# A hierarchical Bayesian inference model for volatile multivariate exponentially distributed signals

## TL;DR

This paper introduces a hierarchical Bayesian model to analyze complex, time-varying brain signals that follow exponential distributions.

## Contribution

A novel hierarchical Bayesian inference model for multivariate exponential signals with time-varying rate parameters and interactions.

## Key findings

- The model successfully estimates time-varying rate parameters of multivariate exponential signals.
- It captures the underlying correlation structure of volatile exponentially distributed data.
- The model provides closed-form update equations for efficient inference.

## Abstract

Brain activities often follow an exponential family of distributions. The exponential distribution is the maximum entropy distribution of continuous random variables in the presence of a mean. The memoryless and peakless properties of an exponential distribution impose difficulties for data analysis methods. To estimate the rate parameter of multivariate exponential distribution from a time series of sensory inputs (i.e., observations), we constructed a hierarchical Bayesian inference model based on a variant of general hierarchical Brownian filter (GHBF). To account for the complex interactions among multivariate exponential random variables, the model estimates the second-order interaction of the rate intensity parameter in logarithmic space. Using variational Bayesian scheme, a family of closed-form and analytical update equations are introduced. These update equations also constitute a complete predictive coding framework. The simulation study shows that our model has the ability to evaluate the time-varying rate parameters and the underlying correlation structure of volatile multivariate exponentially distributed signals. The proposed hierarchical Bayesian inference model is of practical utility in analyzing high-dimensional neural activities.