What Is a Natural Probability Distribution on the Class of All Continuous Functions: Maximum Entropy Approach Leads to Wiener Measure

While many data processing techniques assume that we know the probability distributions, in practice, we often only have a partial information about these probabilities—so that several different distributions are consistent with our knowledge. Thus, to apply these data processing techniques, we need to select one of the possible probability distributions. There is a reasonable approach for such selection—the Maximum Entropy approach. This approach selects a uniform distribution if all we know is that the random variable if located in an interval; it selects a normal distribution if all we know is the mean and the variance. In this paper, we show that the Maximum Entropy approach can also be applied if what we do not know is a continuous function. It turns out that among all probability distributions on the class of such functions, this approach selects the Wiener measure—the probability distribution corresponding to Brownian motion.