Alpha-Beta HMM: Interpretable Low-Parameter Hidden Markov Filtering in Dynamic Environments

Practical online inference in dynamic environments requires a lightweight filtering mechanism that remains adaptive to state changes while retaining reliable information from past noisy observations. To address this challenge, we propose the $\alpha\beta$-HMM, an interpretable low-parameter hidden Markov filtering framework that replaces the full transition matrix with an equal-exit surrogate governed by an exit-probability parameter $\alpha$, and introduces a step-size parameter $\beta$ through a generalized measurement update to regulate the influence of observational evidence. A central feature of the proposed method is that it preserves the nonlinear log-belief-ratio dynamics of HMM-type filtering, which turn out to be critical for strong performance. To analyze this nonlinear recursion, we develop a dynamical-systems framework and a deterministic reference system, through which we characterize adaptation capability, learning performance, and practical guidance for selecting the two proposed parameters. In parallel, we study the approximation error induced by the equal-exit surrogate and show that the resulting low-parameter filter remains competitive with the oracle HMM across a broad range of environments. These results reveal an explicit learning-adaptation trade-off induced by the two proposed parameters, provide principled guidance for parameter tuning, and show that strong filtering performance can be achieved within a tractable and interpretable low-parameter framework.