Sequential Monte Carlo Samplers are a class of stochastic algorithms for Monte Carlo integral estimation w.r.t. probability distributions, which combine elements of Markov chain Monte Carlo methods and importance sampling/resampling schemes. We develop a stability analysis by functional inequalities for a nonlinear flow of probability measures describing the limit behavior of the algorithms as the number of particles tends to infinity. Stability results are derived both under global and local assumptions on the generator of the underlying Metropolis dynamics. This allows us to prove that the combined methods sometimes have good asymptotic stability properties in multimodal setups where traditional MCMC methods mix extremely slowly. For example, this holds for the mean field Ising model at all temperatures.
We study approximations of evolving probability measures by an interacting particle system. The particle system dynamics is a combination of independent Markov chain moves and importance sampling/resampling steps. Under global regularity conditions, we derive non-asymptotic error bounds for the particle system approximation. In a few simple examples, including high dimensional product measures, bounds with explicit constants of feasible size are obtained. Our main motivation are applications to sequential MCMC methods for Monte Carlo integral estimation.
Sequential Monte Carlo methods which involve sequential importance sampling and resampling are shown to provide a versatile approach to computing probabilities of rare events. By making use of martingale representations of the sequential Monte Carlo estimators, we show how resampling weights can be chosen to yield logarithmically efficient Monte Carlo estimates of large deviation probabilities for multidimensional Markov random walks.
We investigate the convergence in distribution of sequential empirical processes of dependent data indexed by a class of functions F. Our technique is suitable for processes that satisfy a multiple mixing condition on a space of functions which differs from the class F. This situation occurs in the case of data arising from dynamical systems or Markov chains, for which the Perron--Frobenius or Markov operator, respectively, has a spectral gap on a restricted space. We provide applications to iterative Lipschitz models that contract on average.
This paper proposes a new algorithm for computing the stationary distribution vector in continuous-time upper block-Hessenberg Markov chains. To this end, we consider the last-block-column-linearly-augmented (LBCL-augmented) truncation of the (infinitesimal) generator of the upper block-Hessenberg Markov chain. The LBCL-augmented truncation is a linearly-augmented truncation such that the augmentation distribution has its probability mass only on the last block column. We first derive an upper bound for the total variation distance between the respective stationary distribution vectors of the …
We consider the Sequential Probability Ratio Test applied to Hidden Markov Models. Given two Hidden Markov Models and a sequence of observations generated by one of them, the Sequential Probability Ratio Test attempts to decide which model produced the sequence. We show relationships between the execution time of such an algorithm and Lyapunov exponents of random matrix systems. Further, we give complexity results about the execution time taken by the Sequential Probability Ratio Test.
We consider an optimal control problem, where a Brownian motion with drift is sequentially observed, and the sign of the drift coefficient changes at jump times of a symmetric two-state Markov process. The Markov process itself is not observable, and the problem consist in finding a {-1,1}-valued process that tracks the unobservable process as close as possible. We present an explicit construction of such a process.
We study finite horizon optimal switching problems for hidden Markov chain models under partially observable Poisson processes. The controller possesses a finite range of strategies and attempts to track the state of the unobserved state variable using Bayesian updates over the discrete observations. Such a model has applications in economic policy making, staffing under variable demand levels and generalized Poisson disorder problems. We show regularity of the value function and explicitly characterize an optimal strategy. We also provide an efficient numerical scheme and illustrate our resul…
We study sequential Bayesian inference in stochastic kinetic models with latent factors. Assuming continuous observation of all the reactions, our focus is on joint inference of the unknown reaction rates and the dynamic latent states, modeled as a hidden Markov factor. Using insights from nonlinear filtering of continuous-time jump Markov processes we develop a novel sequential Monte Carlo algorithm for this purpose. Our approach applies the ideas of particle learning to minimize particle degeneracy and exploit the analytical jump Markov structure. A motivating application of our methods is m…
We investigate the properties of a sequential Monte Carlo method where the particle weight that appears in the algorithm is estimated by a positive, unbiased estimator. We present broadly-applicable convergence results, including a central limit theorem, and we discuss their relevance for applications in statistical physics. Using these results, we show that the resampling step reduces the impact of the randomness of the weights on the asymptotic variance of the estimator. In addition, we explore the limits of convergence of the sequential Monte Carlo method, with a focus on almost sure conver…
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.
We present bounds for the finite sample error of sequential Monte Carlo samplers on static spaces. Our approach explicitly relates the performance of the algorithm to properties of the chosen sequence of distributions and mixing properties of the associated Markov kernels. This allows us to give the first finite sample comparison to other Monte Carlo schemes. We obtain bounds for the complexity of sequential Monte Carlo approximations for a variety of target distributions including finite spaces, product measures, and log-concave distributions including Bayesian logistic regression. The bounds…
We present simple conditions under which the limiting genealogical process associated with a class of interacting particle systems with non-neutral selection mechanisms, as the number of particles grows, is a time-rescaled Kingman coalescent. Sequential Monte Carlo algorithms are popular methods for approximating integrals in problems such as non-linear filtering and smoothing which employ this type of particle system. Their performance depends strongly on the properties of the induced genealogical process. We verify the conditions of our main result for standard sequential Monte Carlo algorit…
We prove finite sample complexities for sequential Monte Carlo (SMC) algorithms which require only local mixing times of the associated Markov kernels. Our bounds are particularly useful when the target distribution is multimodal and global mixing of the Markov kernel is slow; in such cases our approach establishes the benefits of SMC over the corresponding Markov chain Monte Carlo (MCMC) estimator. The lack of global mixing is addressed by sequentially controlling the bias introduced by SMC resampling procedures. We apply these results to obtain complexity bounds for approximating expectation…
We construct a new Markov chain Monte Carlo method on finite states with optimal choices of acceptance-rejection ratio functions. We prove that the constructed continuous time Markov jumping process has a global in-time convergence rate in $L^1$ distance. The convergence rate is no less than one-half and is independent of the target distribution. For example, our method recovers the Metropolis-Hastings algorithm on a two-point state. And it forms a new algorithm for sampling general target distributions. Numerical examples are presented to demonstrate the effectiveness of the proposed algorith…
The main question concerns another recent advance in sequential Monte Carlo, the use of a mixture transition kernel that automatically adapts to the target distribution (Douc et al. 2006). Is there a class of static inference problems for which the backward-kernel approach is better suited, or is it too early to predict which method may have better performance in a particular situation?
This paper studies the mixing time of certain adaptive Markov Chain Monte Carlo algorithms. Under some regularity conditions, we show that the convergence rate of Importance Resampling MCMC (IRMCMC) algorithm, measured in terms of the total variation distance is $O(n^{-1})$, and by means of an example, we establish that in general, this algorithm does not converge at a faster rate. We also study the Equi-Energy sampler and establish that its mixing time is of order $O(n^{-1/2})$.
In the last decade, sequential Monte-Carlo methods (SMC) emerged as a key tool in computational statistics. These algorithms approximate a sequence of distributions by a sequence of weighted empirical measures associated to a weighted population of particles. These particles and weights are generated recursively according to elementary transformations: mutation and selection. Examples of applications include the sequential Monte-Carlo techniques to solve optimal non-linear filtering problems in state-space models, molecular simulation, genetic optimization, etc. Despite many theoretical advanc…
UC Berkeley（加州大学伯克利分校）；University of Washington（华盛顿大学）
Strongly Rayleigh distributions are natural generalizations of product and determinantal probability distributions and satisfy strongest form of negative dependence properties. We show that the "natural" Monte Carlo Markov Chain (MCMC) is rapidly mixing in the support of a {\em homogeneous} strongly Rayleigh distribution. As a byproduct, our proof implies Markov chains can be used to efficiently generate approximate samples of a $k$-determinantal point process. This answers an open question raised by Deshpande and Rademacher.
We show fundamental properties of the Markov semigroup of recently proposed MCMC algorithms based on Piecewise-deterministic Markov processes (PDMPs) such as the Bouncy Particle Sampler, the Zig-Zag process or the Randomized Hamiltonian Monte Carlo method. Under assumptions typically satisfied in MCMC settings, we prove that PDMPs are Feller and that their generator admits the space of infinitely differentiable functions with compact support as a core. As we illustrate via martingale problems and a simplified proof of the invariance of target distributions, these results provide a fundamental …
This paper presents a novel theoretical Monte Carlo Markov chain procedure in the framework of graphs. It specifically deals with the construction of a Markov chain whose empirical distribution converges to a given reference one. The Markov chain is constrained over an underlying graph, so that states are viewed as vertices and the transition between two states can have positive probability only in presence of an edge connecting them. The analysis is carried out on the basis of the relationship between the support of the target distribution and the connectedness of the graph.
We present a sequential Monte Carlo algorithm for Markov chain trajectories with proposals constructed in reverse time, which is advantageous when paths are conditioned to end in a rare set. The reverse time proposal distribution is constructed by approximating the ratio of Green's functions in Nagasawa's formula. Conditioning arguments can be used to interpret these ratios as low-dimensional conditional sampling distributions of some coordinates of the process given the others. Hence the difficulty in designing SMC proposals in high dimension is greatly reduced. We illustrate our method on es…
Historically time-reversibility of the transitions or processes underpinning Markov chain Monte Carlo methods (MCMC) has played a key rôle in their development, while the self-adjointness of associated operators together with the use of classical functional analysis techniques on Hilbert spaces have led to powerful and practically successful tools to characterize and compare their performance. Similar results for algorithms relying on nonreversible Markov processes are scarce. We show that for a type of nonreversible Monte Carlo Markov chains and processes, of current or renewed interest in th…
We introduce an efficient nonreversible Markov chain Monte Carlo algorithm to generate self-avoiding walks with a variable endpoint. In two dimensions, the new algorithm slightly outperforms the two-move nonreversible Berretti-Sokal algorithm introduced by H.~Hu, X.~Chen, and Y.~Deng in \cite{old}, while for three-dimensional walks, it is 3--5 times faster. The new algorithm introduces nonreversible Markov chains that obey global balance and allows for three types of elementary moves on the existing self-avoiding walk: shorten, extend or alter conformation without changing the walk's length.
We present a multivariate central limit theorem for a general class of interacting Markov chain Monte Carlo algorithms used to solve nonlinear measure-valued equations. These algorithms generate stochastic processes which belong to the class of nonlinear Markov chains interacting with their empirical occupation measures. We develop an original theoretical analysis based on resolvent operators and semigroup techniques to analyze the fluctuations of their occupation measures around their limiting values.
