A Pragmatic Bayesian Acceptance Theory

Bayesian approaches to reasoning under uncertainty have gained substantially in popularity recently, due to the efforts of AI researchers and statisticians to find tractable means of probability propagation in restricted domains, namely those properly described by causal (Bayesian) networks (cf. Pearl 1988 and Neapolitan 1990). Most Bayesians are satisfied with such an approach to automating Bayesian inference: it is normatively correct, occasionally tractable, and supplies precisely what is needed for Bayesian decision theory--the theory that supplies normatively correct means for decision making under uncertainty. Some, however, are concerned to find additional means for reasoning under uncertainty, particularly on the grounds that often causal networks are too complex given some pragmatic problem-solving context. The question is whether there might be some qualitative, computationally simpler, reasoning strategy that can be justified as a cooperative supplement to Bayesian inference. Judea Pearl has offered one such method, using his e-semantics for default reasoning (Pearl, 1989). Pearl’s technique is to base qualitative inferential rules leading from A to B upon "extreme" conditional probabilities--roughly, if infinitesimally many cases of A are not cases of B, then we can adopt the default rule to infer that B is the case given A. Pearl demonstrates that his e-semantics corresponds to a qualitative inferential theory using a simple set of inference rules. So we achieve the desired inferential simplicity for some set of propositions satisfying this particular extreme probability relation. Unfortunately, this relationship is rarely satisfied; as Pearl admits, "probabilities that are infinitesimally close to 0 and 1 are very rare in the real world" (1988, p. 493). So, "why not develop a logic that characterizes moderately high probabilities, say probabilities higher than 0.5 or 0.9--or more ambitiously, higher than a where a is a parameter chosen to fit the domains of the predicates involved?" (ibid.). Pearl’s answer is that such a logic would be extremely complicated. And that would surely be true, if the object were to construct a purely qualitative logic to deal with probabilistic relations between propositions. No qualitative logic, left to its own resources, can approximate probability theory beyond some fairly limited number of inferential steps: this is not just complicated, it is impossible. But if we modify the goal slightly, we seem to have a more plausible target: we can take the value of (~ to specify a probability threshold relative to a particular problem context--including our goals--beyond which we are prepared to employ qualitative methods of inference, under a set of constraints. If the probability of h is greater than a, we accept h and reason with h, for a certain limited number of qualitative inferential steps. The idea here is to simplify the reasoning process by avoiding unnecessary probability calculations when the risk of error introduced thereby is tolerable (a pragmatic, goal-based judgment), without abandoning probabilistic reasoning or attempting to capture such reasoning qualitatively. This is just a version of Bayesian acceptance theory, much debated by philosophers of science in the 1960s and widely, but mistakenly, taken for dead since then. A major reason why probabilistic acceptance theory has not been explored adequately is that Henry Kyburg’s lottery paradox (1961) has been thought to pose an impenetrable barrier to such explorations. The lottery paradox points out that acceptance at a fixed probabilistic threshold leads