Comparative Analysis of Bayesian vs. Frequentist Methods in Small-Sample Inference

Thesis Abstract

In small-sample inference, statistical practitioners face a persistent tension between Bayesian and Frequentist methodologies regarding reliability, interpretability, and decision-making under uncertainty, particularly in domains where data collection is costly or restricted. This study addresses the comparative performance of Bayesian and Frequentist approaches when sample sizes are limited to 30–50 observations, focusing on parameter estimation, interval estimation, and hypothesis testing under common likelihood-based models. The aim is to delineate conditions under which each paradigm yields superior inference quality and to provide actionable guidance for practitioners encountering small datasets. Specific objectives are (1) to evaluate the bias, mean squared error, and coverage probabilities of posterior credible intervals versus classical confidence intervals across linear regression, generalized linear models, and t-test scenarios; (2) to compare decision-theoretic properties through Type I/II error rates, Bayes factors, and likelihood ratio tests under varying prior specifications and sample sizes; (3) to assess sensitivity to prior choice, model misspecification, and prior-to-data information ratios using robust and weakly informative priors; (4) to examine computational efficiency, convergence diagnostics, and usability considerations of standard software implementations (e.g., Stan, JAGS, and frequentist counterparts in R) in small-sample contexts; and (5) to synthesize findings into practical recommendations for researchers in medicine, psychology, and social sciences where small samples are common. Methodologically, the study adopts a mixed-methods design anchored in quantitative simulation and empirical analysis. A Monte Carlo simulation study with 1,000 replications per scenario will generate synthetic data under three data-generating processes a normal linear model, a logistic regression framework, and a two-sample t-test setting. Each scenario will manipulate sample sizes (n = 20, 30, 50), effect sizes (small, medium, large), and prior informativeness (uninformative, weakly informative, informative) for Bayesian procedures. The population consists of latent population parameters drawn from known distributions to enable precise evaluation of estimator properties, while the sample is drawn from the specified models with controlled variance structures. Data collection instruments are simulated data generators implemented in Python and R, ensuring replicability and standardization across analyses. For the Bayesian analyses, hierarchical priors will be employed with emphasis on robustness through posterior predictive checks and prior sensitivity analyses. Frequentist analyses will include ordinary least squares, maximum likelihood estimation, and classical t-tests, with exact and approximate interval calculations as appropriate. Analytical techniques comprise a comprehensive comparison of estimation accuracy (bias, variance, mean squared error), interval performance (coverage and interval width for credible and confidence intervals), and decision metrics (Type I/II error rates, Bayes factors, and likelihood ratio statistics). Model comparison will utilize information criteria such as WAIC and LOO-CV for Bayesian models and AIC/BIC for frequentist models. Convergence diagnostics (R-hat, effective sample size) and posterior predictive checks will be reported to ensure valid Bayesian inference. Theoretical grounding will leverage the Bernstein–von Mises theorem for asymptotic alignment and decision theory to interpret paradoxes in small samples. The study will consider theoretical expectations from two named theories Fisherian (frequentist) inference and Bayesian decision theory, contrasting their implications for inference under limited data. Expected findings include (i) Bayesian methods with weakly informative priors will frequently yield narrower, better-calibrated intervals than classical methods in small samples with moderate signal-to-noise ratios, (ii) in low-information priors, Bayesian and frequentist conclusions may diverge on hypothesis testing, highlighting the influence of prior assumptions, (iii) model misspecification adversely affects both paradigms but Bayesian methods with robust priors exhibit greater resilience in estimation accuracy, (iv) computational demands are higher for Bayesian analyses but modern probabilistic programming tools render practical implementation feasible for typical postgraduate workloads. The study contributes to knowledge by clarifying the practical trade-offs between Bayesian and Frequentist approaches in small-sample contexts, offering evidence-based guidance on prior selection, model specification, and method choice across common statistical models. It informs methodological standards for researchers dealing with small datasets and contributes to the refinement of best practices in statistical inference under data scarcity. Final recommendations emphasize transparent reporting of prior assumptions, conducting sensitivity analyses, and pairing inference with predictive validation to strengthen decision-making in applied research settings.