Approximation of Nonlinear Filters for Markov Systems with Delayed Observations

Abstract

The aim of this paper is to give some approximation results for a class of nonlinear filtering problems with delay in the observation. First, we point out some general results on the approximation problem for the filter in nonlinear filtering. In particular we give a general procedure to obtain some upper bounds for the different approximations we consider. This procedure is then applied in the case of nonlinear filtering problems with delay (X, Y ), which can be represented by means of a Markov system (X,Ŷ ), in the sense that Yt =Ŷ a(t) . Finally these upper bounds are computed explicitly in the particular case of Markov jump process with counting observations.

Stochastic filtering for multiscale stochastic reaction networks based on hybrid approximations

“…To prove the result, we borrow the proof framework proposed in [21], which deals with the convergence of filters via constructing auxiliary probability spaces that have nice properties. Specifically, the framework establishes a common probability space, on which some random variables are constructed to mimic the system state and observations on the natural probability space.…”

“…It is worth noting that the idea of applying time-scale separation techniques to developing computationally manageable particle filters has already been proposed in [35][36][37] for diffusion type stochastic models, and its efficiency is proven in the literature [38]. Compared with these works, our paper considers a different type of underlying models, the Markov jump process, and provides a much-simplified proof for the convergence of the approximate filters (based on the framework by [21]). All these references and our paper show the efficiency of applying the time-scale separation technique to the filtering problem for multi-scale systems.…”

“…Especially for multi-scale reaction systems where reactions fire at different timescales, the computational complexity of exact simulation methods (e.g., Gillespie stochastic simulation algorithm [18,19] and the next reaction method [20]) is proportional to the rate of the fastest reaction, and, hence, the algorithm takes a very long time to output an estimate. A typical way to mitigate this type of problems is to replace the underlying system by a more tractable approximation so that a computationally efficient filter can be constructed based on the reduced model while the filter's accuracy is still preserved (see [21,Section 2]). Following this idea, a method was proposed to obtain computationally efficient filters for biological systems by approximating the underlying Markov jump process by a diffusion process [22][23][24][25].…”

Nonlinear Filtering for Markov Systems with Delayed Observations

“…In general, the feasible filter cannot be computed explicitly and an approximation may be necessary. This approximation problem was studied in (Calzolari et al, 2006) for the jump case, i.e., when (X,Ŷ ) is a Markov process with generator L of the form…”

“…On the contrary, we focus on the explicit expression of the filter for the system X, Y with delayed observations, in terms of the feasible version of the filter of the partially observed Markov system X,Ŷ and of its associated semigroup. To obtain an explicit representation of the filter is interesting on its own and, moreover, it plays a key role in the connected filtering approximation problem, see (Calzolari et al, 2006).…”