分布值协变量的直接贝叶斯线性回归。

IF 1 4区 数学 Q3 STATISTICS & PROBABILITY Electronic Journal of Statistics Pub Date : 2024-01-01 Epub Date: 2024-08-27 DOI:10.1214/24-ejs2275
Bohao Tang, Sandipan Pramanik, Yi Zhao, Brian Caffo, Abhirup Datta
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引用次数: 0

摘要

在本手稿中,我们研究的是标量-分布回归,即以特定受试者的分布或密度作为协变量,通过回归模型与标量结果相关联的情况。在实践中,只能从这些协变量分布中观察到重复测量值,常用的方法首先使用这些协变量分布来估计特定受试者的密度函数,然后在标准的标量-函数回归中将其用作协变量。我们提出了一种简单直接的线性标量-分布回归方法,该方法避开了估计特定受试者协变量密度这一中间步骤。我们证明,可以直接使用观测到的重复测量值作为协变量,并为回归函数赋予高斯过程先验,从而获得封闭形式或共轭贝叶斯推断。我们的方法将使用高斯过程的标准贝叶斯非参数回归归为特例,与协变量为狄拉克分布相对应。该模型还不受重复测量的任何变换或排序的影响。我们从理论上证明,尽管只使用了从产生数据的真实密度值协变量中观察到的重复测量值,该方法仍能实现回归函数的最优估计误差约束。该理论超越了 i.i.d.设置,以适应重复测量中某些形式的受试者内依赖性。据我们所知,这是首次对使用分布值协变量的贝叶斯回归进行理论研究。我们提出了许多扩展建议,包括使用低秩高斯过程的可扩展实现,以及对分布上非线性标量回归的概括。通过模拟研究,我们证明了我们的方法比那些需要中间密度估计步骤的方法要好得多,尤其是在每个研究对象重复测量次数较少的情况下。我们将我们的方法应用于研究年龄与活动计数的关联。
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Direct Bayesian linear regression for distribution-valued covariates.

In this manuscript, we study scalar-on-distribution regression; that is, instances where subject-specific distributions or densities are the covariates, related to a scalar outcome via a regression model. In practice, only repeated measures are observed from those covariate distributions and common approaches first use these to estimate subject-specific density functions, which are then used as covariates in standard scalar-on-function regression. We propose a simple and direct method for linear scalar-on-distribution regression that circumvents the intermediate step of estimating subject-specific covariate densities. We show that one can directly use the observed repeated measures as covariates and endow the regression function with a Gaussian process prior to obtain a closed form or conjugate Bayesian inference. Our method subsumes the standard Bayesian non-parametric regression using Gaussian processes as a special case, corresponding to covariates being Dirac-distributions. The model is also invariant to any transformation or ordering of the repeated measures. Theoretically, we show that, despite only using the observed repeated measures from the true density-valued covariates that generated the data, the method can achieve an optimal estimation error bound of the regression function. The theory extends beyond i.i.d. settings to accommodate certain forms of within-subject dependence among the repeated measures. To our knowledge, this is the first theoretical study on Bayesian regression using distribution-valued covariates. We propose numerous extensions including a scalable implementation using low-rank Gaussian processes and a generalization to non-linear scalar-on-distribution regression. Through simulation studies, we demonstrate that our method performs substantially better than approaches that require an intermediate density estimation step especially with a small number of repeated measures per subject. We apply our method to study association of age with activity counts.

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来源期刊
Electronic Journal of Statistics
Electronic Journal of Statistics STATISTICS & PROBABILITY-
CiteScore
1.80
自引率
9.10%
发文量
100
审稿时长
3 months
期刊介绍: The Electronic Journal of Statistics (EJS) publishes research articles and short notes on theoretical, computational and applied statistics. The journal is open access. Articles are refereed and are held to the same standard as articles in other IMS journals. Articles become publicly available shortly after they are accepted.
期刊最新文献
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