Page 136 - DCAP310_INTRODUCTION_TO_ARTIFICIAL_INTELLIGENCE_AND_EXPERT_SYSTEMS

Introduction to Artificial Intelligence & Expert Systems

Notes Dutch Book Approach
The Dutch book argument was proposed by de Finetti, and is based on betting. A Dutch book is
made when a clever gambler places a set of bets that guarantee a profit, no matter what the
outcome of the bets. If a bookmaker follows the rules of the Bayesian calculus in the construction
of his odds, a Dutch book cannot be made.
However, Ian Hacking noted that traditional Dutch book arguments did not specify Bayesian
updating: they left open the possibility that non-Bayesian updating rules could avoid Dutch
books. For example, Hacking writes” And neither the Dutch book argument, nor any other in
the personalist arsenal of proofs of the probability axioms, entails the dynamic assumption. Not
one entails Bayesianism. So the personalist requires the dynamic assumption to be Bayesian. It
is true that in consistency a personalist could abandon the Bayesian model of learning from
experience. Salt could lose its savour.”

!
Caution There are non-Bayesian updating rules that also avoid Dutch books. The additional
hypotheses sufficient to (uniquely) specify Bayesian updating are substantial, complicated,
and unsatisfactory.

Decision Theory Approach

A decision-theoretic justification of the use of Bayesian inference (and hence of Bayesian
probabilities) was given by Abraham Wald, who proved that every admissible statistical
procedure is either a Bayesian procedure or a limit of Bayesian procedures. Conversely, every
Bayesian procedure is admissible.

7.1.3 An Alternative to Bayes: The Stanford Certainty Factor Algebra

It is a more general approach to representing uncertainty than the Bayesian approach. Particularly
useful when decision to be made is based on the amount of evidence that has been collected. It
is appropriate for combining expert opinions, since experts do differ in their opinions with a
certain degree of ignorance. This Approach distinguishes between uncertainty and ignorance by
creating belief functions. This also assumes that the sources of information to be combined are
statistically independent. The basic idea in representing uncertainty in this model is:
Set up a confidence interval – an interval of probabilities within which the true probability lies
with a certain confidence – based on the Belief B and plausibility PL provided by some evidence
E for a proposition P.
MB and MD can be tied together
CF(H|E) = MB(H|E) - MD(H|E)
As CF approaches 1 the confidence for H is stronger
As CF approaches -1 the confidence against H is stronger
As CF approaches 0 there is little support for belief or disbelief in H

A basic probability m(S), a belief BELIEF(S) and a plausible belief PLAUSIBILITY(S) all have
value in the interval [0,1].

Combining Beliefs

To combine multiple sources of evidence to a single (or multiple) hypothesis do the following:
Suppose M and M are two belief functions. Let X be the set of subsets of W to which M assigns
1 2 1

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