Average Prior Distribution of All Possible Probability Density Distributions

Abstract

Bayes’ rule is universally applied in artificial intelligence and especially in Bayes reasoning, Bayes networks, in decision–making, in generating rules for probabilistic knowledge bases. However, its application requires knowledge about a priori distribution of probability or probability density that frequently is not given. Then, to find at least an approximate solution to a problem, the uniform a priori distribution is used. Do we always have to use this distribution? The paper shows that it is not true. The uniform prior should only be used if there is no knowledge about the real distribution. If however, we possess certain qualitative knowledge, e.g. that the real distribution is the unimodal one, or that its expected value is less than 0.5, then we can use this knowledge and apply a priori distribution being the average distribution of all possible unimodal distributions, instead of the uniform distribution. As a result we will usually get better approximation of the problem solution and will avoid large approximation errors. The paper explains the concept of average distributions and shows how they can be determined with a special method of granulation diminution of elementary events and probability.