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 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 $n$. We enrich the first group by working out and testing Jeffreys' prior in the context of the DPM framework, and by evaluating its behaviour. 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.

翻译：考虑一个具有随机精度参数 $\alpha$ 的狄利克雷过程混合模型（DPM），该参数通过其潜在随机划分在 $n$ 个观测值上诱导出 $K_n$ 个聚类。我们的目标是以一种有意义的方式指定先验分布 $p\left(\alpha\mid\boldsymbol{\eta}\right)$，包括其固定参数向量 $\boldsymbol{\eta}$。现有方法大致可分为三类。第一类方法依赖于 $p\left(\alpha\mid\boldsymbol{\eta}\right)$ 与 $p\left(K_n\right)$ 之间的关联，以得出如何最佳选择 $\boldsymbol{\eta}$ 来反映对 $K_{n}$ 的先验知识；我们称其为样本量依赖型。第二类和第三类方法则分别采用准退化先验或不恰当先验。在本文中，我们展示了所有这三种方法均存在局限性，尤其对于大样本量 $n$。我们通过推导并测试 DPM 框架下的 Jeffreys 先验，并评估其行为，从而丰富了第一类方法。随后，我们提出一种替代方法，该方法不依赖于 $K_n$ 或可用样本的规模，而是依赖于 DPM 的折棍构造中最大棍长之间的关系；并将这些先验信念反映在 $p\left(\alpha\mid\boldsymbol{\eta}\right)$ 中。最后，我们通过一个示例说明，在现有样本量依赖型方法失效的情况下，我们提出的样本量独立方法仍然可行。