# Long-tailed distributions of inter-event times as mixtures of exponential distributions

## TL;DR

This paper investigates the long-tailed distributions of human inter-event times, proposing that they can be modeled as mixtures of exponential distributions, and introduces a novel model selection method based on the minimum description length principle.

## Contribution

It introduces a new approach using the minimum description length principle for selecting mixture models of exponential distributions, addressing challenges due to non-identifiability.

## Key findings

- Mixtures with few components are preferred over complex ones.
- Fitting accuracy is comparable to power-law distribution models.
- Results support Poissonian explanations of human behavior.

## Abstract

Inter-event times of various human behavior are apparently non-Poissonian and obey long-tailed distributions as opposed to exponential distributions, which correspond to Poisson processes. It has been suggested that human individuals may switch between different states in each of which they are regarded to generate events obeying a Poisson process. If this is the case, inter-event times should approximately obey a mixture of exponential distributions with different parameter values. In the present study, we introduce the minimum description length principle to compare mixtures of exponential distributions with different numbers of components (i.e., constituent exponential distributions). Because these distributions violate the identifiability property, one is mathematically not allowed to apply the Akaike or Bayes information criteria to their maximum likelihood estimator to carry out model selection. We overcome this theoretical barrier by applying a minimum description principle to joint likelihoods of the data and latent variables. We show that mixtures of exponential distributions with a few components are selected as opposed to more complex mixtures in various data sets and that the fitting accuracy is comparable to that of state-of-the-art algorithms to fit power-law distributions to data. Our results lend support to Poissonian explanations of apparently non-Poissonian human behavior.