发表机构
Duke University; Eli Lilly and Company; University of Wisconsin-Madison; Virginia Commonwealth University（杜克大学; 礼来公司; 威斯康星大学麦迪逊分校; 弗吉尼亚联邦大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出ExFR-LGP方法，通过有限秩高斯过程先验实现条件密度估计的精确似然，获得闭式解，并证明其最优收缩速率，在合成数据和ADNI数据上验证了有效性。
AI 中文摘要
条件密度估计描述了响应的整个分布如何随协变量变化，在影像研究中还随位置变化。逻辑高斯过程（LGP）为此类密度提供了灵活的先验。然而，LGP的归一化常数没有闭式形式，因此现有方法会近似或替换似然。我们提出了精确似然有限秩LGP（ExFR-LGP），它将对数密度写成两个二元函数之和：一个是响应与协变量的位置变化线性指数的函数，另一个是响应与位置的函数。每个函数被赋予一个有限秩高斯过程先验，该先验在规则网格上分段线性，使得归一化常数、条件均值和分位数具有闭式表示。后验样本通过吉布斯采样器从该先验下的精确后验中抽取，该采样器使用椭圆切片采样更新高斯分量。当真对数密度是两个此类二元函数之和时，我们证明后验以α光滑二元密度的极小极大速率收缩，直至对数因子。使用合成数据的数值实验证明了ExFR-LGP在条件密度估计中的有效性。此外，将其应用于阿尔茨海默病神经影像学倡议数据中胼胝体沿线的分数各向异性响应，得到了带不确定性的协变量调整百分位带，阐明了诊断如何改变沿束的分布。
英文摘要
Conditional density estimation describes how the entire distribution of a response changes with covariates, and in imaging studies also with location. Logistic Gaussian processes (LGP) give a flexible prior for such densities. However, the normalizing constant of an LGP has no closed form, so existing methods approximate or replace the likelihood. We propose the exact likelihood finite-rank LGP (ExFR-LGP), which writes the log density as the sum of two bivariate functions, one of the response and a location-varying linear index of the covariates, and one of the response and the location. Each function is assigned a finite-rank Gaussian process prior that is piecewise linear on a regular grid, enabling the normalizing constant, the conditional mean and the quantiles to enjoy closed form representations. Posterior samples are drawn from the exact posterior under this prior via a Gibbs sampler that updates the Gaussian components by elliptical slice sampling. When the true log density is the sum of two such bivariate functions, we show that the posterior contracts at the minimax rate of an $α$-smooth bivariate density up to a logarithmic factor. Numerical experiments using synthetic data demonstrate the effectiveness of ExFR-LGP in conditional density estimation. Furthermore, application to fractional anisotropy responses along the corpus callosum in the Alzheimer's Disease Neuroimaging Initiative data yields covariate-adjusted percentile bands with uncertainty, elucidating how diagnosis changes the distribution along the tract.
Comments47 pages, 4 figures
