Prior selection for the precision parameter of Dirichlet Process Mixtures
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| author | Vicentini, Carlo Jermyn, Ian Hyla |
| author_facet | Vicentini, Carlo Jermyn, Ian Hyla |
| contents | Consider a Dirichlet process mixture model (DPM) with random precision parameter $α$, inducing $K_n$ clusters over $n$ observations through its latent random partition. Our goal is to specify the prior distribution $p\left(α\mid\boldsymbolη\right)$, including its fixed parameter vector $\boldsymbolη$, 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 $p\left(α\mid\boldsymbolη\right)$ and $p\left(K_n\right)$ to draw conclusions on how to best choose $\boldsymbolη$ 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 $p\left(α\mid\boldsymbolη\right)$. We conclude with an example where existing sample-size-dependent approaches fail, while our sample-size-independent approach continues to be feasible. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_00864 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Prior selection for the precision parameter of Dirichlet Process Mixtures Vicentini, Carlo Jermyn, Ian Hyla Methodology Applications Consider a Dirichlet process mixture model (DPM) with random precision parameter $α$, inducing $K_n$ clusters over $n$ observations through its latent random partition. Our goal is to specify the prior distribution $p\left(α\mid\boldsymbolη\right)$, including its fixed parameter vector $\boldsymbolη$, 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 $p\left(α\mid\boldsymbolη\right)$ and $p\left(K_n\right)$ to draw conclusions on how to best choose $\boldsymbolη$ 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 $p\left(α\mid\boldsymbolη\right)$. We conclude with an example where existing sample-size-dependent approaches fail, while our sample-size-independent approach continues to be feasible. |
| title | Prior selection for the precision parameter of Dirichlet Process Mixtures |
| topic | Methodology Applications |
| url | https://arxiv.org/abs/2502.00864 |