Abstract

We present the Bayesian Discrepancy Measure (BDM), a novel approach for the assessment of a single precise hypothesis in parametric models. This measure allows to evaluate the suitability of a given hypothesis with respect to the available information (prior law and data).

In the scalar case, the definition of the BDM is based on the evaluation of the discrepancy between the posterior median and the fixed hypothesis. The measure can be generalized to more complex settings, the extension is straightforward in presence of nuisance parameters, challenges arise in multivariate settings. BDM has several advantageous properties: it is invariant to reparametrizations, does not suffer from the Jeffreys-Lindley paradox, and is asymptotically consistent under regularity conditions making it a robust tool for evaluating hypotheses in the Bayesian framework. Furthermore, unlike traditional frequentist methods, BDM does not depend on asymptotic assumptions.

To show the simplicity and interpretability of the BDM several examples will be presented. In the case of regular models, some of these examples will involve parameters and scenarios that are rarely addressed in the existing literature.
We also explore BDM’s versatility in non-regular models, focusing on those in- volving non-continuous parameters or in which the support depends on parameters, areas often underexplored in existing literature. Simulation studies provide insights into BDM’s performance under non-standard conditions, revealing its strengths and limitations when regularity assumptions are violated.
Finally, we will discuss problems linked to the extension of the BDM to the multivariate setting.

Bertolino F., Columbu S., Manca M., Musio M. (2023) Comparison of two coefficients of variation: a new Bayesian approach. Communications in Statistics - Simulation and Computation, 1--14, DOI: 10.1080/03610918.2023.2231179

Bertolino F., Manca M., Musio M., Racugno W., Ventura L. (2024) A new Bayesian discrepancy measure. Statistical Methods & Applications, 33, 381--405