arXiv subjects

Galin L. Jones

Publications and source records attributed to Galin L. Jones.

At least 19 recordsLinked to original sources

arXiv2026

A New Measure of Dependence Between Continuous and Multinomial Random Variables

A novel measure of dependence between a continuous random variable and a multinomial random variable is introduced. The proposed measure is based on the Hellinger distance between conditional distributions. It satisfies the desiderata for a dependence measure without making specific distributional assumptions about the continuous random variable or assuming that the discrete random variable arises from a latent continuous random variable. An estimator of the dependence measure based on data splitting and kernel density estimation is developed. The asymptotic distribution of the estimator has a simple form with a convergence rate of \sqrt{n}, making confidence intervals for the dependence measure and a test for independence straightforward and computationally convenient.

stat.ME↗

arXiv2026

A scalable Bayesian framework for galaxy emission line detection and redshift estimation

Estimating galaxy redshifts is crucial for constraining key physical quantities like those in the equation of state of dark energy. Modern telescopes such as the James Webb Space Telescope, the Euclid Space Telescope, and the NASA Nancy Grace Roman Space Telescope are producing massive amounts of spectroscopic data that enable precise redshift estimation. However, a galaxy's redshift can be estimated only when emission lines are present in the observed spectrum, which is unknown a priori. A novel Bayesian approach to estimating redshift and simultaneously testing for the presence of emission lines is developed. Although modern spectroscopic surveys involve millions of spectra and give rise to highly multimodal posterior distributions, the proposed framework remains computationally efficient, admitting a parallelizable implementation suitable for large-scale inference.

astro-ph.IM↗

arXiv2026

Testing models for angular power spectra: A distribution-free approach

A novel goodness-of-fit strategy is introduced for testing models of angular power spectra with unknown parameters. Using this strategy, it is possible to assess the validity of such models without specifying the distribution of the angular power spectrum estimators. This holds under general conditions, ensuring the method's applicability in diverse applications. Moreover, the proposed solution overcomes the need for case-by-case simulations when testing different models, leading to notable computational advantages.

physics.data-an↗

arXiv2025

Supernova Siblings and Spectroscopic Host-Galaxy Properties

Over the past century, supernova (SN) searches have detected multiple supernovae (SNe) in hundreds of individual galaxies. So-called SN siblings discovered in the same galaxy present an opportunity to constrain the dependence of the properties of SNe on those of their host galaxies. To investigate whether there is a connection between sibling SNe in galaxies that have hosted multiple SNe and the properties of galaxies, we have acquired integrated optical spectroscopy of 59 galaxies with multiple core-collapse SNe. Perhaps surprisingly, a strong majority of host-galaxy spectra fall within the composite region of the Baldwin-Phillips-Terlevich (BPT) diagram. We find a statistically significant difference (KS test p-value = 0.044) between the distributions of the [NII] $λ$6583/H$α$ of galaxies that have hosted a majority SN Ibc and those that have hosted a majority SN II, where the majority SN Ibc galaxies have, on average, higher ratios. The difference between the distributions of [NII] $λ$6583/H$α$ may arise from either increased contribution from AGN or LINERs in SN Ibc host galaxies, greater metallicity for SN Ibc host galaxies, or both. When comparing the inferred oxygen abundance and the ionization parameter for the galaxies in the Star-Forming region on the BPT diagram, we find statistically significant differences between the distributions for SN Ibc hosts and SN II hosts (p=0.008 and p=0.001, respectively), as well as SN Ib hosts and SN II hosts (p=0.030 and p=0.006, respectively). We also compare the H$α$ equivalent width distributions, also integrated across the galaxies, and find no significant difference.

astro-ph.GA↗

arXiv2024

A Bayesian Generalized Bridge Regression Approach to Covariance Estimation in the Presence of Covariates

A hierarchical Bayesian approach that permits simultaneous inference for the regression coefficient matrix and the error precision (inverse covariance) matrix in the multivariate linear model is proposed. Assuming a natural ordering of the elements of the response, the precision matrix is reparameterized so it can be estimated with univariate-response linear regression techniques. A novel generalized bridge regression prior that accommodates both sparse and dense settings and is competitive with alternative methods for univariate-response regression is proposed and used in this framework. Two component-wise Markov chain Monte Carlo algorithms are developed for sampling, including a data augmentation algorithm based on a scale mixture of normals representation. Numerical examples demonstrate that the proposed method is competitive with comparable joint mean-covariance models, particularly in estimation of the precision matrix. The method is also used to estimate the 253 by 253 precision matrix of 90,670 spectra extracted from images taken by the Hubble Space Telescope, demonstrating its computational feasibility for problems with large n and q.

stat.ME↗

arXiv2024

Lower bounds on the rate of convergence for accept-reject-based Markov chains in Wasserstein and total variation distances

To avoid poor empirical performance in Metropolis-Hastings and other accept-reject-based algorithms practitioners often tune them by trial and error. Lower bounds on the convergence rate are developed in both total variation and Wasserstein distances in order to identify how the simulations will fail so these settings can be avoided, providing guidance on tuning. Particular attention is paid to using the lower bounds to study the convergence complexity of accept-reject-based Markov chains and to constrain the rate of convergence for geometrically ergodic Markov chains. The theory is applied in several settings. For example, if the target density concentrates with a parameter n (e.g. posterior concentration, Laplace approximations), it is demonstrated that the convergence rate of a Metropolis-Hastings chain can be arbitrarily slow if the tuning parameters do not depend carefully on n. This is demonstrated with Bayesian logistic regression with Zellner's g-prior when the dimension and sample increase together and flat prior Bayesian logistic regression as n tends to infinity.

math.ST↗

arXiv2023

Explicit Constraints on the Geometric Rate of Convergence of Random Walk Metropolis-Hastings

Convergence rate analyses of random walk Metropolis-Hastings Markov chains on general state spaces have largely focused on establishing sufficient conditions for geometric ergodicity or on analysis of mixing times. Geometric ergodicity is a key sufficient condition for the Markov chain Central Limit Theorem and allows rigorous approaches to assessing Monte Carlo error. The sufficient conditions for geometric ergodicity of the random walk Metropolis-Hastings Markov chain are refined and extended, which allows the analysis of previously inaccessible settings such as Bayesian Poisson regression. The key technical innovation is the development of explicit drift and minorization conditions for random walk Metropolis-Hastings, which allows explicit upper and lower bounds on the geometric rate of convergence. Further, lower bounds on the geometric rate of convergence are also developed using spectral theory. The existing sufficient conditions for geometric ergodicity, to date, have not provided explicit constraints on the rate of geometric rate of convergence because the method used only implies the existence of drift and minorization conditions. The theoretical results are applied to random walk Metropolis-Hastings algorithms for a class of exponential families and generalized linear models that address Bayesian Regression problems.

math.ST↗

arXiv2023

Convergence Analysis of Data Augmentation Algorithms for Bayesian Robust Multivariate Linear Regression with Incomplete Data

Gaussian mixtures are commonly used for modeling heavy-tailed error distributions in robust linear regression. Combining the likelihood of a multivariate robust linear regression model with a standard improper prior distribution yields an analytically intractable posterior distribution that can be sampled using a data augmentation algorithm. When the response matrix has missing entries, there are unique challenges to the application and analysis of the convergence properties of the algorithm. Conditions for geometric ergodicity are provided when the incomplete data have a "monotone" structure. In the absence of a monotone structure, an intermediate imputation step is necessary for implementing the algorithm. In this case, we provide sufficient conditions for the algorithm to be Harris ergodic. Finally, we show that, when there is a monotone structure and intermediate imputation is unnecessary, intermediate imputation slows the convergence of the underlying Monte Carlo Markov chain, while post hoc imputation does not. An R package for the data augmentation algorithm is provided.

math.ST↗

arXiv2022

Exact Convergence Analysis for Metropolis-Hastings Independence Samplers in Wasserstein Distances

Under mild assumptions, we show the exact convergence rate in total variation is also exact in weaker Wasserstein distances for the Metropolis-Hastings independence sampler. We develop a new upper and lower bound on the worst-case Wasserstein distance when initialized from points. For an arbitrary point initialization, we show the convergence rate is the same and matches the convergence rate in total variation. We derive exact convergence expressions for more general Wasserstein distances when initialization is at a specific point. Using optimization, we construct a novel centered independent proposal to develop exact convergence rates in Bayesian quantile regression and many generalized linear model settings. We show the exact convergence rate can be upper bounded in Bayesian binary response regression (e.g. logistic and probit) when the sample size and dimension grow together.

math.ST↗

arXiv2022

Understanding Linchpin Variables in Markov Chain Monte Carlo

An introduction to the use of linchpin variables in Markov chain Monte Carlo (MCMC) is provided. Before the widespread adoption of MCMC methods, conditional sampling using linchpin variables was essentially the only practical approach for simulating from multivariate distributions. With the advent of MCMC, linchpin variables were largely ignored. However, there has been a resurgence of interest in using them in conjunction with MCMC methods and there are good reasons for doing so. A simple derivation of the method is provided, its validity, benefits, and limitations are discussed, and some examples in the research literature are presented.

stat.CO↗

arXiv2021

Convergence Rates of Two-Component MCMC Samplers

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.

math.ST↗

arXiv2020

Convergence Analysis of a Collapsed Gibbs Sampler for Bayesian Vector Autoregressions

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.

math.ST↗

arXiv2020

Assessing and Visualizing Simultaneous Simulation Error

Monte Carlo experiments produce samples in order to estimate features of a given distribution. However, simultaneous estimation of means and quantiles has received little attention, despite being common practice. In this setting we establish a multivariate central limit theorem for any finite combination of sample means and quantiles under the assumption of a strongly mixing process, which includes the standard Monte Carlo and Markov chain Monte Carlo settings. We build on this to provide a fast algorithm for constructing hyperrectangular confidence regions having the desired simultaneous coverage probability and a convenient marginal interpretation. The methods are incorporated into standard ways of visualizing the results of Monte Carlo experiments enabling the practitioner to more easily assess the reliability of the results. We demonstrate the utility of this approach in various Monte Carlo settings including simulation studies based on independent and identically distributed samples and Bayesian analyses using Markov chain Monte Carlo sampling.

stat.CO↗

arXiv2019

Ensuring Reliable Monte Carlo Estimates of Network Properties

The literature in social network analysis has largely focused on methods and models which require complete network data; however there exist many networks which can only be studied via sampling methods due to the scale or complexity of the network, access limitations, or the population of interest is hard to reach. In such cases, the application of random walk-based Markov chain Monte Carlo (MCMC) methods to estimate multiple network features is common. However, the reliability of these estimates has been largely ignored. We consider and further develop multivariate MCMC output analysis methods in the context of network sampling to directly address the reliability of the multivariate estimation. This approach yields principled, computationally efficient, and broadly applicable methods for assessing the Monte Carlo estimation procedure. In particular, with respect to two random-walk algorithms, a simple random walk and a Metropolis-Hastings random walk, we construct and compare network parameter estimates, effective sample sizes, coverage probabilities, and stopping rules, all of which speaks to the estimation reliability.

stat.AP↗

arXiv2019

Fully Bayesian Penalized Regression with a Generalized Bridge Prior

We consider penalized regression models under a unified framework where the particular method is determined by the form of the penalty term. We propose a fully Bayesian approach that incorporates both sparse and dense settings and show how to use a type of model averaging approach to eliminate the nuisance penalty parameters and perform inference through the marginal posterior distribution of the regression coefficients. We establish tail robustness of the resulting estimator as well as conditional and marginal posterior consistency. We develop an efficient component-wise Markov chain Monte Carlo algorithm for sampling. Numerical results show that the method tends to select the optimal penalty and performs well in both variable selection and prediction and is comparable to, and often better than alternative methods. Both simulated and real data examples are provided.

stat.ME↗

arXiv2019

Consistent Maximum Likelihood Estimation Using Subsets with Applications to Multivariate Mixed Models

We present new results for consistency of maximum likelihood estimators with a focus on multivariate mixed models. Our theory builds on the idea of using subsets of the full data to establish consistency of estimators based on the full data. It requires neither that the data consist of independent observations, nor that the observations can be modeled as a stationary stochastic process. Compared to existing asymptotic theory using the idea of subsets we substantially weaken the assumptions, bringing them closer to what suffices in classical settings. We apply our theory in two multivariate mixed models for which it was unknown whether maximum likelihood estimators are consistent. The models we consider have non-stochastic predictors and multivariate responses which are possibly mixed-type (some discrete and some continuous).

math.ST↗

arXiv2017

Directional Metropolis-Hastings

We propose a new kernel for Metropolis Hastings called Directional Metropolis Hastings (DMH) with multivariate update where the proposal kernel has state dependent covariance matrix. We use the derivative of the target distribution at the current state to change the orientation of the proposal distribution, therefore producing a more plausible proposal. We study the conditions for geometric ergodicity of our algorithm and provide necessary and sufficient conditions for convergence. We also suggest a scheme for adaptively update the variance parameter and study the conditions of ergodicity of the adaptive algorithm. We demonstrate the performance of our algorithm in a Bayesian generalized linear model problem.

stat.CO↗

arXiv2017

Multivariate Output Analysis for Markov chain Monte Carlo

Markov chain Monte Carlo (MCMC) produces a correlated sample for estimating expectations with respect to a target distribution. A fundamental question is when should sampling stop so that we have good estimates of the desired quantities? The key to answering this question lies in assessing the Monte Carlo error through a multivariate Markov chain central limit theorem (CLT). The multivariate nature of this Monte Carlo error largely has been ignored in the MCMC literature. We present a multivariate framework for terminating simulation in MCMC. We define a multivariate effective sample size, estimating which requires strongly consistent estimators of the covariance matrix in the Markov chain CLT; a property we show for the multivariate batch means estimator. We then provide a lower bound on the number of minimum effective samples required for a desired level of precision. This lower bound depends on the problem only in the dimension of the expectation being estimated, and not on the underlying stochastic process. This result is obtained by drawing a connection between terminating simulation via effective sample size and terminating simulation using a relative standard deviation fixed-volume sequential stopping rule; which we demonstrate is an asymptotically valid procedure. The finite sample properties of the proposed method are demonstrated in a variety of examples.

math.ST↗