# 45th John H. Barrett Memorial Lectures

## Stochastic Filtering, Computations and Their Applications

### Main Lectures:

Dan Crisan, Imperial College
Tom Kurtz

## Main Lectures

Dan Crisan, Imperial College London
Robust Filtering, Part I
The first talk is concerned with the description and analysis of the robust representation formula for the conditional expectation operator of nonlinear filtering. The formula, robust in the sense that its dependence on the process of observations is continuous, was introduced by Clark in the late seventies. The formula is “almost obvious” as it can be derived at a formal level by a process of integration-by-parts applied to the stochastic integrals that appear in the integral representation formula. However, the rigorous justification of the formula is quite subtle, as it hinges on a measurability argument the necessity of which is easy to miss at first glance.

Robust Filtering, Part II
In the second talk, I discuss the case when the signal and the observation noise are correlated, and the observation process is multidimensional. In this case such a representation does not exist. By using the theory of rough paths, in recent work we provide a solution to this deficiency: the observation process is “lifted” to a process that consists of of the original process and its corresponding Levy area process. Then the solution of the filtering problem can be represented as a continuous map ?defined on a suitably chosen space of Holder continuous paths on the lifted space. If time permits, I will also discuss some recent approximations developments to the solution of the filtering problem.

The talks are based on the following papers:

1. On a robust version of the integral representation formula of nonlinear filtering, JMC Clark, D Crisan, Probability theory and related fields 133 (1), 43-56
2. Robust filtering: correlated noise and multidimensional observation, D Crisan, J Diehl, PK Friz, H Oberhauser, The Annals of Applied Probability 23 (5), 2139-2160
3. A second order time discretization of the solution of the non-linear filtering problem, D Crisan, S Ortiz-Latorre, arXiv preprint arXiv:1408.5678

Tom Kurtz, University of Wisconsin
Martingale problems for conditional distributions
Let X be a Markov process characterized as the solution of a martingale problem with generator A, and let Y(t) be given by a function of X(t). The conditional distribution of X(t) given observations of Y up to time t is characterized as the solution of a filtered martingale problem. Uniqueness for the original martingale problem implies uniqueness for the filtered martingale problem which in turn implies the Markov property for the conditional distribution considered as a probability-measure-valued process.

Derivation and uniqueness for filtering equations
The conditional distribution of a partially observed Markov process can be characterized as a solution of a filtered martingale problem. In a variety of settings, this characterization in turn implies that the conditional distribution is given as the unique solution of a filtering equation. Previous results will be reviewed, and new uniqueness results based on local martingale problems and a local forward equation will be presented.

## Plenary Speakers

Christophe Andrieu, University of Bristol
Some applications of asymmetric Metropolis-Hastings updates in the context of exact approximate algorithms
There has recently been some interest in approximating MH updates in situations where quantities required for their implementation are not tractable. This is often the case for the acceptance ratio involved in the MH accept/reject mechanism, for examples in situations where the target density is not tractable. In this scenario one can for example resort to pseudo-marginal strategies which exploit the possibility to unbiasedly estimate a function proportional to the target density. Other more general strategies are however available, cover more scenarios, and the aim of the talk is to review such scenarios and show how “asymmetric” MH updates can be useful, and in particular allow for the reduction of the variability of these approximations and improve performance at virtually no cost on a parallel computer. This is joint work with A. Doucet and S. Yildirim.

Iterated filtering algorithms
This talk deals with iterated filtering-type algorithms for computing maximum likelihood estimates in partially observed Markov models. The talk will present the basic iterated filtering algorithm, as well as several extensions, including a recently proposed extension in terms of iterated Bayes maps. We will show that these algorithms are intimately connected with the proximal algorithm, well-known in optimization. This connection is exploited to establish new, and stronger convergence results for iterated filtering algorithms. Based on a joint work with Ed Ionides, and Dao Nguyen.

Evangelos Evangelou, University of Bath
Online filtering and estimation for dynamic spatiotemporal processes
Motivated by the analysis of spatial data arriving in real-time, I will present an online algorithm for the prediction of the underlying spatiotemporal process and estimation of the associated model parameters within a Bayesian framework. It is well-known that standard sequential Monte-Carlo (SMC) methods suffer from sample degeneracy in high dimensions. Furthermore, online estimation of the model parameters is possible only in limited situations e.g. using sufficient statistics. The methodology presented here incorporates importance resampling and empirical Bayes estimation to address these issues. Specifically, the parameter space is first discretised and a mixture sampling distribution for the spatial random field is defined conditioned on the parameters taking values on the discrete set. Monte-Carlo (MC) samples for the spatial random field are drawn for this mixture distribution which allows estimation of the parameters. The MC sample is then reweighted given the parameter estimates and the sample is used to predict the spatial random field. I will discuss theoretical properties and present simulation results which demonstrate the accuracy of the proposed method. Finally, I will close with an analysis of the concentration of radioactive material caused after the accident at the Fukushima power plant station in 2011. Joint work with Vasileios Maroulas (Tennessee).

Xiaoying Han, Auburn University
An implicit scheme for nonlinear filtering problems
In this talk we introduce a novel numerical algorithm that is constructed based on samples of the current state obtained by solving the state equation implicitly. Numerical experiments demonstrate that this algorithm is more accurate than the Kalman filter and more stable than the particle filter.

Lane Hughston, Brunel University London
Signal processing with Lévy information, with applications
Lévy processes are ideal for modelling the various types of noise that can arise in communication channels. If a Lévy process admits exponential moments, then there exists a canonical parametric family of measure changes called Esscher transformations. If the parameter is replaced with an independent random variable, the true value of which represents a “message”, then under the transformed measure the original Lévy process takes on the character of an “information process”. In this work we develop a theory of such Lévy information processes. The underlying Lévy process, which we call the fiducial process, represents the “noise type”. Each noise type is capable of carrying a message of a certain specification. A number of examples are worked out in detail, including information processes of the Brownian, Poisson, gamma, variance gamma, and negative binomial type. Although in general there is no additive decomposition of information into signal and noise, one is led nevertheless for each noise type to a well-defined scheme for signal detection and enhancement relevant to a variety of practical situations. Applications can be found in diverse areas such as (a) information-based asset pricing and (b) the theory of measurement in quantum mechanics. The talk will be based in part on the following paper: D.C. Brody, L.P. Hughston & X. Yang (2013) Signal Processing with Lévy Information, Proceedings of the Royal Society A 469, 20120433

Kayo Ide, University of Maryland
Multi-scale data assimilation for oceans and atmosphere
Geophysical fluid systems exhibit a wide range of spatial and temporal scales. Observing network of geophysical fluid systems, consisting in-situ and remote sensing, are sampled at multi-resolution. This talk addresses the filtering of multi-scale/resolution issues for data assimilation. Using an idealized set-up, we demonstrate the effectiveness of the multi-scale scheme. We then apply the scheme to the California coast ocean, and also discuss the multi-scale issues in the context of the global atmosphere numerical weather prediction.

Nikolas Kantas, Imperial College London
Sequential Monte Carlo Methods for High-Dimensional Inverse Problems
We consider the inverse problem of estimating the initial condition of a partial differential equation, which is only observed through noisy measurements at discrete time intervals. In particular, we focus on the case where Eulerian measurements are obtained from the time and space evolving vector field, whose evolution obeys the two-dimensional Navier-Stokes equations defined on a torus. We will adopt a Bayesian formulation resulting from a particular regularisation that ensures the problem is well posed. In the context of Monte Carlo based inference, it is a challenging task to obtain samples from the resulting high dimensional posterior on the initial condition. Often, in data assimilation applications it is common for computational methods to invoke the use of heuristics and Gaussian approximations. In the presence of non-linear dynamics and observations, the resulting inferences are biased and not well-justified from a theoretical perspective. On the other hand, Monte Carlo methods can be used to assimilate data in a principled manner, but are often perceived as inefficient in this context due to the high-dimensionality of the problem. In this work we will propose a generic adaptive Sequential Monte Carlo (SMC) sampling approach for high dimensional inverse problems that overcomes some of these difficulties. The method builds upon appropriate Markov chain Monte Carlo (MCMC) techniques, which are currently considered as benchmarks for evaluating data assimilation algorithms used in practice. In our numerical examples, the proposed SMC approach achieves the same accuracy as MCMC but in a much more efficient manner. If time permits we will discuss some extensions of these ideas for high dimensional non-linear filtering problems. The talk is based on joint work with Alexandros Beskos (UCL), Ajay Jasra (NUS), Alexandre Thiery (NUS) and Dan Crisan (Imperial).

Michael A. Kouritzin, University of Alberta
Exchangeable branching processes for filtering and model selection
We will start by building a case for branching particle filters. Then, we will discuss some of the convergence properties for a class of such filters with a flexible resampling scheme as the number of particles increases. The key tools are exchangeability and coupling to a McKean-Vlasov particle system that can also be used to predict performance of the branching filter.

Sergey Lototsky, University of Southern California
Optimal linear filtering of stochastic partial differential equations
Consider a stochastic evolution equation, hyperbolic or parabolic. Assume that the equation is diagonalizable, that is, can be solved by the Fourier series method, and that the coefficients in the equation are unobservable Gaussian processes. If the solution of the equation is observable, then the first N Fourier coefficients of the solution become the observation process in a conditionally Gaussian filtering model. The filter estimate of the coefficients is then constructed using a generalized Kalman-Bucy filter, and the variance of the filter is shown to converge to zero as N grows to infinity.

Patrick Rebeschini, Yale University
Filtering compressed signal dynamics in high dimension.
Compressed sensing deals with recovering sparse high-dimensional signals from compressed measurements. In its Bayesian formulation, compressed sensing postulates that the signal comes from a prior distribution with a given sparsity structure, and that the observations are noisy functions of linear measurements of the signal in a lower dimensional space. In this setting, one is interested in computing some statistics of the posterior distribution, typically its mode. Nowadays, many systems that are of interest in compressed sensing are dynamical systems, and the problem of online estimation motivates the development of approximations of the whole posterior distribution that can be reliably propagated over time. Many authors have empirically investigated the application of the sequential Monte Carlo paradigm to compressed sensing, but to date a principled and rigorous approach is still missing, essentially due to the curse of dimensionality that affects classical particle filters in high dimensions. Building on the framework of local particle filters recently proposed by Rebeschini and Van Handel, we show that the structure of the measurement matrix through which the signal is compressed can be exploited to develop particle filters that can avoid the curse of dimensionality. For the simplest possible algorithm of this type we prove an approximation error bound that is uniform both in time and in the model dimensions. (Joint work with Kavita Ramanan)

Juan Restrepo, Oregon State University
Taking Uncertainties into Account in Geosciences, Physics, and Engineering
Accounting for uncertainties has led us to alter our expectations of what is predictable and how such predictions compare to nature. A significant effort, in recent years, has been placed on creating new uncertainty quantification techniques, rediscovering old ones, and the appropriation of existing ones to account for uncertainties in modeling and simulations. Is this nothing more than a greater reliance on statistics techniques in our regular business? Some of it is. However, as this presentation will recount and illustrate, there are important changes on how we perform the business of modeling and predicting natural phenomena: Bayesian inference is used to combine models and data (not just to compare models and data); sensitivity analyses and projection techniques influence mean-field modeling; data classification techniques allow us to work with the more general state variables, which subsume dynamic physical variables; we exploit complex stochastic representations to better capture multi-scale phenomena or to capture the small-scale correlations of big data sets.

Panos Stinis, Pacific Nortwest National Labs
Stochastic global optimization as a filtering problem
For many optimization problems we cannot evaluate exactly the objective function to be optimized. Similarly, we may not be able to evaluate exactly the functions involved in iterative optimization algorithms. For example, we may only have access to noisy measurements of the functions or statistical estimates provided through Monte Carlo sampling. This makes iterative optimization algorithms behave like stochastic maps. Naive global optimization amounts to evolving a collection of realizations of this stochastic map and picking the realization with the best properties. This motivates the use of filtering techniques to allow focusing on realizations that are more promising than others. We present a filtering reformulation of global optimization in terms of a special case of sequential importance sampling methods called particle filters. The increasing popularity of particle filters is based on the simplicity of their implementation and their flexibility. We utilize the flexibility of particle filters to construct a stochastic global optimization algorithm which can converge to the optimal solution appreciably faster than naive global optimization. Several examples of parametric exponential density estimation are provided to demonstrate the efficiency of the approach.

Kasia Taylor, University of Warwick
Exact sampling of one-dimensional diffusions with discontinuous drift.
We discuss new methods for sampling diffusions with discontinuous drift. The suggested algorithms extend the class of Exact Algorithms for simulation of diffusions. These methods use retrospective rejection sampling which allows for using only finite information about paths in order to accept or reject them. The candidate paths are realisations of stochastic processes with known transition densities and we can sample them on an arbitrarily fine time grid. There are no approximation methods involved in the rejection-acceptance step. It results in ’exact’ sampling of diffusions, i.e. using correct distributions of finite dimensional projections of diffusions. An additional advantage is that the realisation of diffusion on a refined time grid can be easily obtained after the path has been already accepted. The focus of this talk will be on one-dimensional diffusions with discontinuous drift. To address the problem for this class of diffusions we use specially tailored candidate probability measures. We provide methodology for sampling a range of functionals of Brownian motion and its local time which allows us to sample from the candidate measures and perform rejection sampling. It is joint work with Omiros Papaspiliopoulos and Gareth O. Roberts.

Jonathan Weare, University of Chicago
Stratification of Markov processes for rare event simulation
I will discuss an ensemble sampling scheme based on a decomposition of the target average of interest into subproblems that are each individually easier to solve and can be solved in parallel. The most basic version of the scheme computes averages with respect to a given density and is a generalization of the Umbrella Sampling method for the calculation of free energies. Our framework and a detailed perturbation analysis for Markov Chains clearly reveal the utility of the approach. I will also discuss extensions of the stratification philosophy to the calculation of dynamic averages with respect a given Markov process. The scheme is capable of computing very general dynamic averages and offers a natural way to parallelize in both time and space.

Clayton Webster, Oak Ridge National Lab
A hybrid sparse-grid approach for nonlinear filtering problems based on adaptive-domain of the Zakai equation approximations
In this talk we present a hybrid finite difference algorithm for the Zakai equation to solve nonlinear filtering problems. The algorithm combines the splitting-up finite difference scheme and hierarchical sparse grid method to solve moderately high-dimensional nonlinear filtering problems. When applying the hierarchical sparse grid method to approximate bell-shaped solutions in most applications of nonlinear filtering problem, we introduce a logarithmic approximation to reduce the approximation errors. Some space adaptive methods are also introduced to make the algorithm more efficient. Numerical experiments are carried out to demonstrate the performance and efficiency of our algorithm. Joint work with F. Bao (ORNL), Y. Cao (Auburn) and G. Zhang (ORNL).

Jie Xiong, University of Macau
Leader-Follower Stochastic Differential Game with Asymmetric Information and Applications
This talk is concerned with a leader-follower stochastic differential game with asymmetric information, where the information available to the follower is based on some sub-$\sigma$-algebra of that available to the leader. Such kind of game problems has wide applications in finance, economics and management engineering such as newsvendor problems, cooperative advertising and pricing problems. Stochastic maximum principles and verification theorems with partial information will be presented. As an application, a linear-quadratic leader-follower stochastic differential game with asymmetric information is studied. It is shown that the open-loop Stackelberg equilibrium admits a state feedback representation if some system of Riccati equations is solvable.

Yong Zeng, University of Missouri at Kansas City
Bayesian Inference via Filtering Equations for Financial Ultra-High Frequency Data
We propose a general partially-observed framework of Markov processes with marked point process observations for ultra-high frequency (UHF) transaction price data, allowing other observable economic or market factors. We develop the corresponding Bayesian inference via filtering equations to quantify parameter and model uncertainty. Specifically, we derive filtering equations to characterize the evolution of the statistical foundation such as likelihoods, posteriors, Bayes factors and posterior model probabilities. Given the computational challenge, we provide a convergence theorem, enabling us to employ the Markov chain approximation method to construct consistent, easily-parallelizable, recursive algorithms. The algorithms calculate the fundamental statistical characteristics and are capable of implementing the Bayesian inference in real-time for streaming UHF data, via parallel computing for sophisticated models. The general theory is illustrated by specific models built for U.S. Treasury Notes transactions data from GovPX and by Heston stochastic volatility model for stock transactions data. This talk consists joint works with B. Bundick, X. Hu, D. Kuipers and J. Yin.

Guannan Zhang, Oak Ridge National Lab
An efficient meshfree implicit filter for nonlinear filtering problems
We developed an efficient meshfree implicit filter, which is a novel numerical algorithm for nonlinear filtering problems. The implicit filter approximates conditional distributions in the optimal filter over a deterministic state space grid and is developed from samples of the current state obtained by solving the state equation implicitly. The purpose of the meshfree approximation is to improve the efficiency of the implicit filter in moderately high-dimensional problems. The construction of the algorithm includes generation of random state space points and a meshfree interpolation method. Numerical experiments show the effectiveness and efficiency of our algorithm. Joint work with Feng Bao, Yanzhao Cao, Clayton Webster

Feng Bao, Oak Ridge National Lab
A Meshfree Implicit Filter for Solving Non- linear Filtering Problems
We consider a nonlinear filtering problem where a signal process is modeled by a stochastic differential equation and the observation is perturbed by a white noise. The goal of nonlinear filtering is to find the best estimation of the signal process based on the observation. An implicit Bayesian filter was proposed to improve the long term and stable computation of particle filter. In this presentation, we shall present an algorithm to improve the efficiency of the implicit filter through the use of meshfree approximations to the solutions of nonlinear filtering problems.

Forward Backward Doubly Stochastic Differential Equations and Applications to the Optimal Filtering Problem
We consider the classical filter problem where a signal process is modeled by a stochastic differential equation and the observation is perturbed by a white noise. The goal is to find the best estimation of the signal process based on the observation. Kalman Filter, Particle Filter, Zakai equations are some well-known approaches to solve optimal filter problems. In this talk, we shall show the optimal filter problem can also be solved using forward backward doubly stochastic differential equations. Both theoretical results and numerical experiments will be presented.

Xiaoyang Pan, University of Tennessee
Consistency and Asymptotics of Least-Squares Estimator for Partially Observed Jump-Diffusion Processes
We consider a parameter estimation problem for a partially observed stochastic system, where the signal evolves as a jump-diffusion process and the observation is a diffusion process. A least-squares estimator of the intensity of the Poisson process based on the observed sample data is proposed. Precisely, we establish the consistency and asymptotic normality of the least-squares estimator when a negative drift coefficient for the jump-diffusion process is considered and data are streamed online ad infinitimum as in a big data scenario. We also demonstrate that the variance and the fourth moment of the estimator is bounded but inconsistent when the drift coefficient of the jump diffusion is positive or data is collected within a fixed time horizon. Simulation results are presented to support the theoretical landscape. This is joint work with Seddik Djouadi, Vasileios Maroulas and Jie Xiong.

Kai Kang, University of Tennessee
Computational Stochastic Filtering and Large Deviations
In this talk, we consider a nonlinear/non-Gaussian stochastic filtering problem in a small signal-to-noise ratio environment. A large deviation estimate is established by considering qualitative properties of perturbations of an equivalent observation process. Moreover, the posterior filtering distribution is approximated using a drift homotopy technique for stochastic differential equations (SDE). This computational implementation can be thought of as a stochastic analog of deterministic homotopy methods for solving nonlinear algebraic equations or as an SDE generalization of simulated annealing. A toy example based on small noise double-well stochastic dynamics is also presented.

Ioannis Sgouralis, National Institute of Mathematical and Biological Synthesis (NIMBioS)
A novel algorithm for automatic reconstruction of subcellular motion
Subcellular motion is an essential function of living cells. Experimentally subcellular motion is assessed by imaging the cell surface at regular time intervals. Reconstruction of the individual paths contained in the acquired image stacks is performed off-line. Due to the high density of moving particles with high velocities, automatic path reconstruction is a difficult task that currently none of the existing algorithms can fully resolve. In the presentation, I will describe a novel two stage algorithm for automatic path reconstruction. In the first stage, an approximation of the velocities of the particles is obtained. In the second stage, linking is performed based on the approximate velocities and topological clustering techniques. The algorithm is utilized for the tracking of organelle movement in plant cells. This is joint work with Vasileios Maroulas, Andreas Nebenführ, and Fernando Schwartz

Organizing Committee: Vasileios Maroulas, Jan Rosinski, Jie Xiong

This conference is being sponsored by NSF, IMA – Institute for Mathematics and its Applications,
The University of Tennessee Mathematics Department,
College of Arts & Sciences, and the Office of Research.