arXiv 2401.056932024-01math.STstat.TH

稀疏计数数据的渐进贝叶斯最优性

Asymptotic Bayes Optimality for Sparse Count Data

Sayantan Paul, Arijit Chakrabarti

AI总结：

本文提出了一类全局-局部先验，用于稀疏计数数据的渐进贝叶斯最优性分析。

摘要：

考虑分析高维计数数据的情况，其中包含大量接近零的计数和少量中等或大计数。假设观测由泊松分布建模，我们感兴趣于同时检验第i个观测的均值是否小或大。在本工作中，我们研究了在贝叶斯风险意义上多重检验规则的一些最优性质，当均值参数由两组和一般的一组收缩先验建模时，这些先验由Polson和Scott（2010）提出。首先，我们对每个均值建模为两组先验，并在加法0-1损失函数下，获得最优贝叶斯风险的表达式，这在某种程度上类似于Bogdan等人（2011）的精神。接下来，假设观测确实来自两组混合模型，并对每个均值参数建模为广泛的一组先验，我们研究所选先验类诱导的贝叶斯风险。我们证明，当底层稀疏性水平已知时，在某些假设下，我们广泛先验类对应的贝叶斯风险达到最优贝叶斯风险，最多乘以一个常数。当稀疏性模式未知时，受Yano等人（2021）启发，我们使用全局收缩参数的经验贝叶斯估计。在这种情况下，我们同样证明修改后的决策规则达到最优贝叶斯风险，最多乘以一个常数。因此，作为两组先验的替代方案，我们提出了一类具有类似贝叶斯风险最优性质的全局-局部先验，适用于准稀疏计数数据。最后，通过模拟研究和实际数据分析验证了理论结果。

英文摘要：

Consider a situation of analyzing high-dimensional count data containing an excess of near-zero counts with a small number of moderate or large counts. Assuming that the observations are modeled by a Poisson distribution, we are interested in simultaneous testing of whether the mean of the $i^{\text{th}}$ observation is small or large. In this work, we study some optimal properties (in terms of Bayes risk) of multiple-testing rules when the mean parameter is modeled by both two-group and a general class of one-group shrinkage priors, proposed by Polson and Scott (2010). Here, first, we model each mean by a two-group prior, and under additive $0-1$ loss function, obtain an expression for the optimal Bayes risk under some assumption similar in the spirit of Bogdan et al. (2011). Next, assuming that the observations are truly generated from a two-group mixture model and modelling each mean parameter by the broad class of one-group priors, we study the Bayes risk induced by our chosen class of priors. We have been able to show that, when the underlying level of sparsity is known, under some proposed assumptions, the Bayes risk corresponding to our broad class of priors attains the optimal Bayes risk, upto a multiplicative constant. When this sparsity pattern is unknown, motivated by Yano et al. (2021), we use an empirical Bayes estimate of the global shrinkage parameter. In this case, also, we show that the modified decision rule attains the optimal Bayes risk, upto a multiplicative constant. In this way, as an alternative solution for two-group prior, we propose a broad class of global-local priors having similar optimal properties in terms of Bayes risk for quasi-sparse count data. Finally, the theoretical results are verified using simulation studies followed by a real data analysis.

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