Exchangeability and Bayesian Inference: A Theoretical and Computational Framework for Reliable Experimental Data Analysis

Exchangeability is a foundational concept in Bayesian statistics, crucial for ensuring the validity and generalizability of inferences from experimental data. This paper presents a theoretical and computational framework for understanding the role of exchangeability in the reliability of scientific conclusions, with specific reference to psychology, neuroimaging, and clinical trials. We build on de Finetti's representation theorem to show how exchangeability enables using hierarchical Bayesian models, and we analyze how its violation can lead to interpretative errors and paradoxes, such as Simpson's Paradox. In addition to theoretical discussion, we present practical strategies to evaluate and enforce exchangeability, including randomization, matching, stratification, and hierarchical modeling. We also introduce computational tools-such as the Shuffle Test and Stratified Bootstrap-to empirically test for exchangeability and detect latent structures in the data. The novelty of this work lies in unifying theoretical reasoning and empirical testing within a single framework that bridges de Finetti's representation theorem and resampling-based diagnostics. By providing concrete tools to evaluate exchangeability prior to model fitting, the proposed approach introduces a pre-analysis verification step that strengthens the reliability and transparency of Bayesian inference. Our results emphasize that exchangeability is not merely a technical assumption, but a structural property that governs the coherence and informational integrity of the data. This framework provides both conceptual clarity and operational tools for researchers aiming to perform robust Bayesian inference in complex and heterogeneous datasets.