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# Robust Bayes Factor for Independent Two-Sample Comparisons under Imprecise Prior Information

*Proceedings of the Eleventh International Symposium on Imprecise Probabilities: Theories and Applications*, PMLR 103:167-174, 2019.

#### Abstract

This paper proposes the robust Bayes Factor as a direct generalization of the conventional Bayes Factor for a special case of independent two-sample comparisons. Such comparisons are of great importance in psychological research, and more generally wherever the scientific endeavour is to ascertain a potential group effect. The conventional Bayes Factor as the ratio of the marginal likelihoods under two considered hypotheses demands for a precise, subjective specification of the prior distribution for the parameter of interest. Thus, it lacks the possibility of incorporating prior knowledge that is only available partially. Drawing on the theory of Imprecise Probabilities, the

*robust*Bayes Factor is presented in view of lifting the restrictions on the specification of the prior distribution as being precise. In practice, the robust Bayes Factor approach enables an analyst to specify hyperparameter*intervals*, whose lengths correspond to the degree of subjective prior uncertainty. Based thereon, a set of (infinitely) many subjective prior distributions is established to substitute one precise prior distribution. Finally, the robust Bayes Factor is defined as an interval, bounded by the minimal and the maximal resultant Bayes Factor values. Latter are obtained by optimizing the conventional Bayes Factor over the predefined set of prior distributions. This explicit incorporation of incomplete prior knowledge increases the feasibility of applying a Bayesian approach to hypothesis comparisons in scientific practice. It reduces error-proneness, enables for an inclusion of multiple perspectives and encourages cautious, more realistic conclusions in hypothesis comparisons.