Dempster-shafer Inference with Weak Beliefs

Beliefs specified for predicting an unobserved realization of pivotal variables in the context of the fiducial and Dempster-Shafer (DS) inference can be weakened for credible inference. We consider predictive random sets for predicting an unobserved random sample from a known distribution, e.g., the uniform distribution U(0, 1). More specifically, we choose our beliefs for inference in two steps: (i) define a class of weak beliefs in terms of DS models for predicting an unobserved sample, and (ii) seek a belief within that class to balance the trade-off between credibility and efficiency of the resulting DS inference. We call this approach the Maximal Belief (MB) method. The MB method is illustrated with two examples: (1) inference about μ based on a sample n from the Gaussian model N(μ, 1), and (2) inference about the number of outliers (μi 6= 0) based on the observed data X1, ..., Xn with the model Xi ind ∼ N(μi, 1). The first example shows that MB-DS analysis does a type of conditional inference. The second example demonstrates that MB posterior probabilities are easy to interpret for hypothesis testing.