Reasoning with Consistent Probabilities

The objective of the present paper is twofold: first, to establish explicit conditions for the elicitation of consistent a priori and conditional probabilities for a set of events representing pieces of evidence and hypotheses. Furthermore, an algorithm is proposed which uses these consistent input probabilities to compute lower and upper bounds for higher order joint probabilities. Secondly, problems concerning the aggregation and propagation of probabilistic estimates are considered. It is shown how these could be solved by using the higher order joint probabilities obtained for the elements of the so-called complete set of mutually exclusive atomic events. representing a rule as How to determine a consistent higher order joint probability such as etc. These are usually answered under the very much restrictive assumption that all E. 's represent independent evidence. Also it is often forgotten that although the input probabilities might have been elicited in a consistent way, the resulting higher order joint probabilities might not be consistent at all.