Quantitative convergence assessment for Markov chain Monte Carlo via cusums

Yu (1995) provides a novel convergence diagnostic for Markov chain Monte Carlo (MCMC) which provides a qualitative measure of mixing for Markov chains via a cusum path plot for univariate parameters of interest. The method is based upon the output of a single replication of an MCMC sampler and is therefore widely applicable and simple to use. One criticism of the method is that it is subjective in its interpretation, since it is based upon a graphical comparison of two cusum path plots. In this paper, we develop a quantitative measure of smoothness which we can associate with any given cusum path, and show how we can use this measure to obtain a quantitative measure of mixing. In particular, we derive the large sample distribution of this smoothness measure, so that objective inference is possible. In addition, we show how this quantitative measure may also be used to provide an estimate of the burn-in length for any given sampler. We discuss the utility of this quantitative approach, and highlight a problem which may occur if the chain is able to remain in any one state for some period of time. We provide a more general implementation of the method to overcome the problem in such cases.     

Covariance changes detection in multivariate time series

This paper studies the detection of step changes in the variances and in the correlation structure of the components of a vector of time series. Two procedures based on the likelihood ratio test (LRT) statistic and on a cumulative sums (cusum) statistic are considered and compared in a simulation study. We conclude that for a single covariance change the cusum procedure is more powerful in small and medium samples, whereas the likelihood ratio test is more powerful in large samples. However, for several covariance changes the cusum procedure works clearly better. The procedures are illustrated in two real data examples.     

Coordinate sampler: a non-reversible Gibbs-like MCMC sampler

We derive a novel non-reversible, continuous-time Markov chain Monte Carlo sampler, called Coordinate Sampler, based on a piecewise deterministic Markov process, which is a variant of the Zigzag sampler of Bierkens et al. (Ann Stat 47(3):1288–1320, 2019). In addition to providing a theoretical validation for this new simulation algorithm, we show that the Markov chain it induces exhibits geometrical ergodicity convergence, for distributions whose tails decay at least as fast as an exponential distribution and at most as fast as a Gaussian distribution. Several numerical examples highlight that our coordinate sampler is more efficient than the Zigzag sampler, in terms of effective sample size.     

Trace Class Markov Chains for Bayesian Inference with Generalized Double Pareto Shrinkage Priors

Bayesian shrinkage methods have generated a lot of interest in recent years, especially in the context of high‐dimensional linear regression. In recent work, a Bayesian shrinkage approach using generalized double Pareto priors has been proposed. Several useful properties of this approach, including the derivation of a tractable three‐block Gibbs sampler to sample from the resulting posterior density, have been established. We show that the Markov operator corresponding to this three‐block Gibbs sampler is not Hilbert–Schmidt. We propose a simpler two‐block Gibbs sampler and show that the corresponding Markov operator is trace class (and hence Hilbert–Schmidt). Establishing the trace class property for the proposed two‐block Gibbs sampler has several useful consequences. Firstly, it implies that the corresponding Markov chain is geometrically ergodic, thereby implying the existence of a Markov chain central limit theorem, which in turn enables computation of asymptotic standard errors for Markov chain‐based estimates of posterior quantities. Secondly, because the proposed Gibbs sampler uses two blocks, standard recipes in the literature can be used to construct a sandwich Markov chain (by inserting an appropriate extra step) to gain further efficiency and to achieve faster convergence. The trace class property for the two‐block sampler implies that the corresponding sandwich Markov chain is also trace class and thereby geometrically ergodic. Finally, it also guarantees that all eigenvalues of the sandwich chain are dominated by the corresponding eigenvalues of the Gibbs sampling chain (with at least one strict domination). Our results demonstrate that a minor change in the structure of a Markov chain can lead to fundamental changes in its theoretical properties. We illustrate the improvement in efficiency resulting from our proposed Markov chains using simulated and real examples.     

Automating and evaluating reversible jump MCMC proposal distributions

The reversible jump Markov chain Monte Carlo (MCMC) sampler (Green in Biometrika 82:711–732, 1995) has become an invaluable device for Bayesian practitioners. However, the primary difficulty with the sampler lies with the efficient construction of transitions between competing models of possibly differing dimensionality and interpretation. We propose the use of a marginal density estimator to construct between-model proposal distributions. This provides both a step towards black-box simulation for reversible jump samplers, and a tool to examine the utility of common between-model mapping strategies. We compare the performance of our approach to well established alternatives in both time series and mixture model examples.     

Variational Bayes with synthetic likelihood

Synthetic likelihood is an attractive approach to likelihood-free inference when an approximately Gaussian summary statistic for the data, informative for inference about the parameters, is available. The synthetic likelihood method derives an approximate likelihood function from a plug-in normal density estimate for the summary statistic, with plug-in mean and covariance matrix obtained by Monte Carlo simulation from the model. In this article, we develop alternatives to Markov chain Monte Carlo implementations of Bayesian synthetic likelihoods with reduced computational overheads. Our approach uses stochastic gradient variational inference methods for posterior approximation in the synthetic likelihood context, employing unbiased estimates of the log likelihood. We compare the new method with a related likelihood-free variational inference technique in the literature, while at the same time improving the implementation of that approach in a number of ways. These new algorithms are feasible to implement in situations which are challenging for conventional approximate Bayesian computation methods, in terms of the dimensionality of the parameter and summary statistic.     

Over-relaxation methods and coupled Markov chains for Monte Carlo simulation

This paper is concerned with improving the performance of certain Markov chain algorithms for Monte Carlo simulation. We propose a new algorithm for simulating from multivariate Gaussian densities. This algorithm combines ideas from coupled Markov chain methods and from an existing algorithm based only on over-relaxation. The rate of convergence of the proposed and existing algorithms can be measured in terms of the square of the spectral radius of certain matrices. We present examples in which the proposed algorithm converges faster than the existing algorithm and the Gibbs sampler. We also derive an expression for the asymptotic variance of any linear combination of the variables simulated by the proposed algorithm. We outline how the proposed algorithm can be extended to non-Gaussian densities.     

The Cusum of Squares Test for Scale Changes in Infinite Order Moving Average Processes

In this paper we consider the problem of testing for a scale change in the infinite order moving average process X _j = _{i =0} a _{i j i} , where _j are i.i.d. r.v.s with E ₁ < for some > 0. In performing the test, a cusum of squares test statistic analogous to Inclan & Tiao's (1994) statistic is considered. It is well-known from the literature that outliers affect test procedures leading to false conclusions. In order to remedy this, a cusum of squares test based on trimmed observations is considered. It is demonstrated that this test is robust against outliers, is valid for infinite variance processes as well. Simulation results are given for illustration.     

Stochastic adaptation of importance sampler

Improving efficiency of the importance sampler is at the centre of research on Monte Carlo methods. While the adaptive approach is usually not so straightforward within the Markov chain Monte Carlo framework, the counterpart in importance sampling can be justified and validated easily. We propose an iterative adaptation method for learning the proposal distribution of an importance sampler based on stochastic approximation. The stochastic approximation method can recruit general iterative optimization techniques like the minorization–maximization algorithm. The effectiveness of the approach in optimizing the Kullback divergence between the proposal distribution and the target is demonstrated using several examples.     

Monte Carlo integration with Markov chain

There are two conceptually distinct tasks in Markov chain Monte Carlo (MCMC): a sampler is designed for simulating a Markov chain and then an estimator is constructed on the Markov chain for computing integrals and expectations. In this article, we aim to address the second task by extending the likelihood approach of Kong et al. for Monte Carlo integration. We consider a general Markov chain scheme and use partial likelihood for estimation. Basically, the Markov chain scheme is treated as a random design and a stratified estimator is defined for the baseline measure. Further, we propose useful techniques including subsampling, regulation, and amplification for achieving overall computational efficiency. Finally, we introduce approximate variance estimators for the point estimators. The method can yield substantially improved accuracy compared with Chib's estimator and the crude Monte Carlo estimator, as illustrated with three examples.     

On the Cusum test for parameter changes in garch(1,1) Models

This paper considers the problem of testing parameter constancy in GARCH(1,1) models. A cusum of squares test is propesed in analogy Of Incl´n and Tiao (1394)'s statistic. its limiting distribution is derived via using the invariance principle for mixingaie sequences obtained by McLeish(1975). Simulation results are illustrated to demonstrate the validity of the cusum test.     

Analysis of the Gibbs sampler for a model related to James-Stein estimators

We analyse a hierarchical Bayes model which is related to the usual empirical Bayes formulation of James-Stein estimators. We consider running a Gibbs sampler on this model. Using previous results about convergence rates of Markov chains, we provide rigorous, numerical, reasonable bounds on the running time of the Gibbs sampler, for a suitable range of prior distributions. We apply these results to baseball data from Efron and Morris (1975). For a different range of prior distributions, we prove that the Gibbs sampler will fail to converge, and use this information to prove that in this case the associated posterior distribution is non-normalizable.     

Finite sample distributions of statistics in change point analysis

This paper is concerned with derivation of finite sampling distributions of some statistics which appear frequently in change point analysis. The exact distribution of cusum test statistic is approximated by two methods. Approximations are presented and their accuracies are measured. We first consider the change point in mean problem and we study the exact distribution of change point estimator. Finally, we consider the change point in variance case.     

Wald's approximations to the average run length in cusum procedures

Wald's approximation to the ARL(average run length in cusum) (cumulative sum) procedures are given for an exponential family of densities. From these approximations it is shown that Page's (1954) cusum procedure is (in a sense) identical with a cusum procedure defined in terms of likelihood ratios. Moreover, these approximations are improved by estimating the excess over the boundary and their closeness is examined by numerical comparisons with some exact results. Some examples are also given.     

Efficient MCMC sampling in dynamic mixture models

We show how to improve the efficiency of Markov Chain Monte Carlo (MCMC) simulations in dynamic mixture models by block-sampling the discrete latent variables. Two algorithms are proposed: the first is a multi-move extension of the single-move Gibbs sampler devised by Gerlach, Carter and Kohn (in J. Am. Stat. Assoc. 95, 819–828, 2000); the second is an adaptive Metropolis-Hastings scheme that performs well even when the number of discrete states is large. Three empirical examples illustrate the gain in efficiency achieved. We also show that visual inspection of sample partial autocorrelations of the discrete latent variables helps anticipating whether blocking can be effective.     

DRAM: Efficient adaptive MCMC

We propose to combine two quite powerful ideas that have recently appeared in the Markov chain Monte Carlo literature: adaptive Metropolis samplers and delayed rejection. The ergodicity of the resulting non-Markovian sampler is proved, and the efficiency of the combination is demonstrated with various examples. We present situations where the combination outperforms the original methods: adaptation clearly enhances efficiency of the delayed rejection algorithm in cases where good proposal distributions are not available. Similarly, delayed rejection provides a systematic remedy when the adaptation process has a slow start.     

C219. Comment on ”calculating binomial probabilities when the trial probabilities are unequal“

In complex models like hidden Markov chains, the convergence of the MCMC algorithms used to approximate the posterior distribution and the Bayes estimates of the parameters of interest must be controlled in a robust manner. We propose in this paper a series of online controls, which rely on classical non-parametric tests, to evaluate independence from the start-up distribution, stability of the Markov chain, and asymptotic normality. These tests lead to graphical control spreadsheets which arepresentedin the set-up of normalmixture hidden Markov chains to compare the full Gibbs sampler with an aggregated Gibbs sampler based on the forward – backward formulas.     

Exploring Cook's Statistic Graphically

An index plot of Cook's statistic is frequently used to highlight influential observations. In this article we illustrate how enhanced higher dimensional plots of Cook's statistic can provide further useful information about sets of influential observations. We provide examples using normal and generalized linear models.     

Choosing Principal Components: A New Graphical Method Based on Bayesian Model Selection

This article approaches the problem of selecting significant principal components from a Bayesian model selection perspective. The resulting Bayes rule provides a simple graphical technique that can be used instead of (or together with) the popular scree plot to determine the number of significant components to retain. We study the theoretical properties of the new method and show, by examples and simulation, that it provides more clear-cut answers than the scree plot in many interesting situations.     

Test for Parameter Change in Linear Processes Based on Whittle's Estimator

In this article, we develop a cusum test for testing for parameter changes in linear processes based on Whittle's estimator. It is shown that under regularity conditions, the test statistic converges to the sup of a Brownian bridge. The result is particularly useful in handling the change point test in stationary ARMA processes. A simulation result is provided for illustration.