markov Models (HMMs) with finite state space and finite observation space. The state space is aggregated/partitioned to reduce the complexity of the HMM. The optimal aggregation is obtained by minimizing the Kullback-Leibler divergence rate between the laws of the observation process. The optimal aggregated HMM is given as a function of the partition function of the state space. The optimal partition is obtained by using a recursive stochastic approximation learning algorithm, which can be implemented through a single sample path of the HMM. Convergence of the algorithm is established using ergodicity of the filtering process and standard stochastic approximation arguments.