pARIX: An index for causal induction as extended pARIs/JACCARD index

Intensity of human causal induction is expressed by statistical models in cognitive science. A statistical index, pARIs, which is one of the models of human causal induction, has good performance on some tests. The form of pARIs is nearly equal to one of Jaccard index. However, it has not been known yet why pARIs can be related to intensity of human causal induction. In this study, we propose pARIX (i.e. pARIs-eXtended), in which explanatory variables of pARIs are replaced from Bernoulli random variables to general continuous random variables. pARIX is applied to Gaussian random variables, and is compared with correlation coefficient and mutual information. Moreover, response of pARIX for parameter damping in which a cubic function gets close to a linear function is researched. When the given data pattern is regarded as the thing based on some function, high value of pARIX implies high degree of “single-valued-ness” and “injectivity (one-to-one-ness)” in neighborhood of a selected domain.