Bayesian Theory Testing

• Bayesian theory testing is a formal framework that evaluates scientific models by integrating prior beliefs with observed data to compute posterior probabilities.
• It employs Bayes factors, posterior model probabilities, and model evidence based on proper prior specification to compare competing hypotheses.
• It adapts to complex and high-dimensional scenarios, offering rigorous evidence quantification and error control across diverse scientific domains.

Bayesian theory testing is a formal framework for evaluating scientific models, hypotheses, or parameter constraints in light of observed data and explicit prior beliefs. Core to Bayesian theory testing is the allocation of posterior probability mass across alternative theories, models, or parameter regimes, conditioned on both the likelihood of observed data under competing hypotheses and the structure of prior knowledge. This approach encompasses Bayes factors, posterior model probabilities, mixture and distance-based frameworks, Bayesian discrepancy measures, sequential and high-dimensional scenarios, and connections to frequentist error control. Rigorous control of inference operating characteristics, adaptation to model complexity, and principled probabilistic quantification of evidence—rather than arbitrary thresholding of p-values—distinguish Bayesian theory testing as a technically robust methodology with broad applications in the physical sciences, social sciences, experimental design, and model selection.

1. Foundations and Core Frameworks

Bayesian theory testing roots all model comparison in conditional probabilities: the probability of a theory $T$ given data $D$ and background $B$ is

$$P(T\mid D,B) = \frac{P(D\mid T,B) \, P(T\mid B)}{P(D\mid B)},$$

where $P(D\mid T,B)$ is the model evidence or marginal likelihood. When comparing two competing theories $T_1$ and $T_2$, the Bayes factor,

$B_{12} = \frac{P(D\mid T_1,B)}{P(D\mid T_2,B)},$

quantifies the evidence provided by the data in favor of $T_1$ over $T_2$ after marginalizing over the model’s parameters using a normalizable prior distribution (Barnes, 2017).

The model evidence is always a marginalization,

$D$0

and the prior $D$1 must be proper (normalizable) to make Bayesian testing well-defined. Proper prior specification and careful attention to parameter support (e.g., physical bounds such as the Planck scale for dimensional constants) are required to avoid pathological inferences or indeterminate evidences (Barnes, 2017).

Bayesian theory testing yields not only direct model probabilities but also posterior odds, Bayes factors, and, in many modern approaches, full posterior distributions on quantities such as mixture weights or distances to a null set, accommodating both sharp (point) and composite hypotheses (Kamary et al., 2014, Salomond, 2013).

2. Methodological Variants: Bayes Factors, Mixture Models, and Discrepancy Measures

Bayes Factors act as the central default for model selection and are widely used across domains such as regression, network analysis, and scientific theory testing. For example, in regression or ERGM settings with equality or order constraints, the Bayes factor, often computed by Savage–Dickey ratios or encompassing-prior methods, directly quantifies evidence for constraint hypotheses versus unconstrained or competing alternatives (Mulder et al., 2023, Mulder et al., 2018). With proper unit-information priors or default fractional priors,