The Logic of Prediction

Abstract

We consider the predictions provided by the inductive theories. For these theories predictions are performed by the Inductive Statistical (I-S) inferences. It was noted by Hempel that the IS inference is statistically ambiguous. To avoid this ambiguity we need to use the rules that satisfy the Requirement of Maximum Specificity (RMS). The formal definition of the RMS wasn't given by Hempel. We define the notions of law and probabilistic law, and also the sets of all laws L, and probabilistic laws LP. We prove that the set SPL of Strongest Probabilistic Laws (with the maximum values of conditional probability) contains the set L, so we have L ⊂ SPL ⊂ LP. We prove that the maximum specific rules-the strongest SPL rules for prediction of atoms-satisfy the RMS condition. The maximum specific rules may be used in IS inference. We prove that the set MSR of all Maximum Specific Rules is consistent and the IS inferences based on MSR rules avoid the problem of statistical ambiguity. We define Semantic Probabilistic Inferences (SP-inference) that infer the sets L, LP, SPL, MSR. Finally, we mention the program system 'Discovery', which realize the SP-inference and discovers the sets L, LP, SPL, MSR. This system was applied for solution of many practical tasks (see website www.math.nsc.ru/AP/ScientificDiscovery).

Probabilistic dynamic logic of cognition

“…We have introduced a new concept of prediction Vityaev, 2006b; which is not use a logical inference and replace the ''true'' and ''false'' values by probability. Instead of logical inference we introduced semantic probabilistic inference.…”

“…Expanded definitions in the language of first-order logic (in several different versions) can be found in Vityaev, 2006b;.…”

“…By semantic probabilistic inference (Vityaev, 2006a;Vityaev, 2006b), we shall mean the maximal (that cannot be extended) sequence of probability laws in relation of strict probabilistic inference R 1 @R 2 @ . .…”

“…In this paper, we interpret these concepts in terms of logic and probability: the uncertainty we interpret as a probability, and the process of training as a semantic probabilistic inference Vityaev, 2006a;Vityaev, 2006b;). Obtained Probabilistic Dynamic Logic of Cognition (PDLC) belongs to the field of probabilistic models of cognition (Probabilistic Inductive Logic Programming, 2008;Probabilistic models of cognition, 2006;The Probabilistic Mind, 2008).…”

Comparison of “Discovery” Software System Versus Generic Algorithms Embedded in Microsoft SQL Server Analysis Services

“…1. ) симальные условные вероятности, может обнаруживать непротиворечивую вероятностную аппроксимацию теории предметной области [11], обнаруживает все максимально специфи-ческие правила, позволяющие предсказывать без противоречий [1,12].…”