arXiv subjects

Nick Whiteley

Publications and source records attributed to Nick Whiteley.

44 records · Page 3Linked to original sources

arXiv2014

Twisted particle filters

We investigate sampling laws for particle algorithms and the influence of these laws on the efficiency of particle approximations of marginal likelihoods in hidden Markov models. Among a broad class of candidates we characterize the essentially unique family of particle system transition kernels which is optimal with respect to an asymptotic-in-time variance growth rate criterion. The sampling structure of the algorithm defined by these optimal transitions turns out to be only subtly different from standard algorithms and yet the fluctuation properties of the estimates it provides can be dramatically different. The structure of the optimal transition suggests a new class of algorithms, which we term "twisted" particle filters and which we validate with asymptotic analysis of a more traditional nature, in the regime where the number of particles tends to infinity.

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arXiv2013

Stability properties of some particle filters

Under multiplicative drift and other regularity conditions, it is established that the asymptotic variance associated with a particle filter approximation of the prediction filter is bounded uniformly in time, and the nonasymptotic, relative variance associated with a particle approximation of the normalizing constant is bounded linearly in time. The conditions are demonstrated to hold for some hidden Markov models on noncompact state spaces. The particle stability results are obtained by proving $v$-norm multiplicative stability and exponential moment results for the underlying Feynman-Kac formulas.

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arXiv2012

Bayesian learning of noisy Markov decision processes

We consider the inverse reinforcement learning problem, that is, the problem of learning from, and then predicting or mimicking a controller based on state/action data. We propose a statistical model for such data, derived from the structure of a Markov decision process. Adopting a Bayesian approach to inference, we show how latent variables of the model can be estimated, and how predictions about actions can be made, in a unified framework. A new Markov chain Monte Carlo (MCMC) sampler is devised for simulation from the posterior distribution. This step includes a parameter expansion step, which is shown to be essential for good convergence properties of the MCMC sampler. As an illustration, the method is applied to learning a human controller.

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arXiv2012

Approximate Bayesian Computation for Smoothing

We consider a method for approximate inference in hidden Markov models (HMMs). The method circumvents the need to evaluate conditional densities of observations given the hidden states. It may be considered an instance of Approximate Bayesian Computation (ABC) and it involves the introduction of auxiliary variables valued in the same space as the observations. The quality of the approximation may be controlled to arbitrary precision through a parameter ε>0 . We provide theoretical results which quantify, in terms of ε, the ABC error in approximation of expectations of additive functionals with respect to the smoothing distributions. Under regularity assumptions, this error is O(nε), where n is the number of time steps over which smoothing is performed. For numerical implementation we adopt the forward-only sequential Monte Carlo (SMC) scheme of [16] and quantify the combined error from the ABC and SMC approximations. This forms some of the first quantitative results for ABC methods which jointly treat the ABC and simulation errors, with a finite number of data and simulated samples. When the HMM has unknown static parameters, we consider particle Markov chain Monte Carlo [2] (PMCMC) methods for batch statistical inference.

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arXiv2012

Linear Variance Bounds for Particle Approximations of Time-Homogeneous Feynman-Kac Formulae

This article establishes sufficient conditions for a linear-in-time bound on the non-asymptotic variance of particle approximations of time-homogeneous Feynman-Kac formulae. These formulae appear in a wide variety of applications including option pricing in finance and risk sensitive control in engineering. In direct Monte Carlo approximation of these formulae, the non-asymptotic variance typically increases at an exponential rate in the time parameter. It is shown that a linear bound holds when a non-negative kernel, defined by the logarithmic potential function and Markov kernel which specify the Feynman-Kac model, satisfies a type of multiplicative drift condition and other regularity assumptions. Examples illustrate that these conditions are general and flexible enough to accommodate two rather extreme cases, which can occur in the context of a non-compact state space: 1) when the potential function is bounded above, not bounded below and the Markov kernel is not ergodic; and 2) when the potential function is not bounded above, but the Markov kernel itself satisfies a multiplicative drift condition.

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arXiv2011

Error Bounds and Normalizing Constants for Sequential Monte Carlo in High Dimensions

In a recent paper Beskos et al (2011), the Sequential Monte Carlo (SMC) sampler introduced in Del Moral et al (2006), Neal (2001) has been shown to be asymptotically stable in the dimension of the state space d at a cost that is only polynomial in d, when N the number of Monte Carlo samples, is fixed. More precisely, it has been established that the effective sample size (ESS) of the ensuing (approximate) sample and the Monte Carlo error of fixed dimensional marginals will converge as $d$ grows, with a computational cost of $\mathcal{O}(Nd^2)$. In the present work, further results on SMC methods in high dimensions are provided as $d\to\infty$ and with $N$ fixed. We deduce an explicit bound on the Monte-Carlo error for estimates derived using the SMC sampler and the exact asymptotic relative $\mathbb{L}_2$-error of the estimate of the normalizing constant. We also establish marginal propagation of chaos properties of the algorithm. The accuracy in high-dimensions of some approximate SMC-based filtering schemes is also discussed.

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arXiv2011

Sequential Monte Carlo samplers: error bounds and insensitivity to initial conditions

This paper addresses finite sample stability properties of sequential Monte Carlo methods for approximating sequences of probability distributions. The results presented herein are applicable in the scenario where the start and end distributions in the sequence are fixed and the number of intermediate steps is a parameter of the algorithm. Under assumptions which hold on non-compact spaces, it is shown that the effect of the initial distribution decays exponentially fast in the number of intermediate steps and the corresponding stochastic error is stable in \mathbb{L}_{p} norm.

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arXiv2010

Efficient Bayesian Inference for Switching State-Space Models using Discrete Particle Markov Chain Monte Carlo Methods

Switching state-space models (SSSM) are a very popular class of time series models that have found many applications in statistics, econometrics and advanced signal processing. Bayesian inference for these models typically relies on Markov chain Monte Carlo (MCMC) techniques. However, even sophisticated MCMC methods dedicated to SSSM can prove quite inefficient as they update potentially strongly correlated discrete-valued latent variables one-at-a-time (Carter and Kohn, 1996; Gerlach et al., 2000; Giordani and Kohn, 2008). Particle Markov chain Monte Carlo (PMCMC) methods are a recently developed class of MCMC algorithms which use particle filters to build efficient proposal distributions in high-dimensions (Andrieu et al., 2010). The existing PMCMC methods of Andrieu et al. (2010) are applicable to SSSM, but are restricted to employing standard particle filtering techniques. Yet, in the context of discrete-valued latent variables, specialised particle techniques have been developed which can outperform by up to an order of magnitude standard methods (Fearnhead, 1998; Fearnhead and Clifford, 2003; Fearnhead, 2004). In this paper we develop a novel class of PMCMC methods relying on these very efficient particle algorithms. We establish the theoretical validy of this new generic methodology referred to as discrete PMCMC and demonstrate it on a variety of examples including a multiple change-points model for well-log data and a model for U.S./U.K. exchange rate data. Discrete PMCMC algorithms are shown to outperform experimentally state-of-the-art MCMC techniques for a fixed computational complexity. Additionally they can be easily parallelized (Lee et al., 2010) which allows further substantial gains.

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