Metaprobability and Dempster-Shafer in Evidential Reasoning

Evidentia.l rea.. '>oning in ex pert systems has often used ad-hoc uncertainty calculi. Although it is generally accepted that probability theory provides a firm theoretical fo undat ion, researchers have found some problems with its usc as a workable uncertainty calculus. Among these problems arc representation ol' ignorance, consistency of probabilistic judgements, and adjustment of a priori judgements with experience. The application of metaprobability theory to evidential reasoning is a new approaeh to solving these problems. We use the Dempstcr-Shafer theory, an altcrnativP theory of evidential reasoning to judge mctaprobability theory as a theory of evidential reasoning. This pap<>r will compare how metaprobability theory and Dcmpster-Shafer theory handle the adjustment of judgements with experience. Section 2 and 3 describe the basics of the metaprobability and Dempster-Shafer theories. Mctaprobability theory deals with higher order probabilities applied to evidential ren.'>oning. Dempster-Shafer theory is a generalization of probability theory which has evolved from a theory of upper�.a. nd lower probabilities. Section 4 describes a thought experiment and the metaprobability and Dempstcr-Shafer ·analysis of the experiment. The thought experiment focuses on forming beliefs about a die from evidence accrued from two sensors: an odd-even sensor, a' nd a large-small sensor. For a large number of tosses, the odd­ even sensor sees half the tosses come up even, and the other half come up odd. For a different set of tosses of the same number, the large-small sensor sees half the tosses come up large and the other half small. Based on these two pieces of evidence, what should be the beliefs about the die? 2. Metaprobability Theory Metaprobability theory deals with probability measures on the space of first-order probability distributions. In turn, the first-order distributions are defined over some domain state-space. In the field of probability theory, metaprobability has been known as "higher order probability" or "hierarchical probability". However, it is more appropriate to refer to the application of higher order probability to evidential reasoning as metaprobability because the order of the probability distribution eorresponds to the order of the meta-levei at which the probability theory is attempting to provide a quantifiable model of the evidential accrual process. Many theorists have considered it only briefly due to the lack of practical applications to rnot.ivatc the usc of metaprobabilities as well as a lack of computational resources to implerrwnt t.hcrn. We believe that evidential reasoning is an appropriate application of metaprobabilitics and t.hat the computational …