Prior selection for the precision parameter of Dirichlet Process Mixtures

Abstract: Consider a Dirichlet process mixture model (DPM) with random precision parameter $\alpha$, inducing $K_n$ clusters over $n$ observations through its latent random partition. Our goal is to specify the prior distribution $p\left(\alpha\mid\boldsymbol{\eta}\right)$, including its fixed parameter vector $\boldsymbol{\eta}$, in a way that is meaningful. Existing approaches can be broadly categorised into three groups. Those in the first group depend on the sample size $n$, and often rely on the linkage between $p\left(\alpha\mid\boldsymbol{\eta}\right)$ and $p\left(K_n\right)$ to draw conclusions on how to best choose $\boldsymbol{\eta}$ to reflect one's prior knowledge of $K_{n}$; we call them sample-size-dependent. Those in the second and third group consist instead of using quasi-degenerate or improper priors, respectively. In this article, we show how all three methods have limitations, especially for large $n$. Then we propose an alternative methodology which does not depend on $K_n$ or on the size of the available sample, but rather on the relationship between the largest stick lengths in the stick-breaking construction of the DPM; and which reflects those prior beliefs in $p\left(\alpha\mid\boldsymbol{\eta}\right)$. We conclude with an example where existing sample-size-dependent approaches fail, while our sample-size-independent approach continues to be feasible.