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Decision under Uncertainty
Philippe Smets
TL;DR
Existing uncertainty models often specify beliefs without explaining how to derive decisions from them. This paper axiomatizes a transformation from credibility functions to pignistic probabilities, justifying generalized insufficient reason; decisions then use expected utility with those probabilities.
Problem
The paper addresses how decisions should be derived from belief models when uncertainty is represented by credibility functions rather than probabilities.
Method
The paper uses coherence requirements and special bets to axiomatize a unique transformation that distributes each proposition’s belief equally among its atoms.
Results
The derived pignistic probability function supports expected-utility decisions across belief, possibility, and upper- or lower-probability models.
Takeaways & Limitations
The generalized insufficient reason principle receives an axiomatic justification for converting belief functions into probabilities used in decision-making.
Takeaways & Limitations
The justification applies to belief measures whose major property is monotonicity for sets and relies on coherence between combined bets.
Abstract
from arXiv · showhide
We derive axiomatically the probability function that should be used to make decisions given any form of underlying uncertainty.
1. Introduction.
The paper distinguishes a credal level, where beliefs are represented by general credibility functions, from a pignistic level, where beliefs guide decisions through probability functions. It derives the pignistic probability underlying expected utility for any credibility function, rather than only a particular uncertainty model.
- The expected utility model can be derived from alternative models proposed to quantify beliefs.
- Beliefs manifest at a credal level for entertaining them and a pignistic level for using them to make decisions.
- At the credal level, credibility functions generically include probability, belief, lower-probability, necessity, and dual functions.
- The derived pignistic probability function applies to any credibility function, including the Dempster-Shafer model but not restricted to it.
2. The credibility function.
The credibility function quantifies beliefs over propositions subject to domain, monotonicity, boundary, and equivalence requirements. Coherence requirements for combining credibility spaces yield a unique transformation characterized by a continuous, strictly monotone function.
- Axioms: The axioms require a credibility function to exist uniquely, map propositions to a real interval, be monotonic, satisfy lower and upper limits, and treat doxastically equivalent propositions equally.The lower limit applies to the empty proposition, while the upper limit applies when the total proposition is equivalent to a tautology.
- Credibility spaces: Credibility functions assign belief degrees to propositions in a Boolean algebra over a finite frame of discernment.The credibility space consists of the frame, its algebra, and the credibility function.
- Axioms: Doxastically equivalent propositions receive the same credibility regardless of the algebraic structure in which they occur.This invariance also supports the anonymity theorem under permutations of the frame.
- a-combined credibility spaces: For an a-combined credibility space, the combined credibility is a weighted average: Cr12(At2I) = a Cr1(A1I) + (1-a) Cr2(A2I).The weights correspond to the random generator selecting between the two component spaces.
4. The pignistic probability function.
The section defines a transformation from any credibility function to a pignistic probability function for optimal expected-utility decisions. Its axioms force linearity, preserve existing probabilities, and determine the transformation’s coefficients.
- Definition: The transformation r_9 maps any credibility function Cr on a Boolean algebra to a pignistic probability function P used to maximize expected utility.It depends only on the credibility function and the algebra’s cardinality, not on the nature of its atoms.
- Pignistic axioms: Axiom P2 requires probabilities under combined credibility spaces to mix according to the probability of each context.This axiom implies that the function f in Theorem 2 satisfies f(x) = x.
- Probability consistency: If credibility is already represented by a probability function, the transformation leaves it unchanged: r_9(P) = P.This is the content of Axiom Q3.
- Coefficient determination: Axioms Q1–Q3 determine all coefficients in the linear representation of the pignistic probability function.Q1 ensures atom probabilities add to one when the algebra contains a tautology, Q2 preserves credibility after adding an impossible proposition, and Q3 equates pignistic probabilities with existing probabilities.
- Linearity: Theorem 3 shows that each atom’s pignistic probability is a linear function of the credibility values Cr(A) over the algebra.The coefficients may depend only on the algebra’s size and the atom.
5. C o-credibility function.
The section defines the co-credibility function as the dual of a credibility function and shows that theorem 3 yields the same probability function when either function is used. The resulting probability expression depends on atom counts and credibility values.
- Definition: CoCr is defined as the co-credibility function associated with a credibility function Cr on a propositional space.
- Probability representation: The probability function combines credibility values with coefficients determined by the numbers of atoms in the relevant propositions.The expression uses coefficients a and b that depend only on the number of atoms in A and its complement.
- Theorem 4: Theorem 4 specializes relation (3) using theorem 3, theorem 1, and anonymity axiom P3.
- Duality: Replacing Cr with its dual CoCr in theorem 3 leads to the same probability function P.For any pair (Cr, CoCr), using Cr or its dual is equivalent.
7. Conclusions.
The paper axiomatizes the generalized insufficient reason principle, enabling quantified belief models to produce pignistic probabilities for decision-making. It argues that maintaining separate credal and pignistic levels can yield decisions different from those based on probability alone, especially after conditioning on new evidence.
- 7. Conclusions.: The generalized insufficient reason principle receives an axiomatic justification for belief measures that are monotonic for sets and coherent between combined bets.The principle had previously been proposed intuitively but had not been justified.
- 7. Conclusions.: Any quantified-belief model can transform credal-level beliefs into a pignistic probability for decisions, addressing criticism of belief-, possibility-, and upper/lower-probability models.Decisions then use expected utility theory with the pignistic probability to compute expectations.
- 7. Conclusions.: Given a credibility function Cr, applying the generalized insufficient reason principle yields a pignistic probability that supports selecting the optimal decision through classical expected utility theory.The paper presents this link to practical decision problems as straightforward.
- 7. Conclusions.: A two-level model can produce decisions different from those obtained when only one probability level is considered.The paper points to examples developed in Smets (1989a) to support this claim.
- 7. Conclusions.: Conditioning new evidence should occur at the credal level, because the pignistic probability derived afterward is usually different from the probability obtained by conditioning the original pignistic function.The passage describes conditioning Cr into CrA and deriving the pignistic probability from CrA.