Mathematics

Building probabilistic models that are able to capture high-order statistics from data, with limited sample sizes, is a challenging problem. This is important in inferring dependencies and estimating information represented by large dimensional interacting systems. I will propose a nonparametric copula method combined with a graphical and sequential representation known as vine, to estimate the probability density function of large dimensional problems from limited sample sizes approximating all the statistical interaction orders. I will show that using nonparametric vine-copula models, we can generate realistic samples of the data which can then be used to estimate mutual information between large dimensional data. I will then talk about a neural network implementation of vine-copula models based on variational autoencoders. In the end, I will show some results of applying this method in modeling mice’s neuronal activities collected during a navigation decision-making experiment in virtual reality environment. I will show how vine-copula tools can be used to estimate and decompose mutual information of neuronal activities about a high dimensional set of variables. Also, using the sequential structure of the vine-copula, I will show how the information represented by an interacting population of neurons can be decomposed into components corresponding to individual network interaction orders such as pairs, triplets, and higher-order motifs. The results suggest that there is a significant high order information representation in the cortical networks and also the method outperforms existing approaches in modeling the data and inferring its information components.