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An inductive inference problem will often be structured so that the target variable (hypotheses) are logically distant from the observable events (data). In this situation it may be difficult or impossible to assess the probabilistic connection between them, but it may be possible to decompose the problem through the use of intermediate or explanatory variables. That is, it will often be possible to assess the likelihood of the observed data given some intermediate variable, and the likelihood of that intermediate variable given another, and so on, until the hypotheses of interest are reached. Inferences which incorporate one or more intermediate variables are called hierarchical, cascaded, or multistage inferences. The present paper presents a normative model for the solution of the general hierarchical inference problem. The formulation begins with a formal description of the hierarchical inference tree, including a discussion of various simplifying conditional independence assumptions. The solution is first derived for three special case models of differing structure, and then the algorithm for the general solution is given for two cases: one in which the conditional independence assumptions have been made and one in which they have not. [ABSTRACT FROM AUTHOR]

In many cases Bayes' theorem is an appropriate algorithm for the aggregation of probabilistic evidence. As with other statistical procedures, there are restrictions that must be taken into account. In the present paper we shall comment on several approaches that have been devoted to one of these restrictions; the incorporation of uncertainty about the true state of a datum. A datum is a variable which can be partitioned into equivalence classes. These classes represent the possible data states which will be also called events. In Bayes' theorem an event is an item of information which will be used for revising the opinion about the relative likelihood of hypotheses. In any specific situation only one of the possible events will be the true event. An event may come from a source whose reporting or observational accuracy is not perfect. An example may illustrate the issue. A medical doctor wants to come to a diagnosis. To achieve this he considers several data. One datum might be the result of a medical test, which has three possible states: positive, negative, inconclusive. If the doctor reports the state of the datum to be positive, he may be wrong by whatever reasons. That is, the report of an event must not necessarily coincide with the actual or true event of the datum under consideration. This kind of uncertainty about the true event in any specific situation is a characteristic of the source. In most cases it will reduce the diagnostic impact of an event. Whenever the report of an event and the true event coincide imperfectly, measures of source inaccuracy must be incorporated into Bayes' theorem. The task can be considered as two stage probabilistic induction. The first step is induction from the reported to the actual event, the second is induction from the actual event to the creditation of hypotheses. This is the reason why Gettys and Willke (1969) speak about "cascaded inference." [ABSTRACT FROM AUTHOR]

This paper discusses the application of Bayes's theorem to those cases where the true state of the world is not known with certainty. An algorithm is proposed that relaxes the requirement of Bayes's theorem that the true data state be known with certainty by postulating a true but unobservable elementary event, ω, which gives rise to posterior probabilities which reflect the uncertainty of the data. A derivation is presented for the calculation of Bayesian posterior probabilities which uses as its input these probabilities, rather than the true event, o, which is assumed to be unavailable. Suggestions are made as to the application of this modification of Bayes's theorem to cascaded inference processes. [ABSTRACT FROM AUTHOR]