  Linguistic typology studies the range of structures present in human
language. The main goal of the field is to discover which sets of possible
phenomena are universal, and which are merely frequent. For example, all
languages have vowels, while most---but not all---languages have an /u/ sound.
In this paper we present the first probabilistic treatment of a basic question
in phonological typology: What makes a natural vowel inventory? We introduce a
series of deep stochastic point processes, and contrast them with previous
computational, simulation-based approaches. We provide a comprehensive suite of
experiments on over 200 distinct languages.
