Backward Map for Filter Stability Analysis

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Abstract

Abstract: A backward map is introduced for the purposes of analysis of nonlinear (stochastic) filter stability. The backward map is important because filter-stability, in the sense of $\chi^2$ divergence, follows from a certain variance decay property associated with the backward map. To show this property requires additional assumptions on the hidden Markov model (HMM). The analysis in this paper is based on introducing a Poincaré Inequality (PI) for HMMs with white noise observations. In finite state-space settings, PI is related to both the ergodicity of the Markov process as well as the observability of the HMM. It is shown that the Poincaré constant is positive if and only if the HMM is detectable.

Citation

@INPROCEEDINGS{10886772,
  author={Kim, Jin Won and Joshi, Anant A. and Mehta, Prashant G.},
  booktitle={2024 IEEE 63rd Conference on Decision and Control (CDC)}, 
  title={Backward Map for Filter Stability Analysis}, 
  year={2024},
  volume={},
  number={},
  pages={4070-4077},
  keywords={Hidden Markov models;White noise;Markov processes;Stability analysis;Observability},
  abstract = {Abstract: A backward map is introduced for the purposes of analysis of nonlinear (stochastic) filter stability. The backward map is important because filter-stability, in the sense of $\chi^2$ divergence, follows from a certain variance decay property associated with the backward map. To show this property requires additional assumptions on the hidden Markov model (HMM). The analysis in this paper is based on introducing a Poincaré Inequality (PI) for HMMs with white noise observations. In finite state-space settings, PI is related to both the ergodicity of the Markov process as well as the observability of the HMM. It is shown that the Poincaré constant is positive if and only if the HMM is detectable.},
  doi={10.1109/CDC56724.2024.10886772}
}