Tuesday, Aug 6: 8:30 AM - 10:20 AM
1756
Topic-Contributed Paper Session
Oregon Convention Center
Room: CC-F150
This session highlights recent advancements in Markov chain Monte Carlo (MCMC) methodology and computation. Speakers will present innovative approaches such as Riemannian geometric frameworks for MCMC algorithms, bandwidth selection for covariance estimation, Bayesian generalized linear models for correlated data, simultaneous confidence bands for functional parameters, and shape-constrained inference for covariance function estimation. Attendees can expect to gain valuable insights into cutting-edge techniques shaping the future of MCMC research.
Applied
Yes
Main Sponsor
Section on Bayesian Statistical Science
Co Sponsors
International Society for Bayesian Analysis (ISBA)
Section on Statistical Computing
Presentations
Simulated tempering Markov chain Monte Carlo method has long been an important tool for exploring multimodal distributions that are difficult to sample from. The method creates 'tempered' distributions that help to move the Markov chain between modes of the target density. While one can ignore the samples not coming from the target distribution, an importance sampling estimator that uses all the samples generated is usually more numerically stable. The current practice requires repeated pilot runs to estimate normalizing constants of these 'tempered' distributions, yet often overlooks the inherent errors of those estimates, which leads to under-reporting of the standard errors. We provide an effective method for implementing simulated tempering efficiently using the available computing resources and derive formulas for asymptotically valid standard errors of the estimators. The proposed methods are illustrated using several examples.
Test statistics, confidence intervals, and p-values all typically rely on an estimate for variance. For data sets that are not independent and identically distributed (iid) caution must be used when selecting a variance estimator. If the dependence structure is unknown but stationary, a robust long run variance (LRV) estimator can be used which can handle a wide variety of scenarios. Spectral variance (SV) estimators are one of the most common LRV estimation methods, but they suffer from a negative bias in the presence of positive correlation. An alternative zero lugsail estimator has been proposed to combat this issue which has a zero asymptotic bias regardless of correlation. Both SV and zero lugsail estimators rely on a bandwidth parameter, a critical component for the estimation process. Currently no guidelines exist for selecting a bandwidth for the zero lugsail estimator. We propose an optimal bandwidth rule for zero lugsail estimators when relying on nonstandard limiting distributions. With this procedure we can greatly improve bias, account for variability, and obtain an estimator optimized for inference.
I will introduce nonparametric, shape-constrained estimation for covariance functions, with an emphasis on a shape-constrained estimator of the autocovariance sequence from a reversible Markov chain. The estimator will be shown to lead to strongly consistent estimates of the asymptotic variance of the sample mean from an MCMC sample, as well as to \ell_2 consistent estimates of the autocovariance sequence. An algorithm for computing our estimator will be presented, and some empirical applications will be shown. The proposed shape-constrained estimator exploits a mixture representation of the autocovariance sequence from a reversible Markov chain. Similar mixture representations exist for stationary covariance functions in spatial statistics, including for the Matérn covariance as a special case, and I will highlight some possible extensions of shape-constrained approaches for estimating covariance functions in spatial statistics.
There are many challenges that arise when simulating from Bayesian generalized linear model posterior distributions in practice, especially when the observed data is assumed to be dependent. We focus on two challenges that stem from the introduction of one or more auxiliary latent variables for each observation. First, several popular methods for simulating from Bayesian generalized linear model posterior distributions rely on the introducing of an auxiliary random variable for each observation. These methods can scale poorly when the number of observations is large because they require additional posterior draws and repeated expensive matrix calculations. Second, many of the most useful approaches for introducing dependence in the observed data do so by introducing a latent random variable with a dense but computationally convenient prior covariance matrix. However, the computational conveniences offered by the prior covariance matrix may be absent (or appear to be absent) from the posterior. We introduce methods for addressing these challenges that take advantage of simple reparameterizations of the problem, advances in posterior mode computation, and modern sampling.
We consider the construction of simultaneous confidence bands (CBs) for functional parameters over uncountable sets in Monte Carlo simulations. This setting arises naturally in estimating one or more marginal posterior densities, which may be done nonparametrically or via Rao-Blackwellisation. Although our motivation is primarily in estimating and visualizing densities, our approach can be applied in more general functional estimation, such as in estimation of likelihood functions for likelihood-based inference and when functional parameters are of interest in Bayesian inference. In short, we provide principled uncertainty quantification in the form of a simultaneous CB and hence increase the reliability of the resulting functional inference.