A Bayesian Framework for Robust Small-Sample Inference under Model Uncertainty

Thesis Abstract

In many applied statistical settings, inference from small samples is severely compromised by model misspecification and the absence of a universal likelihood model, leading to biased estimators, inflated Type I error, and unreliable predictive accuracy. This study develops a Bayesian framework for robust small-sample inference under model uncertainty, addressing the dual challenges of limited data and competing candidate models. The aim is to formulate a principled approach that combines robust prior elicitation, scalable computation, and principled model averaging to yield reliable parameter estimates and predictive inferences when standard likelihood-based methods fail. Specific objectives are (i) to construct a hierarchical Bayesian model class that integrates objective priors, robust loss-based adjustments, and explicit model uncertainty via Bayesian model averaging (BMA); (ii) to derive computationally efficient algorithms for posterior inference using sequential Monte Carlo and variational Bayes with calibration for small-sample regimes; (iii) to establish diagnostic criteria and sensitivity analyses for prior robustness and model-averaging weight stability; (iv) to evaluate performance on synthetic data under controlled misspecification and on real-world small-sample datasets from biomedical and environmental contexts; and (v) to formulate practical guidelines for practitioners on reporting uncertainty under model uncertainty in small samples. The methodology adopts a mixed-methods structure anchored in Bayesian decision-theoretic principles and information-theoretic robustness. The research design comprises simulation experiments, perturbation analyses, and empirical case studies. The population of interest includes statistical models applicable to small-sample inference, such as generalized linear models, hierarchical mixed-effects models, and time-series models, under competing likelihood specifications. A minimum of 200 synthetic datasets per scenario, spanning sample sizes n = 20 to n = 150, will be generated to assess estimator bias, posterior coverage, and predictive accuracy across misspecification degrees. Real-data applications will involve up to 15 case studies drawn from oncology clinical trials with early-stopping designs, environmental monitoring with sparse events, and econometric small-sample policy evaluations. Data collection instruments comprise simulated data generators for model misspecification scenarios, publicly available biomedical and environmental datasets, and standardized data preparation pipelines. Validity and reliability will be ensured through replication across multiple seeds, cross-validation for predictive checks, and pre-registration of analysis plans. Analytical techniques include the specification of a robust Bayesian model class built from a family of candidate likelihoods with shared prior structure, selection of objective priors (e.g., reference priors) augmented by loss-based robustifications, and implementation of Bayesian model averaging to quantify model uncertainty. Inference will be conducted via particle-based methods, notably sequential Monte Carlo samplers, complemented by variational Bayes approximations to scale to higher-dimensional settings. Model specification will incorporate a hierarchical prior on model weights and model-specific hyperparameters to capture between-model uncertainty, with posterior predictive checks used to evaluate adequacy. Hypothesis testing will be reframed as posterior probability statements, and decision rules will be guided by anticipated utility under small-sample constraints. Theoretical contributions include proofs of finite-sample posterior concentration under bounded misspecification and bounds on posterior risk for model-averaged estimators. Expected findings indicate that the proposed Bayesian framework yields superior posterior coverage and reduced mean squared error relative to single-model Bayesian and frequentist methods in severe small-sample and misspecification scenarios. The framework is anticipated to display stable model-weighting behavior under prior perturbations and provide robust predictive intervals that adapt to the degree of model uncertainty. The study will contribute to knowledge by formalizing a robust, scalable Bayesian approach for small-sample inference that explicitly accounts for model uncertainty, offering practical guidelines for practitioners on prior specification, model averaging, and uncertainty reporting. The main conclusion is that robust small-sample inference under model uncertainty benefits from integrating model-averaged Bayesian decision theory with robust priors and scalable computation, yielding improved inferential reliability and interpretability. Recommendations include adopting the framework in preliminary analyses for high-stakes decisions with limited data, standardizing reporting of posterior model probabilities and predictive intervals, and extending the methodology to nonparametric and multivariate settings where small samples and model uncertainty interact.