Abstract

In contemporary statistical practice, many parameters are often introduced into the model to enhance its flexibility and realism, even though the main focus of inference usually revolves around a much lower number of parameters of interest. Dealing with many nuisance parameters in a Bayesian framework entails the cumbersome task of eliciting prior distributions over all of them and then performing high-dimensional integration, which can significantly increase computational complexity. Moreover, it is desirable that the chosen priors for the nuisance components have minimal impact on the conclusion on inference of the parameters of interest, but at the same time the choice of improper priors in such scenarios render the use Bayes Factor impractical. We consider the problem of hypothesis testing and examine two Bayesian evidence measures, namely the Bayesian evidence value (ev), proposed by Pereira and Stern (1999) and the Bayesian Discrepancy Measure (BDM) introduced by Bertolino et al. (2023, 2024), for the purpose of testing precise null hypotheses within a potentially high-dimensional setting that includes nuisance parameters, as also considered by Cabras et al. (2015). We specifically focus on some approximations of the two evidence measures, necessitated by the difficulty of computing high-dimensional integrals and driven by desirable invariance properties. These approximations, which are built on asymptotic expansions of the posterior distribution (see e.g. Ventura and Reid, 2014), are fast-converging to the original evidence measures, demonstrate good calibration, and furthermore ensure that the results are invariant to reparameterizations of the model. The main advantage of the proposed approaches is that no high-dimensional integrals need to be computed. We discuss properties and differences between the two methods as well as their advantages and disadvantages. We also present computational strategies that enable the practical implementation of the proposed methods. This is based on joint works with Francesco Bertolino, Monica Musio, Laura Ventura.

References

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Bertolino, F., Manca, M., Musio, M., Racugno, W., Ventura, L. (2024) A new Bayesian
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Cabras, S., Racugno, W., Ventura, L. (2015). Higher order asymptotic computation of Bayesian significance tests for precise null hypotheses in the presence of nuisance parameters.
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Pereira, C. A. D. B., Stern, J. M. (2022). The e-value: a fully Bayesian significance measure for precise statistical hypotheses and its research program. São Paulo Journal of Mathematical Sciences, 16(1), 566-584.

Ventura, L., Reid, N. (2014). Approximate Bayesian computation with modified loglikelihood ratios. Metron, 7, 231-245