Uncertain Reasoning and

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Uncertain Reasoningand Decision TheoryWolfgang Spohn

Multi-disciplinary approaches to reasoning with imperfect information and knowledge – a synthesis and a roadmap of challengesDagstuhl Seminar, May 25-29, 2015

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Many Measures of UncertaintyBrief historical remarks:Probability (ca.

1650, Pascal, Huygens)Sub-additive probability (18th century, Bernoulli, Lambert)Frank Knight, Risk, Uncertainty, and Profit, 1921George Shackle, Expectation in Economics, 1949 (distributional and non-distributional uncertainty = probability and possibility)And then there is an explosion of theories and measures in the 1970’s and 1980’s, which we are still about to explore.2Slide3

Many Measures of UncertaintyPoint measures (taking specific values):Probability measures

Capacities (Choquet 1953)Functions of potential surprise (Shackle)DS belief functions (Dempster 1967, Shafer 1976)Prospect theory (Kahneman, Tversky)Fuzzy logic (Zadeh)Possibility measures (Dubois, Prade) (almost = Shackle)Ranking functions (Spohn) (formally, but not interpretationally equivalent to possibility measures)Plausibility measures (Halpern, Reasoning about Uncertainty, 2003)3Slide4

Many Measures of UncertaintyInterval Measures (taking convex sets of specific values):Convex sets of probability measures

Representable as sub-additive probabilities, or as upper and lo-wer probabilities, and as DS belief functionsThe idea does not seem to be pursued for other point measures of uncertainty.Second order measuresSubjective probabilities for objective or my own subjective proba-bilitiesRanking functions over sets of ranking functionsThe idea is certainly viable for other measures as well, but I am not aware that it has been carried out and provided with a sens-ible interpretation.4Slide5

Mixtures of Uncertainty MeasuresHomogeneous mixturesYou may mix uncertainty measures by a measure of the same kind; this possibility is important for various purposes.

This is indeed presupposed by the second order measures.Such mixtures are feasible for probability measures, ranking functions, hence also possibility measures, and presumably other measures as well; however, one would have to check. Heterogeneous mixturesMore interesting are situations in which two kinds of uncertainty are mixed.White’s (2010) mixture of point and interval probabilities: the coin gameI assume that with a little inventiveness more cases of such he-terogeneous mixtures can be constructed. However, the field seems entirely underexplored.5Slide6

Mandatory Theoretical AchievementsThe StaticsWhen theorizing about uncertainty you have to deliver the structure or the static laws of your uncertainty measure.

In principle this is done simply by defining the measure; however, you have to unfold all the theorems entailed by the definition.This includes determining the structure of the domain of the un-certainty measure. This is usually a Boolean algebra (of proposi-tions), but maybe the propositions have a richer structure (quanti-fication, complete algebras, other sentential operators like modals and conditionals, etc.)6Slide7

Mandatory Theoretical AchievementsThe DynamicsUncertainty changes under the impact of information. Therefore, you also have to state the dynamic laws of your uncertainty measure.Usually, these laws take the form of conditionalization rules, which presuppose, though, that you have defined conditional degrees of uncertainty. (Only (?) exception: DS belief functions.)

You also need to model information. Often it is taken to be just a pro-position.Often, though, it is taken to be an uncertainty measure by itself (Jeff-rey conditionalization for probabilities and for ranks, Dempster’s rule of combination). This can be done only if homogeneous mixtures of that uncertainty measure are defined.Ultimately, this requires getting clear about the nature of evidence. A deeply philosophical issue. And a deeply contested issue, e.g., bet-ween Bayesianism and Dempster-Shafer theory, which has never been cleared up, not even tackled in a satisfying way.7Slide8

The Normative PerspectiveUncertainty measures represent degrees of belief and hence epistemic states. However, epistemic states are essentially normatively constitu-ted. They collapse when the deviate too far from the epistemic norms. They cannot be grasped in a purely empirical way side-stepping the normative issues.Hence, each uncertainty measure must justify the normative force of its basic properties. Deplorably, this issue is explicitly addressed only in philosophy and at best vaguely in other disciplines. Just professing normative intuitions is certainly not good enough.

Probability theory satisfies this requirement. Maybe this virtue trans-mits to all derivative representations like upper and lower probabilities, etc. However, additional arguments seem to be needed here.Ranking theory does as well. So do plausibility measures, if only be-cause they are so weak. I see little normative discourse about possibi-lity theory. Also, the normative foundations of DS belief functions ap-pear quite dim to me.8Slide9

The Empirical PerspectiveThe normative perspective also provides an idealized empirical perspective; and, as mentioned, we may not deviate from the ideal too far without losing the concept of an epistemic state. However, the ideal does not ex-haust the empirical perspective.

Within the empirical perspective, the operationalization of the central notions is of first importance. So, how is belief usually operationalized? As far as I see, it’s just assent and dissent: if you assent to “p”, you believe that p, and if not, not. In philosophy we call this the disquotation prin-ciple, and there is a vigorous discussion about it.9Slide10

The Empirical PerspectiveUsually, we have nothing better to go for, and often all goes well. However, one must be aware how deeply pro-visional this is, how many things can go wrong with this:How serious and sincere is the subject?How good is his linguistic grasp?Assent to “

p” may express quite different things than belief in p simply because of the complex pragmatics of utterances of “p”.What about the internal coherence of that assent? Perhaps the assent would change every 5 minutes and hence violate any persistence conditions for belief?Does assent to “p” agree with the subject’s perceptions? Maybe we have reasons to think that he perceived something incoherent with p?Maybe his actions are not in agreement with the belief in p? May-be he talks this way and acts that way. What does he belief then?10Slide11

The Empirical PerspectiveWe also have to operationalize degrees of belief. Since degrees of belief are introspectively vague, subjects are usually asked to position themselves on a 7-point (or slightly coarser or finer) scale. I am deeply wondering about how one might discriminate between the various theoretical models of uncertainty on such a base.From the theoretical side, however, one might at least ex-pect that such a model provides a rigorous measurement theory for its uncertainty measure (so that we would

theo-retically know how to go about measuring).Probability theory meets this requirement.Ranking theory does so as well.However, I have not seen any measurement theory for all the other uncertainty measures.11Slide12

The Entanglement of the Normative and the Empirical Perspective

Let me simply emphasize once more how deeply entang-led the normative and the empirical perspective are.We have a complicated normative reflective equilibrium.We feed this in as an empirical ideal into empirical consi-derations. And then we try to find an empirical reflective equilibrium.If the ideal does not fit, this may have repercussions on the normative reflective equilibrium. And so on.All in all, this double reflective equilibrium is a most deli-cate affair. But it can neither be avoided nor circumvent-ed. In particular, no empirical account without normative considerations!12Slide13

Implications for Decision TheoryOften we engage in uncertain reasoning only for theoreti-cal purposes. Often we do so also for practical purposes. And most practical deliberations are of this kind;

deci-sions under certainty are a rare exception.Hence, we might expect each uncertainty measure to be extended to a kind of decision theory.Historically, standard decision theory (i.e., the principle of maximizing expected utility) emerged at the same time as probability theory. This is still the paradigm.13Slide14

Implications for Decision TheoryEconomists developed alternative uncertainty measures precisely with this goal in mind. I am not aware that they have developed alternative decision theories in a theore-tical richness comparable to the standard theory.

In particular, I have never heard of an alternative game theory. (Recall that Nash equilibria are a fundamentally probabilistic notion.)As far as I know, the AI community has much less attend-ed the issue. In particular, I am not informed about a DS or a possibilistic decision theory.There is, however, a decision-theoretic extension of ranking theory.14Slide15

Requirements for a Decision-Theoretic ExtensionThe fundamental requirement is that each decision theo-ry must account in some way or other for the weighing of the value and the uncertainty of the consequences of our actions.

This translates into various formal requirements.First, you need not only a representation of uncertainty, but also of values. The paradigm are standard utility functions measured on an interval scale (because they smoothly combine with probability measures).There is no reason why this should be the only paradigm.15Slide16

Requirements for a Decision-Theoretic ExtensionFor, secondly, the notion of utility must formally fit to your uncertainty measure. That is, your uncertainty measure must be such as to allow for theory of expectation or inte-gration, and the utility function must be

integrable or al-low for expected values wrt to the uncertainty measure.This requirement is satisfied for standard utility functions and probability measures; both combine to standard decision theory.Most alternative uncertainty measures seem to satisfy this re-quirement, apparently on the basis of standard utility functions. Ranking theory, though, also combines with a different kind of utility functions.The formal possibilities seem severely underexplored.16Slide17

Making Sense of a Decision-Theoretic ExtensionFormal feasibility is only a precondition. Again, you have to observe all the points discussed above wrt to uncertain-ty measures.

You have to account for the (static) structure of decision models and for their dynamics, i.e., a theory of dynamic choice.You have to normatively justify your decision rule, i.e., whatever replaces the maximization if expected utility.You have to fit in the normative ideal into your empirical investigations,also by appropriately operationalizing your notion of utility.17Slide18

Making Sense of a Decision-Theoretic ExtensionI am aware that those alternative decision theories are a very experimental field: very scattered with various interesting ideas and no leading paradigms.Maybe I am not sufficiently informed. But this field still seems thoroughly underresearched. I wish more would be done here in a systematic way.

Thanks for your attention!18