Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userDimension free convergence rates for Gibbs samplers for Bayesian linear mixed models
215
0
0.0
(
0
)
Added by
Zhumengmeng Jin
Publication date
2021
fields
Mathematical Statistics
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
The emergence of big data has led to a growing interest in so-called convergence complexity analysis, which is the study of how the convergence rate of a Monte Carlo Markov chain (for an intractable Bayesian posterior distribution) scales as the underlying data set grows in size. Convergence complexity analysis of practical Monte Carlo Markov chains on continuous state spaces is quite challenging, and there have been very few successful analyses of such chains. One fruitful analysis was recently presented by Qin and Hobert (2021b), who studied a Gibbs sampler for a simple Bayesian random effects model. These authors showed that, under regularity conditions, the geometric convergence rate of this Gibbs sampler converges to zero as the data set grows in size. It is shown herein that similar behavior is exhibited by Gibbs samplers for more general Bayesian models that possess both random effects and traditional continuous covariates, the so-called mixed models. The analysis employs the Wasserstein-based techniques introduced by Qin and Hobert (2021b).
rate research
Read More
The Bayesian probit regression model (Albert and Chib (1993)) is popular and widely used for binary regression. While the improper flat prior for the regression coefficients is an appropriate choice in the absence of any prior information, a proper normal prior is desirable when prior information is available or in modern high dimensional settings where the number of coefficients ($p$) is greater than the sample size ($n$). For both choices of priors, the resulting posterior density is intractable and a Data Dugmentation (DA) Markov chain is used to generate approximate samples from the posterior distribution. Establishing geometric ergodicity for this DA Markov chain is important as it provides theoretical guarantees for constructing standard errors for Markov chain based estimates of posterior quantities. In this paper, we first show that in case of proper normal priors, the DA Markov chain is geometrically ergodic *for all* choices of the design matrix $X$, $n$ and $p$ (unlike the improper prior case, where $n geq p$ and another condition on $X$ are required for posterior propriety itself). We also derive sufficient conditions under which the DA Markov chain is trace-class, i.e., the eigenvalues of the corresponding operator are summable. In particular, this allows us to conclude that the Haar PX-DA sandwich algorithm (obtained by inserting an inexpensive extra step in between the two steps of the DA algorithm) is strictly better than the DA algorithm in an appropriate sense.
Logistic linear mixed model (LLMM) is one of the most widely used statistical models. Generally, Markov chain Monte Carlo algorithms are used to explore the posterior densities associated with Bayesian LLMMs. Polson, Scott and Windles (2013) Polya-Gamma data augmentation (DA) technique can be used to construct full Gibbs (FG) samplers for LLMMs. Here, we develop efficient block Gibbs (BG) samplers for Bayesian LLMMs using the Polya-Gamma DA method. We compare the FG and BG samplers in the context of simulated and real data examples as the correlation between the fixed and random effects changes as well as when the dimensions of the design matrices vary. These numerical examples demonstrate superior performance of the BG samplers over the FG samplers. We also derive conditions guaranteeing geometric ergodicity of the BG Markov chain when the popular improper uniform prior is assigned on the regression coefficients and proper or improper priors are placed on the variance parameters of the random effects. This theoretical result has important practical implications as it justifies the use of asymptotically valid Monte Carlo standard errors for Markov chain based estimates of posterior quantities.
Component-wise MCMC algorithms, including Gibbs and conditional Metropolis-Hastings samplers, are commonly used for sampling from multivariate probability distributions. A long-standing question regarding Gibbs algorithms is whether a deterministic-scan (systematic-scan) sampler converges faster than its random-scan counterpart. We answer this question when the samplers involve two components by establishing an exact quantitative relationship between the $L^2$ convergence rates of the two samplers. The relationship shows that the deterministic-scan sampler converges faster. We also establish qualitative relations among the convergence rates of two-component Gibbs samplers and some conditional Metropolis-Hastings variants. For instance, it is shown that if some two-component conditional Metropolis-Hastings samplers are geometrically ergodic, then so are the associated Gibbs samplers.
Distances to compact sets are widely used in the field of Topological Data Analysis for inferring geometric and topological features from point clouds. In this context, the distance to a probability measure (DTM) has been introduced by Chazal et al. (2011) as a robust alternative to the distance a compact set. In practice, the DTM can be estimated by its empirical counterpart, that is the distance to the empirical measure (DTEM). In this paper we give a tight control of the deviation of the DTEM. Our analysis relies on a local analysis of empirical processes. In particular, we show that the rates of convergence of the DTEM directly depends on the regularity at zero of a particular quantile fonction which contains some local information about the geometry of the support. This quantile function is the relevant quantity to describe precisely how difficult is a geometric inference problem. Several numerical experiments illustrate the convergence of the DTEM and also confirm that our bounds are tight.
We study the convergence properties of a collapsed Gibbs sampler for Bayesian vector autoregressions with predictors, or exogenous variables. The Markov chain generated by our algorithm is shown to be geometrically ergodic regardless of whether the number of observations in the underlying vector autoregression is small or large in comparison to the order and dimension of it. In a convergence complexity analysis, we also give conditions for when the geometric ergodicity is asymptotically stable as the number of observations tends to infinity. Specifically, the geometric convergence rate is shown to be bounded away from unity asymptotically, either almost surely or with probability tending to one, depending on what is assumed about the data generating process. This result is one of the first of its kind for practically relevant Markov chain Monte Carlo algorithms. Our convergence results hold under close to arbitrary model misspecification.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
