Extension of conditional probability and measures of belief and disbelief in a hypothesis based on uncertain evidence

Conditional probability is extended so as to be conditioned by an uncertain proposition of which truth value is ¿ (0 ¿ ¿ ¿ 1). Using the extended conditional probability, the measure of increased belief and disbelief in a hypothesis resulting from the observation of uncertain evidence are derived from MYCIN's measures based on certain evidence. On the comparison with our measures, it is shown that MYCIN's intuitive measures based on uncertain evidence, called the strength of evidence, utilize affirmative information which increases belief in the uncertain evidence but ignore negative information which causes new doubt in the uncertain evidence. An interpretation of the disregard of the negative information is presented from the viewpoint of cognitive psychology. It is pointed out that this disregard of the negative information is reasonable for a model of human inference but the negative information must also be utilized in order to evaluate a hypothesis correctly, or impartially on the basis of uncertain evidence. Our measures provide a means for utilizing both the affirmative and negative information on uncertain evidence. It is shown that inference based on the negation of evidence, which is contained in one of our measures, is difficult for an expert. A method for estimating the measure is presented which does not demand the difficult inference from an expert. The significance of the method is explained from the viewpoint of cognitive psychology.