Bayesian Inference as Probability Transfer Across Sample Spaces

Sample space in probability theory refers to the set of all possible outcomes of an experiment or all possible values of a random variable. Probability measures are defined with respect to an associated sample space. When there is more than one random variable (and hence more than one sample space) involved, interaction of these sample spaces can give rise to subtle yet important issues in both classical and modern contextual probability theories and their applications to cognitive modeling. Here we investigate Bayesian reasoning in humans from the perspective of transportation of probability measures across two different sample spaces: namely, a hypothesis space and an evidence (or data) space. Taking Bayesian inference as a task of constructing probability measures, we advance the Probability Transfer Theory (PTT) in which we postulate that probability transfer (transportation) between hypothesis and evidence spaces is susceptible to the structural similarity between these 2 sample spaces. We report experiments on Bayesian reasoning using the commonly used Mammography Problem, the Lottery Problem, as well as three versions of a Factory Problem, all with the same numerical values of probability but varying degrees of structural similarity between the hypothesis space and the evidence space. We analyze the pattern of overestimation and underestimation of Bayesian posterior probability through receiver operating characteristic (or ROC) curve analysis (adapted from the Signal Detection Theory). We find empirical evidence in support of our PTT model of contextual influence on Bayesian reasoning as probability transport.