Metaprobability and Dempster-Shafer in Evidential Reasoning
  Evidential reasoning in expert systems has often used ad-hoc uncertainty
calculi. Although it is generally accepted that probability theory provides a
firm theoretical foundation, researchers have found some problems with its use
as a workable uncertainty calculus. Among these problems are representation of
ignorance, consistency of probabilistic judgements, and adjustment of a priori
judgements with experience. The application of metaprobability theory to
evidential reasoning is a new approach to solving these problems.
Metaprobability theory can be viewed as a way to provide soft or hard
constraints on beliefs in much the same manner as the Dempster-Shafer theory
provides constraints on probability masses on subsets of the state space. Thus,
we use the Dempster-Shafer theory, an alternative theory of evidential
reasoning to illuminate metaprobability theory as a theory of evidential
reasoning. The goal of this paper is to compare how metaprobability theory and
Dempster-Shafer theory handle the adjustment of beliefs with evidence with
respect to a particular thought experiment. Sections 2 and 3 give brief
descriptions of the metaprobability and Dempster-Shafer theories.
Metaprobability theory deals with higher order probabilities applied to
evidential reasoning. Dempster-Shafer theory is a generalization of probability
theory which has evolved from a theory of upper and lower probabilities.
Section 4 describes a thought experiment and the metaprobability and
DempsterShafer analysis of the experiment. The thought experiment focuses on
forming beliefs about a population with 6 types of members {1, 2, 3, 4, 5, 6}.
A type is uniquely defined by the values of three features: A, B, C. That is,
if the three features of one member of the population were known then its type
could be ascertained. Each of the three features has two possible values, (e.g.
A can be either "a0" or "al"). Beliefs are formed from evidence accrued from
two sensors: sensor A, and sensor B. Each sensor senses the corresponding
defining feature. Sensor A reports that half of its observations are "a0" and
half the observations are 'al'. Sensor B reports that half of its observations
are ``b0,' and half are "bl". Based on these two pieces of evidence, what
should be the beliefs on the distribution of types in the population? Note that
the third feature is not observed by any sensor.
