# Duality for Nonlinear Filtering I: Observability

arXiv: https://arxiv.org/abs/2208.06586 · HTML: https://mlsanity.com/paper/2208.06586/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 Aug 2022.

## Built on by (most cited first, 1 of 1)

- [Dual Filter: A Transformer-like Inference Architecture for Hidden Markov Models](https://mlsanity.com/paper/2505.00818/) (2025)

## Abstract (arXiv)

This paper is concerned with the development and use of duality theory for a hidden Markov model (HMM) with white noise observations. The main contribution of this work is to introduce a backward stochastic differential equation (BSDE) as a dual control system. A key outcome is that stochastic observability (resp. detectability) of the HMM is expressed in dual terms: as controllability (resp. stabilizability) of the dual control system. All aspects of controllability, namely, definition of controllable space and controllability gramian, along with their properties and explicit formulae, are discussed. The proposed duality is shown to be an exact extension of the classical duality in linear systems theory. One can then relate and compare the linear and the nonlinear systems. A side-by-side summary of this relationship is given in a tabular form (Table~II).