Zipf's law arises naturally in structured, high-dimensional data

Zipf’s law, which states that the probability of an observation is inversely proportional to its rank, has been observed in many different domains. Although there are models that explain Zipf’s law in each of them, there is not yet a general mechanism that covers all, or even most, domains. Here we propose such a mechanism. It relies on the observation that real world data is often generated from some underlying, often low dimensional, causes — low dimensional latent variables. Those latent variables mix together multiple models that do not obey Zipf’s law, giving a model that does obey Zipf’s law. In particular, we show that when observations are high dimensional, latent variable models lead to Zipf’s law under very mild conditions — conditions that are typically satisfied for real world data. We identify an underlying latent variable for language, neural data, and amino acid sequences, and we speculate that yet to be uncovered latent variables are responsible for Zipf’s law in other domains.