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Duality between estimation and control is a foundational concept in Control Theory. Most students learn about the elementary duality -- between observability and controllability -- in their first graduate course in linear systems theory. Therefore, it comes as a surprise that for a more general class of nonlinear stochastic systems (hidden Markov models or HMMs), duality is incomplete. Our objective in writing this article is two-fold: (i) To describe the difficulty in extending duality to HMMs; and (ii) To discuss its recent resolution by the authors. A key message is that the main difficulty in extending duality comes from time reversal in going from estimation to control. The reason for time reversal is explained with the aid of the familiar linear deterministic and linear Gaussian models. The explanation is used to motivate the difference between the linear and the nonlinear models. Once the difference is understood, duality for HMMs is described based on our recent work. The article also includes a comparison and discussion of the different types of duality considered in literature.

In this paper, a backward map is introduced for the purposes of analysis of the nonlinear (stochastic) filter stability. The backward map is important because the filter-stability in the sense of $\chisq$-divergence follows from showing a certain variance decay property for the backward map. To show this property requires additional assumptions on the model properties of the hidden Markov model (HMM). The analysis in this paper is based on introducing a Poincar\'e 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 Poincar\'e constant is positive if and only if the HMM is detectable., Comment: Conference proceeding related to arXiv:2305.12850

This paper is divided into two parts. The first part reviews the formulae for f-divergences in the study of continuous-time Markov processes and explores their applications in areas such as stochastic stability, the second law of thermodynamics, and its non-equilibrium extensions. This sets the foundation for the second part, which focuses on f-divergence in the study of hidden Markov processes. In this context, we present analyses of filter stability and stochastic thermodynamics, with the latter being used to illustrate the concept of a Maxwell demon in an over-damped Langevin model with white noise observations. The paper's expository style and unified formalism for both Markov and hidden Markov processes aim to serve as a valuable resource for researchers working across related fields.

This paper is concerned with the problem of nonlinear (stochastic) filter stability of a hidden Markov model (HMM) with white noise observations. A contribution is the variance decay property which is used to conclude filter stability. For this purpose, a new notion of the Poincar\'e inequality (PI) is introduced for the nonlinear filter. PI is related to both the ergodicity of the Markov process as well as the observability of the HMM. The proofs are based upon a recently discovered minimum variance duality which is used to transform the nonlinear filtering problem into a stochastic optimal control problem for a backward stochastic differential equation (BSDE)., Comment: 16 pages

The work of Kalman and Bucy has established a duality between filtering and optimal estimation in the context of time-continuous linear systems. This duality has recently been extended to time-continuous nonlinear systems in terms of an optimization problem constrained by a backward stochastic partial differential equation. Here we revisit this problem from the perspective of appropriate forward-backward stochastic differential equations. This approach sheds new light on the estimation problem and provides a unifying perspective. It is also demonstrated that certain formulations of the estimation problem lead to deterministic formulations similar to the linear Gaussian case as originally investigated by Kalman and Bucy. Finally, optimal control of partially observed diffusion processes is discussed as an application of the proposed estimators.

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)., Comment: arXiv admin note: text overlap with arXiv:2207.07709

This paper is concerned with the development and use of duality theory for a nonlinear filtering model with white noise observations. The main contribution of this paper is to introduce a stochastic optimal control problem as a dual to the nonlinear filtering problem. The mathematical statement of the dual relationship between the two problems is given in the form of a duality principle. The constraint for the optimal control problem is the backward stochastic differential equation (BSDE) introduced in the companion paper. The optimal control solution is obtained from an application of the maximum principle, and subsequently used to derive the equation of the nonlinear filter. The proposed duality is shown to be an exact extension of the classical Kalman-Bucy duality, and different from other types of optimal control and variational formulations given in literature.

This thesis is concerned with the stochastic filtering problem for a hidden Markov model (HMM) with the white noise observation model. For this filtering problem, we make three types of original contributions: (1) dual controllability characterization of stochastic observability, (2) dual minimum variance optimal control formulation of the stochastic filtering problem, and (3) filter stability analysis using the dual optimal control formulation. For the first contribution of this thesis, a backward stochastic differential equation (BSDE) is proposed as the dual control system. The observability (detectability) of the HMM is shown to be equivalent to the controllability (stabilizability) of the dual control system. For the linear-Gaussian model, the dual relationship reduces to classical duality in linear systems theory. The second contribution is to transform the minimum variance estimation problem into an optimal control problem. The constraint is given by the dual control system. The optimal solution is obtained via two approaches: (1) by an application of maximum principle and (2) by the martingale characterization of the optimal value. The optimal solution is used to derive the nonlinear filter. The third contribution is to carry out filter stability analysis by studying the dual optimal control problem. Two approaches are presented through Chapters 7 and 8. In Chapter 7, conditional Poincar\'e inequality (PI) is introduced. Based on conditional PI, various convergence rates are obtained and related to literature. In Chapter 8, the stabilizability of the dual control system is shown to be a necessary and sufficient condition for filter stability on certain finite state space model., Comment: Ph.D. Thesis of the author

Evolution of disease in a large population is a function of the top-down policy measures from a centralized planner, as well as the self-interested decisions (to be socially active) of individual agents in a large heterogeneous population. This paper is concerned with understanding the latter based on a mean-field type optimal control model. Specifically, the model is used to investigate the role of partial information on an agent's decision-making, and study the impact of such decisions by a large number of agents on the spread of the virus in the population. The motivation comes from the presymptomatic and asymptomatic spread of the COVID-19 virus where an agent unwittingly spreads the virus. We show that even in a setting with fully rational agents, limited information on the viral state can result in an epidemic growth., Comment: arXiv admin note: text overlap with arXiv:2111.10422

This paper is concerned with developing mean-field game models for the evolution of epidemics. Specifically, an agent's decision -- to be socially active in the midst of an epidemic -- is modeled as a mean-field game with health-related costs and activity-related rewards. By considering the fully and partially observed versions of this problem, the role of information in guiding an agent's rational decision is highlighted. The main contributions of the paper are to derive the equations for the mean-field game in both fully and partially observed settings of the problem, to present a complete analysis of the fully observed case, and to present some analytical results for the partially observed case.

This paper build on our recent work where we presented a dual stochastic optimal control formulation of the nonlinear filtering problem [1]. The constraint for the dual problem is a backward stochastic differential equations (BSDE). The solution is obtained via an application of the maximum principle (MP). In the present paper, a dynamic programming (DP) principle is presented for a special class of BSDE-constrained stochastic optimal control problems. The principle is applied to derive the solution of the nonlinear filtering problem.

This paper is concerned with the problem of nonlinear filter stability of ergodic Markov processes. The main contribution is the conditional Poincar\'e inequality (PI), which is shown to yield filter stability. The proof is based upon a recently discovered duality which is used to transform the nonlinear filtering problem into a stochastic optimal control problem for a backward stochastic differential equation (BSDE). Based on these dual formalisms, a comparison is drawn between the stochastic stability of a Markov process and the filter stability. The latter relies on the conditional PI described in this paper, whereas the former relies on the standard form of PI., Comment: 60th IEEE Conference on Decision and Control (CDC)

This paper revisits the classical question of the stability of the nonlinear Wonham filter. The novel contributions of this paper are two-fold: (i) definition of the stabilizability for the (control-theoretic) dual to the nonlinear filter; and (ii) the use of this definition to obtain conclusions on the stability of the Wonham filter. Specifically, it is shown that the stabilizability of the dual system is necessary for filter stability and conversely stabilizability implies that the filter asymptotically detects the correct ergodic class. The formulation and the proofs are based upon a recently discovered duality result whereby the nonlinear filtering problem is cast as a stochastic optimal control problem for a backward stochastic differential equation (BSDE). The control-theoretic proof techniques and results may be viewed as a generalization of the classical work on the stability of the Kalman filter., Comment: 60th IEEE Conference on Decision and Control (CDC)

The purpose of this paper is to describe the feedback particle filter algorithm for problems where there are a large number ($M$) of non-interacting agents (targets) with a large number ($M$) of non-agent specific observations (measurements) that originate from these agents. In its basic form, the problem is characterized by data association uncertainty whereby the association between the observations and agents must be deduced in addition to the agent state. In this paper, the large-$M$ limit is interpreted as a problem of collective inference. This viewpoint is used to derive the equation for the empirical distribution of the hidden agent states. A feedback particle filter (FPF) algorithm for this problem is presented and illustrated via numerical simulations. Results are presented for the Euclidean and the finite state-space cases, both in continuous-time settings. The classical FPF algorithm is shown to be the special case (with $M=1$) of these more general results. The simulations help show that the algorithm well approximates the empirical distribution of the hidden states for large $M$., Comment: 15 pages, 2 figures. Submitted to FoDA