arXiv subjects

Madelyn Clinch

Publications and source records attributed to Madelyn Clinch.

4 recordsLinked to original sources

arXiv2026

Bayesian Analysis Using a Constrained Mixture of Normal-Inverse-Gamma Models

Gaussian mixtures of regressions are commonly implemented via a Gibbs sampler. This Markov chain Monte Carlo (MCMC) algorithm can be computationally burdensome because of the need to update discrete-valued latent component allocation parameters whose dimension increases as the sample size increases. In this article, we propose applying the method of composition to a Gaussian finite mixture model with a Normal-Inverse-Gamma (NIG) prior which allows one to write the posterior distribution as the product of conditional distributions. Namely, the conditional distribution of parameters given the data and mixture labels, times the marginal posterior of the mixture labels. The conditional distribution of parameters given the data and mixture labels, can be sampled from directly, instead of using MCMC. The expression of the marginal posterior of the mixture labels is known up to a proportionality constant and we adapt existing approaches in Bayesian selective inference to constrain the space of component labels to those arising from preliminary estimators, which alleviates a commonly encountered bottleneck. In simulation studies, we consider several settings and compare several versions of our constrained mixture of NIG models to two different MCMC-based strategies and demonstrate their use on natality data from the CDC.

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arXiv2025

An Efficient Class of Bayesian Generalized Quadratic Nonlinear Dynamic Models with Application to Birth Rate Monitoring

Many real-world spatio-temporal processes exhibit nonlinear dynamics that can often be described through stochastic partial differential equations. These models are flexible and scientifically motivated, however, implementing them in a fully Bayesian framework can be computationally challenging. We are motivated by birth rate data, which has important implications for public health and are known to follow nonlinear dynamics. We propose a covariance calibration strategy that specifies the covariance matrix of a linear mixed effects model to be close in Frobenius norm to that of a Generalized Quadratic Nonlinearity (GQN) model. We refer to this as Frobenius norm matching. This allows us to model nonlinear dynamics using an easier to implement linear framework. The calibrated linear model is efficiently implemented using Exact Posterior Regression (EPR), a recently proposed Bayesian model that enables sampling of fixed and random effects directly from the posterior distribution. We provide simulation studies that compare to implementations using MCMC. Finally, we use this approach to analyze Florida county-level birth rate data from 1990-2023. Our results indicate that our non-linear spatio-temporal model outperforms linear dynamic spatio-temporal models for this data, and identifies covariate effects consistent with existing literature, all while avoiding the computational difficulties of MCMC.

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arXiv2024

Exact Bayesian Inference for Multivariate Spatial Data of Any Size with Application to Air Pollution Monitoring

Fine particulate matter and aerosol optical thickness are of interest to atmospheric scientists for understanding air quality and its various health/environmental impacts. The available data are extremely large, making uncertainty quantification in a fully Bayesian framework quite difficult, as traditional implementations do not scale reasonably to the size of the data. We specifically consider roughly 8 million observations obtained from NASA's Moderate Resolution Imaging Spectroradiometer (MODIS) instrument. To analyze data on this scale, we introduce Scalable Multivariate Exact Posterior Regression (SM-EPR) which combines the recently introduced data subset approach and Exact Posterior Regression (EPR). EPR is a new Bayesian hierarchical model where it is possible to sample independent replicates of fixed and random effects directly from the posterior without the use of Markov chain Monte Carlo (MCMC). We extend EPR to the multivariate spatial context, where the multiple variables may be distributed according to different distributions. The combination of the data subset approach with EPR allows one to perform exact Bayesian inference without MCMC for effectively any sample size. Additional motivation is provided via technical results illustrating favorable Kullback-Leibler and covariance properties. We demonstrate SM-EPR using a motivating big remote sensing data application and provide several simulations.

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arXiv2022

Generating Independent Replicates Directly from the Posterior Distribution for a Class of Spatial Latent Gaussian Process Models

Markov chain Monte Carlo (MCMC) allows one to generate dependent replicates from a posterior distribution for effectively any Bayesian hierarchical model. However, MCMC can produce a significant computational burden. This motivates us to consider finding expressions of the posterior distribution that are computationally straightforward to obtain independent replicates from directly. We focus on a broad class of Bayesian latent Gaussian process (LGP) models that allow for spatially dependent data. First, we derive a new class of distributions we refer to as the generalized conjugate multivariate (GCM) distribution. The GCM distribution's theoretical development is similar to that of the CM distribution with two main differences; namely, (1) the GCM allows for latent Gaussian process assumptions, and (2) the GCM explicitly accounts for hyperparameters through marginalization. The development of GCM is needed to obtain independent replicates directly from the exact posterior distribution, which has an efficient projection/regression form. Hence, we refer to our method as Exact Posterior Regression (EPR). Illustrative examples are provided including simulation studies for weakly stationary spatial processes and spatial basis function expansions. An additional analysis of poverty incidence data from the U.S. Census Bureau's American Community Survey (ACS) using a conditional autoregressive model is presented.

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