[论文解读] The ratio of normalizing constants for Bayesian graphical Gaussian model selection

Gérard Letac, Hélène Massam+1 more|arXiv (Cornell University)|Jun 14, 2017

Bayesian Modeling and Causal Inference参考文献 27被引用 7

一句话总结

本文提出了一种贝叶斯图Gaussian模型中归一化常数之比的解析近似方法，采用涉及伽马函数的闭式表达式，其参数为(δ+d)/2和(δ+d+1)/2，其中δ为G- Wishart分布的形状参数，d为边端点间长度为二的路径数。该方法避免了蒙特卡洛计算，计算效率高，且保持了高精度——近似比始终介于0.55至1之间，从而实现了可扩展的高维模型选择。

ABSTRACT

Many graphical Gaussian selection methods in a Bayesian framework use the G-Wishart as the conjugate prior on the precision matrix. The Bayes factor to compare a model governed by a graph G and a model governed by the neighboring graph G-e, derived from G by deleting an edge e, is a function of the ratios of prior and posterior normalizing constants of the G-Wishart for G and G-e. While more recent methods avoid the computation of the posterior ratio, computing the ratio of prior normalizing constants, (2) below, has remained a computational stumbling block. In this paper, we propose an explicit analytic approximation to (2) which is equal to the ratio of two Gamma functions evaluated at (delta+d)/2 and (delta+d+1)/2 respectively, where delta is the shape parameter of the G-Wishart and d is the number of paths of length two between the endpoints of e. This approximation allows us to avoid Monte Carlo methods, is computationally inexpensive and is scalable to high-dimensional problems. We show that the ratio of the approximation to the true value is always between zero and one and so, one cannot incur wild errors. In the particular case where the paths between the endpoints of e are disjoint, we show that the approximation is very good. When the paths between these two endpoints are not disjoint we give a sufficient condition for the approximation to be good. Numerical results show that the ratio of the approximation to the true value of the prior ratio is always between .55 and 1 and very often close to 1. We compare the results obtained with a model search using our approximation and a search using the double Metropolis-Hastings algorithm to compute the prior ratio. The results are extremely close.

研究动机与目标

• 解决由于计算G- Wishart分布归一化常数之比而引起的贝叶斯图Gaussian模型选择中的计算瓶颈问题。
• 为G- Wishart分布的先验归一化常数之比提供一种快速、可扩展的蒙特卡洛方法替代方案。
• 确保近似结果保持高精度且有界，避免在贝叶斯因子比较中出现较大误差。
• 通过提供一种计算轻量但可靠的归一化常数之比方法，实现高维模型选择。

提出的方法

• 提出G- Wishart分布先验归一化常数之比的解析近似，表达为在(δ+d)/2和(δ+d+1)/2处求值的伽马函数之比。
• 定义d为从图G中删除边e的端点之间长度为二的路径数。
• 在G- Wishart分布形状参数δ已知且固定的前提下推导该近似。
• 证明近似值与真实值之比始终介于0和1之间，确保误差有界。
• 给出在端点间路径不互斥时近似仍有效的充分条件。
• 通过数值实验验证该方法，并与双Metropolis-Hastings算法在计算先验比值方面的表现进行比较。

实验结果

研究问题

• RQ1 能否在不依赖蒙特卡洛方法的前提下，为G- Wishart分布的归一化常数之比推导出解析近似？
• RQ2 基于伽马函数的所提近似在估计真实归一化常数之比时的准确性如何？
• RQ3 当边端点间的路径不互斥时，该近似在不同图结构下是否仍保持可靠且有界？
• RQ4 使用解析近似进行模型选择的性能与使用双Metropolis-Hastings算法相比如何？
• RQ5 该近似能否在高维图Gaussian模型中实现有效扩展？

主要发现

• 所提近似与真实先验比值之比始终介于0.55至1之间，表明其具有强有界性与可靠性。
• 当边e端点间的路径互不相交时，该近似特别精确，比值常接近1。
• 提供了在路径不互斥时仍保持良好近似的充分条件。
• 数值结果表明，该近似高度准确，近似值与真实值之比始终高于0.55。
• 使用解析近似进行的模型选择结果与双Metropolis-Hastings算法所得结果极为接近，验证了其实际可用性。
• 该方法计算高效，可扩展至高维问题，避免了昂贵的蒙特卡洛积分计算。