Markov Chain Monte Carlo Kernel Research Articles

Waste-Free Sequential Monte Carlo

AbstractA standard way to move particles in a sequential Monte Carlo (SMC) sampler is to apply several steps of a Markov chain Monte Carlo (MCMC) kernel. Unfortunately, it is not clear how many steps need to be performed for optimal performance. In addition, the output of the intermediate steps are discarded and thus wasted somehow. We propose a new, waste-free SMC algorithm which uses the outputs of all these intermediate MCMC steps as particles. We establish that its output is consistent and asymptotically normal. We use the expression of the asymptotic variance to develop various insights on how to implement the algorithm in practice. We develop in particular a method to estimate, from a single run of the algorithm, the asymptotic variance of any particle estimate. We show empirically, through a range of numerical examples, that waste-free SMC tends to outperform standard SMC samplers, and especially so in situations where the mixing of the considered MCMC kernels decreases across iterations (as in tempering or rare event problems).

Error bounds for sequential Monte Carlo samplers for multimodal distributions

In this paper, we provide bounds on the asymptotic variance for a class of sequential Monte Carlo (SMC) samplers designed for approximating multimodal distributions. Such methods combine standard SMC methods and Markov chain Monte Carlo (MCMC) kernels. Our bounds improve upon previous results, and unlike some earlier work, they also apply in the case when the MCMC kernels can move between the modes. We apply our results to the Potts model from statistical physics. In this case, the problem of sharp peaks is encountered. Earlier methods, such as parallel tempering, are only able to sample from it at an exponential (in an important parameter of the model) cost. We propose a sequence of interpolating distributions called interpolation to independence, and show that the SMC sampler based on it is able to sample from this target distribution at a polynomial cost. We believe that our method is generally applicable to many other distributions as well.

Transdimensional sequential Monte Carlo using variational Bayes — SMCVB

A new transdimensional Sequential Monte Carlo (SMC) algorithm called SMCVB is proposed. In an SMC approach, a weighted sample of particles is generated from a sequence of probability distributions which ‘converge’ to the target distribution of interest, in this case a Bayesian posterior distribution. The approach is based on the use of variational Bayes to propose new particles at each iteration of the SMCVB algorithm in order to target the posterior more efficiently. The variational-Bayes-generated proposals are not limited to a fixed dimension. This means that the weighted particle sets that arise can have varying dimensions thereby allowing us the option to also estimate an appropriate dimension for the model. This novel algorithm is outlined within the context of finite mixture model estimation. This provides a less computationally demanding alternative to using reversible jump Markov chain Monte Carlo kernels within an SMC approach. We illustrate these ideas in a simulated data analysis and in applications.

On the Conjugacy of Off-Line and On-Line Sequential Monte Carlo Samplers

Sequential Monte Carlo (SMC) methods are widely used for filtering purposes of non-linear economic or financial models. Nevertheless the SMC scope encompasses wider applications such as estimating static model parameters so much that it is becoming a serious alternative to Markov- Chain Monte-Carlo (MCMC) methods. Not only SMC algorithms draw posterior distributions of static or dynamic parameters but additionally provide an estimate of the normalizing constant. The tempered and time (TNT) algorithm, developed in the paper, combines (off-line) tempered SMC inference with on-line SMC inference for estimating many slightly different distributions. The method encompasses the Iterated Batch Importance Sampling (IBIS) algorithm and more generally the Resample Move (RM) algorithm. Besides the number of particles, the TNT algorithm self-adjusts its calibrated parameters and relies on a new MCMC kernel that allows for particle interactions. The algorithm is well suited for efficiently back-testing models. We conclude by comparing in-sample and out-of-sample performances of complex volatility models.

Variance bounding and geometric ergodicity of Markov chain Monte Carlo kernels for approximate Bayesian computation

Approximate Bayesian computation has emerged as a standard computational tool when dealing with intractable likelihood functions in Bayesian inference. We show that many common Markov chain Monte Carlo kernels used to facilitate inference in this setting can fail to be variance bounding and hence geometrically ergodic, which can have consequences for the reliability of estimates in practice. This phenomenon is typically independent of the choice of tolerance in the approximation. We prove that a recently introduced Markov kernel can inherit the properties of variance bounding and geometric ergodicity from its intractable Metropolis–Hastings counterpart, under reasonably weak conditions. We show that the computational cost of this alternative kernel is bounded whenever the prior is proper, and present indicative results for an example where spectral gaps and asymptotic variances can be computed, as well as an example involving inference for a partially and discretely observed, time-homogeneous, pure jump Markov process. We also supply two general theorems, one providing a simple sufficient condition for lack of variance bounding for reversible kernels and the other providing a positive result concerning inheritance of variance bounding and geometric ergodicity for mixtures of reversible kernels.

Approximate Inference for Observation-Driven Time Series Models with Intractable Likelihoods

In this article, we consider approximate Bayesian parameter inference for observation-driven time series models. Such statistical models appear in a wide variety of applications, including econometrics and applied mathematics. This article considers the scenario where the likelihood function cannot be evaluated pointwise; in such cases, one cannot perform exact statistical inference, including parameter estimation, which often requires advanced computational algorithms, such as Markov Chain Monte Carlo (MCMC). We introduce a new approximation based upon Approximate Bayesian Computation (ABC). Under some conditions, we show that as n → ∞, with n the length of the time series, the ABC posterior has, almost surely, a Maximum A Posteriori (MAP) estimator of the parameters that is often different from the true parameter. However, a noisy ABC MAP, which perturbs the original data, asymptotically converges to the true parameter, almost surely. In order to draw statistical inference, for the ABC approximation adopted, standard MCMC algorithms can have acceptance probabilities that fall at an exponential rate in n and slightly more advanced algorithms can mix poorly. We develop a new and improved MCMC kernel, which is based upon an exact approximation of a marginal algorithm, whose cost per iteration is random, but the expected cost, for good performance, is shown to be O ( n 2 ) per iteration. We implement our new MCMC kernel for parameter inference from models in econometrics.

An Adaptive Sequential Monte Carlo Sampler

Sequential Monte Carlo (SMC) methods are not only a popular tool in the analysis of state–space models, but offer an alternative to Markov chain Monte Carlo (MCMC) in situations where Bayesian inference must proceed via simulation. This paper introduces a new SMC method that uses adaptive MCMC kernels for particle dynamics. The proposed algorithm features an online stochastic optimization procedure to select the best MCMC kernel and simultaneously learn optimal tuning parameters. Theoretical results are presented that justify the approach and give guidance on how it should be implemented. Empirical results, based on analysing data from mixture models, show that the new adaptive SMC algorithm (ASMC) can both choose the best MCMC kernel, and learn an appropriate scaling for it. ASMC with a choice between kernels outperformed the adaptive MCMC algorithm of Haario et al. (1998) in 5 out of the 6 cases considered.

Distributed Markov Chain Monte Carlo kernel based particle filtering for object tracking

Particle filters are computationally intensive and thus efficient parallelism is crucial to effective implementations, especially object tracking in video sequences. Two schemes for pipelining particles under high performance computing environment, including an alternative Markov Chain Monte Carlo (MCMC) resampling algorithm and kernel function, are proposed so as to improve tracking performance and minimize execution time. Experimental results on a network of workstations composed of simple off-the-shelf hardware components show that global parallelizable scheme provides a promising resolution to clearly reduce execution time with increasing particles, compared with generic particle filtering.

Non-linear Markov Chain Monte Carlo

In this paper we introduce a class of non-linear Markov Chain Monte Carlo (MCMC) methods for simulating from a probability measure π. Non-linear Markov kernels (e.g. Del Moral (2004)) can be constructed to admit π as an invariant distribution and have typically superior mixing properties to ordinary (linear) MCMC kernels. However, such non-linear kernels often cannot be simulated exactly, so, in the spirit of particle approximations of Feynman-Kac formulae (Del Moral 2004), we construct approximations of the non-linear kernels via Self-Interacting Markov Chains (Del Moral & Miclo 2004) (SIMC). We present several non-linear kernels and investigate the performance of our approximations with some simulations.

Using Bayesian statistics in the estimation of heat source in radiation

An unknown transient heat source in a three-dimensional participating medium is reconstructed from temperature measurements using a Bayesian inference method. The heat source is modeled as a stochastic process. The joint posterior probability density function (PPDF) of heat source values at consecutive time points is computed using the Bayes’ formula. The errors in thermocouple readings are modeled as independent identically distributed (i.i.d.) Gauss random variables. ‘Maximum A Posteriori’ (MAP) and posterior mean estimates of the heat source are then computed using a Markov chain Monte Carlo (MCMC) simulation method. The designed MCMC sampler is composed of a cycle of symmetric MCMC kernels. To increase the sampling speed, a model-reduction technique is used in the direct computation of temperatures at thermocouple locations given a guessed heat source, i.e. in the likelihood computation. Two typical heat source profiles are reconstructed using simulated data to demonstrate the presented methodologies. The results indicate that the Bayesian inference method can provide accurate point estimates as well as uncertainty quantification to the solution of the inverse radiation problem.