Estimate exponential memory decay in hidden Markov model and its applications to inference

• Published in: Physica. D Vol. 460; p. 134053
• Main Authors: Ye, Felix X.-F., Ma, Yi-an, Qian, Hong
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.04.2024
• Subjects: Hidden Markov model; Lyapunov exponents; Machine learning; Random dynamical systems; Statistical inference; Hidden Markov model; Statistical inference; Lyapunov exponents; Random dynamical systems; Machine learning
• ISSN: 0167-2789; 1872-8022
• DOI: 10.1016/j.physd.2024.134053

Inference in hidden Markov model has been challenging in terms of scalability due to dependencies in the observation data. In this paper, we utilize the inherent memory decay in hidden Markov models, such that the forward and backward probabilities can be carried out with subsequences, enabling efficient inference over long sequences of observations. We formulate this forward filtering process in the setting of the random dynamical system and there exist Lyapunov exponents in the i.i.d random matrices production. And the rate of the memory decay is known as λ2−λ1, the gap of the top two Lyapunov exponents almost surely. An efficient and accurate algorithm is proposed to numerically estimate the gap after the soft-max parametrization. The length of subsequences B given the controlled error ϵ is B≈log(ϵ)/(λ2−λ1). We theoretically prove the validity of the algorithm and demonstrate the effectiveness with numerical examples. The method developed here can be applied to widely used algorithms, such as mini-batch stochastic gradient method. Moreover, the continuity of Lyapunov spectrum ensures the estimated B could be reused for the nearby parameter during the inference. •We propose an algorithm to estimate the rate of memory decay in hidden Markov models.•We theoretically prove the validity of the algorithm and demonstrate the effectiveness with numerical examples.•We utilize the inherent memory decay to propose a mini-batch stochastic gradient method for inference over long sequences of observations.