CONDITIONAL PROBLEM FOR OBJECTIVE PROBABILITY

Marginal problem (see [5]) consists in finding a joint distribution wh ose marginals are equal to the given less-dimensional distributions. L et's generalize the problem so that there are given not only less-dime nsional distributions but also conditional probabilities. It is necess ary to distinguish between objective (Kolmogorov) probability and subj ective (de Finetti) approach ([4, 10]). In the latter, the coherence p roblem incorporates both probabilities and conditional probabilities i n a unified framework. Different algorithms available for its solution are described e.g. in ([3, 4, 11]). In the context of the former appr oach, it will be shown that it is possible to split the task into solv ing the marginal problem independently and to subsequent solving pure ''conditional'' problem as certain type of optimization. First, an alg orithm (Conditional problem) that generates a distribution whose condi tional probabilities are equal to the given ones is presented. Due to the multimodality of the criterion function, the algorithm is only heu ristical. Due to the computational complexity, it is efficient for sma ll size problems e. g. 5 dichotomical variables. Second, a method is m entioned how to unite marginal and conditional problem to a more gener al consistency problem for objective probability. Due to computational complexity, both algorithms are effective only for limited number of variables and conditionals. The described approach makes possible to i ntegrate in the solution of the consistency problem additional knowled ge contained e.g. in an empirical distribution.