Comparative Analysis of Bayesian vs. Frequentist Interval Estimation Methods in Practice

Thesis Abstract

This study examines the practical performance and implications of Bayesian and Frequentist interval estimation approaches across diverse applied settings, addressing the gap between theoretical appeal and real-world decision requirements in statistical inference. The problem central to this research is the ambiguous guidance practitioners face when selecting interval estimation methods under varying data conditions, model misspecification, and computational constraints. The aim is to provide a rigorous comparative assessment of coverage probability, interval length, and interpretability of Bayesian credible intervals and Frequentist confidence intervals, and to identify contexts in which each paradigm offers superior decision support. Specific objectives include (i) evaluating nominal vs. empirical coverage across simulated and real-world datasets; (ii) comparing interval width, robustness to outliers and model misspecification, and sensitivity to prior choices in Bayesian analyses; (iii) assessing computational efficiency and convergence behavior of standard Markov chain Monte Carlo (MCMC) methods versus analytical or bootstrap-based Frequentist procedures; (iv) examining the impact of model complexity, sample size, and data-generating mechanisms on interval performance; (v) developing practical guidance for practitioners on method selection under explicit risk and information constraints. The methodology adopts a mixed-methods, cross-sectional research design combining simulation experiments with empirical data analysis drawn from multiple domains. The population comprises statistical analysts and practitioners in biomedicine, environmental science, and social sciences who routinely report interval estimates. A stratified random sample of 180 datasets is used for simulation studies, with 60 datasets per domain reflecting varying effect sizes, variance structures, and sample sizes (n = 30, 100, 300). For empirical analysis, 120 published studies with publicly available data are collected, spanning linear regression, generalized linear models, and hierarchical models; within each study, Bayesian credible intervals and Frequentist confidence intervals are re-estimated under consistent modeling assumptions. Data collection instruments include standardized data extraction templates, replication scripts in R and Python (rstan, PyMC3, and StatsModels), and a survey instrument to capture practitioner preferences and perceived interpretability. Analytical methods comprise both quantitative and qualitative components. For the simulation component, interval coverage probabilities, average interval lengths, and the frequency of interval inclusion of true parameters are computed across conditions, with performance compared using paired t-tests and nonparametric equivalents. Regression-based meta-analytic models are employed to identify factors driving differences in interval performance, incorporating predictors such as prior informativeness, sample size, and model misspecification indicators. Bayesian analyses utilize weakly informative and informative priors to assess sensitivity to prior assumptions, with convergence diagnostics (R-hat, effective sample size) and posterior predictive checks. Frequentist procedures include bootstrap-t confidence intervals and profile likelihood intervals, with comparisons extended to alternative asymptotic approximations where applicable. For empirical studies, meta-regression evaluates how domain, model type, and data quality influence the relative performance of the two approaches. Thematic analysis of practitioner feedback and interpretability concerns is conducted on survey responses to elucidate contextual preferences. Expected findings indicate that Bayesian intervals often achieve shorter credible ranges without substantial loss of coverage when priors are reasonably specified, whereas Frequentist intervals tend to be robust under model misspecification in larger samples but may require bootstrap or simulation-based adjustments in complex models. In small-sample contexts or when prior information is strong and credible, Bayesian intervals may provide more informative and actionable inferences, whereas in high-stakes decisions requiring strict coverage guarantees, Frequentist intervals with robust resampling techniques may perform comparably or better. The study anticipates nuanced domain-dependent patterns, with hierarchical and multilevel models showing greater sensitivity to prior choices and computational demands in Bayesian analyses. The contribution to knowledge includes a comprehensive, evidence-based framework for method selection that integrates empirical performance metrics, computational feasibility, and interpretability considerations across disciplines. It also yields practical, domain-specific recommendations and a set of decision rules for practitioners facing trade-offs between coverage, precision, and prior influence. The main conclusion highlights that neither paradigm universally dominates; instead, the optimal interval estimation strategy depends on sample size, model structure, data quality, prior information, and stakeholder tolerance for uncertainty. Recommendations emphasize transparent reporting of priors, sensitivity analyses, and the adoption of hybrid best-practice guidelines that leverage strengths of both approaches to enhance inferential reliability in applied research.