Reward Versus Risk in Uncertain Inference: Theorems and Simulations

Abstract

<jats:title>Abstract</jats:title><jats:p>Systems of<jats:italic>logico-probabilistic</jats:italic>(LP) reasoning characterize inference from conditional assertions that express high conditional probabilities. In this paper we investigate four prominent LP systems, the systems<jats:bold>O, P</jats:bold>,<jats:bold>Z</jats:bold>, and<jats:bold>QC</jats:bold>. These systems differ in the number of inferences they licence (<jats:bold>O</jats:bold>⊂ P ⊂<jats:bold>Z</jats:bold>⊂<jats:bold>QC)</jats:bold>. LP systems that license more inferences enjoy the possible<jats:italic>reward</jats:italic>of deriving more true and informative conclusions, but with this possible reward comes the<jats:italic>risk</jats:italic>of drawing more false or uninformative conclusions. In the first part of the paper, we present the four systems and extend each of them by theorems that allow one to compute almost-tight lower-probability-bounds for the conclusion of an inference, given lower-probability-bounds for its premises. In the second part of the paper, we investigate by means of computer simulations which of the four systems provides the best balance of<jats:italic>reward</jats:italic>versus<jats:italic>risk</jats:italic>. Our results suggest that system<jats:bold>Z</jats:bold>offers the best balance.</jats:p>

Qualitative probabilistic inference under varied entropy levels

“…As described in [16], system P represents the confluence of a number of different semantic criteria. But the feature of system P that is of greatest interest here is its connection with the following consequence relation (cf.…”

“…Following [16], we propose a marriage of system P, and a rule for inferring lower probability bounds that corresponds to the correctness of system P for -i.s.p. , and say that system P licenses the (valid) inference to…”

“…One disadvantage of Z-entailment is that (in the absence of further assumptions) it does not automatically provide information concerning probabilistic reliability, such as provided by the improbability-sum semantics for system P. However, in [16] it is shown how to obtain this desideratum (based on work in [13]):…”

“…As with the evaluations conducted in [16], a central question concerns whether, and in which environments, inference in accordance with the preceding principle tends to yield accurate conclusions.…”

“…In a previous paper [16], we investigated the performance of these reasoning systems in randomly chosen environments, with the unambiguous result that among the four systems, system Z achieved the best balance of reward and risk. In this article, we investigate the degree to which the performance of the four systems depends on the entropy of the environment which is reasoned about.…”