Source-linked AI summary
Decision-Making with Belief Functions: a Review
Thierry Denoeux
TL;DR
Decision-making with belief functions lacks a settled framework for converting uncertainty into preferences and choices. This review synthesizes classical, belief-function, and imprecise-probability approaches, distinguishing complete from incomplete preferences. It finds that methods coincide with maximum expected utility for Bayesian belief functions but diverge in the general case, while foundational and evaluative questions remain open.
Problem
The paper addresses the need to clarify and compare decision methods for uncertainty represented by belief functions after a longstanding gap in decision-making theory.
Method
The review systematically surveys belief-function decision criteria, classical decision principles, imprecise-probability methods, and competing axiomatic approaches.
Results
All reviewed methods reduce to maximum expected utility when the belief function is Bayesian, but they differ substantially in the general case.
Takeaways & Limitations
Decision methods divide between complete preference constructions and approaches that preserve incomparability to reflect lack of information.
Takeaways & Limitations
Some reviewed methods are computationally costly, including strong-dominance comparisons and e-admissibility via linear programming.
Abstract
from arXiv · showhide
Approaches to decision-making under uncertainty in the belief function framework are reviewed. Most methods are shown to blend criteria for decision under ignorance with the maximum expected utility principle of Bayesian decision theory. A distinction is made between methods that construct a complete preference relation among acts, and those that allow incomparability of some acts due to lack of information. Methods developed in the imprecise probability framework are applicable in the Dempster-Shafer context and are also reviewed. Shafer's constructive decision theory, which substitutes the notion of goal for that of utility, is described and contrasted with other approaches. The paper ends by pointing out the need to carry out deeper investigation of fundamental issues related to decision-making with belief functions and to assess the descriptive, normative and prescriptive values of the different approaches.
1. Introduction
The introduction defines decision problems and classical criteria before positioning belief-function decision methods as responses to uncertainty, ignorance, and competing preference relations.
- Belief functions generalize probability and sets, representing logical information, probabilistic information, or combinations of both.
- The review addresses a decision-making vacuum by surveying methods for decisions under belief-function uncertainty and identifying directions for further research.
- A decision problem specifies finite acts, states, and consequences, with each act mapping states of nature to consequences.
- Preference Relations: Decision methods may produce complete or partial preorders, choice sets, or maximal acts, allowing incomparability when preferences are not complete.
- Classical Criteria: Classical criteria aggregate utilities differently: maximax uses the maximum, maximin the minimum, Hurwicz a minimum–maximum convex combination, and Laplace the average.
Ordered Weighted Average Criterion
The OWA criterion unifies several utility aggregation rules through ordered weights and parameterizes attitudes toward optimism, while extending beyond minimum–maximum aggregation.
- Laplace, maximax, maximin, and Hurwicz are OWA operators obtained from average, maximum, minimum, and minimum–maximum weight vectors.
- OWA weights can be interpreted as probabilities assigned to ordered outcomes, with an optimism degree ranging from minimum to maximum behavior.
- In the investment example, Hurwicz and OWA produce similar results, while OWA recovers the Laplace criterion for β = 0.5.
Axiomatic Arguments
Axiomatic analysis evaluates classical criteria by requiring choice behavior to remain stable under changes that should not alter optimality, supporting some criteria while rejecting others.
- The review presents axioms concerning contraction consistency, relabeling, deletion of duplicate states, and dominance for choice operators.
- Minimax regret violates contraction consistency: adding an act can change the preferred act among the original alternatives.
- Laplace violates duplicate-state deletion: splitting one state and later removing a duplicate can change the investment choice.
- Under regularity assumptions, the axioms imply that choices depend only on each act’s worst and best consequences, supporting Hurwicz.
- Decision under Risk: Expected-utility methods evaluate acts using known probabilities and utility matrices, selecting acts with higher expected utility.
Axiomatic justification
Axiomatic decision theory connects rational preference requirements to utility representation and expected utility, while empirical work documents departures from that principle under ambiguity.
- The von Neumann–Morgenstern theorem links complete preorder, continuity, and independence to a utility representation of lottery preferences.
- The theorem provides a formal justification for utility and the maximum expected utility principle in decision under risk.
- Experimental findings associated with Allais and Ellsberg show violations of maximum expected utility or its underlying axioms in some situations.
- Savage argued that rational decision under uncertainty should maximize expected utility using a subjective probability measure and utility function.
Statistical preference and stochastic dominance
The paper contrasts statistical preference with stochastic dominance as ways to compare uncertain utilities. Statistical preference is complete but can be intransitive, whereas stochastic dominance forms a partial order based on exceedance probabilities.
- Statistical preference ranks X over Y when the probability of X exceeding Y is at least the probability of Y exceeding X.
- Although statistical preference is complete, it is not transitive and can produce preference cycles among three gambles.
- Stochastic dominance declares X at least as desirable as Y when X has at least as great a probability of exceeding every utility threshold.
- Because stochastic dominance is a partial order, it is useful when the utility function is known only up to a nondecreasing transformation.
3. Theory of Belief Functions
Belief functions generalize probability by representing precise information, set-valued knowledge, and ignorance, while supporting an imprecise-probability interpretation. In decision problems, acts transform uncertain states and possibly imprecise consequences into evidential lotteries whose preferences can then be evaluated.
- Theory of Belief Functions: The section recalls belief-function definitions, their probabilistic and set-theoretic relationships, and their role in decision-making under weaker information.
- Theory of Belief Functions: A mass function assigns evidence to subsets of the state space, with positive-mass subsets called focal sets.
- Theory of Belief Functions: Belief and plausibility measure, respectively, how strongly evidence supports a proposition and how little it contradicts that proposition.
- Theory of Belief Functions: When focal sets are singletons, belief functions reduce to probability; broader focal sets represent less precise information, including total ignorance.
- Theory of Belief Functions: The credal set contains all probability measures dominating a belief function, and it is convex; belief functions are coherent lower probabilities, but not all coherent lower probabilities are belief functions.
- Theory of Belief Functions: An act transforms a mass function over states into an evidential lottery over consequences, including cases where consequences themselves are set-valued or underspecified.
4. Extensions of Classical Criteria
Decision criteria for belief functions extend classical approaches by combining ignorance-based aggregation with expected utility, while axiomatic and partial-preference methods address how acts should be compared under incomplete information.
- Lower and Upper Expected Utilities: Lower and upper expectations average, respectively, the minimum and maximum utility within each focal set, collapsing to ordinary expectation for Bayesian mass functions.For logical belief functions, they reduce to the minimum and maximum utility over the focal set.
- Lower and Upper Expected Utilities: The resulting complete preference relations generalize maximin and maximax, representing pessimistic and optimistic attitudes toward uncertainty.The lower-expectation relation uses the least favorable consequence in each focal set, whereas the upper-expectation relation uses the most favorable one.
- Generalized Hurwicz Criterion: Generalized Hurwicz criteria introduce a pessimism index, whose value may be fixed, focal-set dependent, or automatically determined from a normalized nonspecificity measure.The nonspecificity-based model makes the decision-maker more cautious as ambiguity increases.
- Pignistic Criterion: The pignistic criterion converts belief functions into probabilities and maximizes expected utility by averaging mean utility within each focal set, thereby extending Laplace’s criterion.Its pignistic probabilities depend on the selected granularity of the frame of discernment, motivating the prior choice of a betting frame.
- Generalized Minimax Regret: Generalized minimax regret coincides with maximum expected utility when the mass function is Bayesian, but utility-based models may require eliciting values for every consequence subset.This elicitation requirement limits the practical use of the general model.
- Axiomatic Foundations: Jaffray’s axioms characterize a linear utility representation for evidential lotteries, while dominance reasoning can reject pignistic and generalized OWA criteria except when OWA coincides with Hurwicz.Related axiomatic results derive similar utility representations from source-based or Savage-style assumptions.
Criteria based on lower and upper expectations
These criteria compare evidential lotteries using partial preference relations that may preserve incomparability. Strong dominance is conservative but often leaves a large choice set, while interval bound dominance compares more alternatives by spanning all pessimism levels.
- Strong dominance may leave many pairs of mass functions incomparable, often making its resulting choice set too large.
- Strong dominance is difficult to justify outside the imprecise-probability setting.
- Interval bound dominance is weaker than strong dominance and therefore compares more pairs of mass functions.
- Interval bound dominance accepts an act when it is at least as desirable under the Hurwicz criterion for every pessimism index.
- In the example, strong dominance retains all four acts, whereas interval bound dominance selects only f2.
Criteria based on extensions of stochastic dominance
Extensions of stochastic dominance compare belief-function lotteries through belief and plausibility bounds. They recover ordinary first-order stochastic dominance for Bayesian lotteries and connect to maximin, maximax, and strong-dominance criteria.
- Preference relations between induced mass functions can be defined by comparing belief or plausibility of upper-tail events.
- These four credal ordering relations reduce to first-order stochastic dominance when the evidential lotteries are Bayesian.
- For bounded non-decreasing utility transformations, the relations correspond respectively to maximin, maximax, and strong-dominance criteria.
- Statistical preference has been extended in imprecise probability, but corresponding extensions in the Dempster-Shafer setting remain to be explored.
5. Imprecise-Probability View
The imprecise-probability perspective interprets belief functions through compatible probability measures and supplies decision criteria that may preserve incomparability. Maximality is more useful than strong dominance but computationally more demanding, while e-admissibility requires linear programming.
- Because belief functions are coherent lower probabilities, imprecise-probability decision criteria can also be applied in the Dempster-Shafer framework.
- Lower and upper expectations represent bounds on expectations over all probability measures compatible with the belief function.
- Strong dominance can produce a large choice set because its comparison may use different compatible probability measures for different gambles.
- Maximality is weaker than strong dominance, so its maximal choice set is included in the strong-dominance choice set.
- Maximality requires n^2 − n lower-expectation computations, whereas strong dominance uses n lower and n upper expectations, giving strong dominance a computational advantage for many alternatives.
- In the example, both X1 and X2 are maximal and e-admissible, while their lower-expectation differences are negative in both directions.
- E-admissibility directly constructs a choice set but is more costly because testing it requires solving a linear programming problem.
6. Shafer’s Constructive Decision Theory
Shafer’s constructive decision theory replaces utility-centered decision-making with goals attached to descriptions of how things may turn out. Its goal-based score accommodates uncertainty and reduces to maximum expected utility when induced mass functions are Bayesian.
- Shafer proposes basing constructive decision theory on goals rather than utilities.
- A goal is a consequence that the decision-maker values and to which utility is attached regardless of what else happens.
- The constructive framework uses one frame of discernment describing mutually exclusive and collectively exhaustive ways things may turn out.
- The score of an act is formed by subtracting the total weight of goals it precludes from the total weight of goals it achieves.
- When act effects are uncertain, belief and plausibility functions weight the expected goals achieved and precluded.
- For Bayesian induced mass functions, the goal-based method reduces to maximum expected utility with a suitable utility definition.
- In the classification example, the method balances correctness against precision through monotonic goals and weighted scoring of candidate class sets.
7. Conclusions
The review finds that belief-function decision methods converge to maximum expected utility for Bayesian beliefs but diverge in the general case, especially over preference completeness. It also identifies unresolved questions about axiomatic foundations, constructive decision theory, and empirical, normative, and prescriptive evaluation.
- All reviewed methods reduce to maximum expected utility when the belief function is Bayesian, but differ substantially for general belief functions.
- Complete preferences: Complete-preference models include pignistic and Jaffray-style criteria, supported by different axiomatic arguments and generally aggregating utilities within focal sets.
- Complete preferences: Jaffray’s axioms constrain each focal set’s contribution to the worst and best utilities, thereby excluding pignistic and OWA criteria and yielding a local pessimism index.
- Partial preferences: Incomplete-preference models preserve incomparability caused by lack of information and include dominance, stochastic-dominance, maximality, and e-admissibility approaches.
- Constructive decision theory: Shafer’s constructive decision theory forms a separate category by replacing pre-existing elicited utilities with constructed goals represented as subsets of a frame.
- The review concludes that the descriptive value of DS theory and the normative and prescriptive values of its decision methods remain insufficiently assessed.
Ackowledgments
The authors thank two anonymous reviewers and acknowledge research support from Labex MS2T through a French Government-funded future-investment program.
- The authors thank two anonymous reviewers for their useful comments.
- The research was supported by Labex MS2T.
- Labex MS2T was funded through the French Government’s “Investments for the future” program by the National Agency for Research.