The paper discusses shrinkage priors which impose increasing shrinkage in a sequence of parameters. We review the cumulative shrinkage process (CUSP) prior of Legramanti et al. (2020), which is a spike-and-slab shrinkage prior where the spike probability is stochastically increasing and constructed from the stick-breaking representation of a Dirichlet process prior. As a first contribution, this CUSP prior is extended by involving arbitrary stick-breaking representations arising from beta distributions. As a second contribution, we prove that exchangeable spike-and-slab priors, which are popular and widely used in sparse Bayesian factor analysis, can be represented as a finite generalized CUSP prior, which is easily obtained from the decreasing order statistics of the slab probabilities. Hence, exchangeable spike-and-slab shrinkage priors imply increasing shrinkage as the column index in the loading matrix increases, without imposing explicit order constraints on the slab probabilities. An application to sparse Bayesian factor analysis illustrates the usefulness of the findings of this paper. A new exchangeable spike-and-slab shrinkage prior based on the triple gamma prior of Cadonna et al. (2020) is introduced and shown to be helpful for estimating the unknown number of factors in a simulation study.

翻译：本文讨论了对参数序列施加递增收缩效应的收缩先验。我们回顾了Legramanti等人(2020)提出的累积收缩过程(CUSP)先验，它是一种尖峰-板状收缩先验，其中尖峰概率随机递增且由狄利克雷过程先验的棍子断裂表示构建。作为第一个贡献，我们通过引入由贝塔分布产生的任意棍子断裂表示，对CUSP先验进行了扩展。作为第二个贡献，我们证明了在稀疏贝叶斯因子分析中流行且广泛使用的可交换尖峰-板状先验可以表示为有限广义CUSP先验，这可以轻松从板状概率的递减顺序统计量中获得。因此，可交换尖峰-板状收缩先验在载荷矩阵列索引递增时隐含了递增的收缩效应，而无需对板状概率施加显式的顺序约束。将本文的发现应用于稀疏贝叶斯因子分析，展示了其有用性。我们引入了一种基于Cadonna等人(2020)三重伽马先验的新型可交换尖峰-板状收缩先验，并在一项模拟研究中证明其有助于估计未知因子数量。