Henry E. Kyburg

Henry E. Kyburg

One purpose -- quite a few thinkers would say the main purpose -- of seeking knowledge about the world is to enhance our ability to make good decisions. An item of knowledge that can make no conceivable difference with regard to anything we might do would strike many as frivolous. Whether or not we want to be philosophical pragmatists in this strong sense with regard to everything we might want to enquire about, it seems a perfectly appropriate attitude to adopt toward artificial knowledge systems. If is granted that we are ultimately concerned with decisions, then some constraints are imposed on our measures of uncertainty at the level of decision making. If our measure of uncertainty is real-valued, then it isn't hard to show that it must satisfy the classical probability axioms. For example, if an act has a real-valued utility U(E) if the event E obtains, and the same real-valued utility if the denial of E obtains, so that U(E) = U(-E), then the expected utility of that act must be U(E), and that must be the same as the uncertainty-weighted average of the returns of the act, p-U(E) + q-U('E), where p and q represent the uncertainty of E and-E respectively. But then we must have p + q = 1.

Henry E. Kyburg

A distinction is sometimes made between "statistical" and "subjective" probabilities. This is based on a distinction between "unique" events and "repeatable" events. We argue that this distinction is untenable, since all events are "unique" and all events belong to "kinds", and offer a conception of probability for A1 in which (1) all probabilities are based on -- possibly vague -- statistical knowledge, and (2) every statement in the language has a probability. This conception of probability can be applied to very rich languages.

Henry E. Kyburg

A number of writers have supposed that for the full specification of belief, higher order probabilities are required. Some have even supposed that there may be an unending sequence of higher order probabilities of probabilities of probabilities.... In the present paper we show that higher order probabilities can always be replaced by the marginal distributions of joint probability distributions. We consider both the case in which higher order probabilities are of the same sort as lower order probabilities and that in which higher order probabilities are distinct in character, as when lower order probabilities are construed as frequencies and higher order probabilities are construed as subjective degrees of belief. In neither case do higher order probabilities appear to offer any advantages, either conceptually or computationally.

Henry E. Kyburg

For many years, at least since McCarthy and Hayes (1969), writers have lamented, and attempted to compensate for, the alleged fact that we often do not have adequate statistical knowledge for governing the uncertainty of belief, for making uncertain inferences, and the like. It is hardly ever spelled out what "adequate statistical knowledge" would be, if we had it, and how adequate statistical knowledge could be used to control and regulate epistemic uncertainty.

Henry E. Kyburg

(l) I have enough evidence to render the sentence S probable. (la) So, relative to what I know, it is rational of me to believe S. (2) Now that I have more evidence, S may no longer be probable. (2a) So now, relative to what I know, it is not rational of me to believe S. These seem a perfectly ordinary, common sense, pair of situations. Generally and vaguely, I take them to embody what I shall call probabilistic inference. This form of inference is clearly non-monotonic. Relatively few people have taken this form of inference, based on high probability, to serve as a foundation for non-monotonic logic or for a logical or defeasible inference. There are exceptions: Jane Nutter [16] thinks that sometimes probability has something to do with non-monotonic reasoning. Judea Pearl [ 17] has recently been exploring the possibility. There are any number of people whom one might call probability enthusiasts who feel that probability provides all the answers by itself, with no need of help from logic. Cheeseman [1], Henrion [5] and others think it useful to look at a distribution of probabilities over a whole algebra of statements, to update that distribution in the light of new evidence, and to use the latest updated distribution of probability over the algebra as a basis for planning and decision making. A slightly weaker form of this approach is captured by Nilsson [15], where one assumes certain probabilities for certain statements, and infers the probabilities, or constraints on the probabilities of other statement. None of this corresponds to what I call probabilistic inference. All of the inference that is taking place, either in Bayesian updating, or in probabilistic logic, is strictly deductive. Deductive inference, particularly that concerned with the distribution of classical probabilities or chances, is of great importance. But this is not to say that there is no important role for what earlier logicians have called "ampliative" or "inductive" or "scientific" inference, in which the conclusion goes beyond the premises, asserts more than do the premises. This depends on what David Israel [6] has called "real rules of inference". It is characteristic of any such logic or inference procedure that it can go wrong: that statements accepted at one point may be rejected at a later point. Research underlying the results reported here has been partially supported by the Signals Warfare Center of the United States Army.

Henry E. Kyburg

Uncertainty enters into human reasoning and inference in at least two ways. It is reasonable to suppose that there will be roles for these distinct uses of uncertainty also in automated reasoning.

Henry E. Kyburg

Surely we want solid foundations. What kind of castle can we build on sand? What is the point of devoting effort to balconies and minarets, if the foundation may be so weak as to allow the structure to collapse of its own weight? We want our foundations set on bedrock, designed to last for generations. Who would want an architect who cannot certify the soundness of the foundations of his buildings?

Bülent Murtezaoğlu, Henry E. Kyburg

Selecting the right reference class and the right interval when faced with conflicting candidates and no possibility of establishing subset style dominance has been a problem for Kyburg's Evidential Probability system. Various methods have been proposed by Loui and Kyburg to solve this problem in a way that is both intuitively appealing and justifiable within Kyburg's framework. The scheme proposed in this paper leads to stronger statistical assertions without sacrificing too much of the intuitive appeal of Kyburg's latest proposal.

Henry E. Kyburg, Michael Pittarelli

We discuss problems for convex Bayesian decision making and uncertainty representation. These include the inability to accommodate various natural and useful constraints and the possibility of an analog of the classical Dutch Book being made against an agent behaving in accordance with convex Bayesian prescriptions. A more general set-based Bayesianism may be as tractable and would avoid the difficulties we raise.

Henry E. Kyburg

A number of writers(Joseph Halpern and Fahiem Bacchus among them) have offered semantics for formal languages in which inferences concerning probabilities can be made. Our concern is different. This paper provides a formalization of nonmonotonic inferences in which the conclusion is supported only to a certain degree. Such inferences are clearly 'invalid' since they must allow the falsity of a conclusion even when the premises are true. Nevertheless, such inferences can be characterized both syntactically and semantically. The 'premises' of probabilistic arguments are sets of statements (as in a database or knowledge base), the conclusions categorical statements in the language. We provide standards for both this form of inference, for which high probability is required, and for an inference in which the conclusion is qualified by an intermediate interval of support.

Henry E. Kyburg

Uncertainty may be taken to characterize inferences, their conclusions, their premises or all three. Under some treatments of uncertainty, the inferences itself is never characterized by uncertainty. We explore both the significance of uncertainty in the premises and in the conclusion of an argument that involves uncertainty. We argue that for uncertainty to characterize the conclusion of an inference is natural, but that there is an interplay between uncertainty in the premises and uncertainty in the procedure of argument itself. We show that it is possible in principle to incorporate all uncertainty in the premises, rendering uncertainty arguments deductively valid. But we then argue (1) that this does not reflect human argument, (2) that it is computationally costly, and (3) that the gain in simplicity obtained by allowing uncertainty inference can sometimes outweigh the loss of flexibility it entails.

Henry E. Kyburg

The idea of fully accepting statements when the evidence has rendered them probable enough faces a number of difficulties. We leave the interpretation of probability largely open, but attempt to suggest a contextual approach to full belief. We show that the difficulties of probabilistic acceptance are not as severe as they are sometimes painted, and that though there are oddities associated with probabilistic acceptance they are in some instances less awkward than the difficulties associated with other nonmonotonic formalisms. We show that the structure at which we arrive provides a natural home for statistical inference.

Henry E. Kyburg, Choh Man Teng

There is available an ever-increasing variety of procedures for managing uncertainty. These methods are discussed in the literature of artificial intelligence, as well as in the literature of philosophy of science. Heretofore these methods have been evaluated by intuition, discussion, and the general philosophical method of argument and counterexample. Almost any method of uncertainty management will have the property that in the long run it will deliver numbers approaching the relative frequency of the kinds of events at issue. To find a measure that will provide a meaningful evaluation of these treatments of uncertainty, we must look, not at the long run, but at the short or intermediate run. Our project attempts to develop such a measure in terms of short or intermediate length performance. We represent the effects of practical choices by the outcomes of bets offered to agents characterized by two uncertainty management approaches: the subjective Bayesian approach and the Classical confidence interval approach. Experimental evaluation suggests that the confidence interval approach can outperform the subjective approach in the relatively short run.