如何使用库尔贝克-莱布勒散度来构造共轭先验，具有定义良好的非信息限制，用于多元高斯分布
How to use KL-divergence to construct conjugate priors, with well-defined non-informative limits, for the multivariate Gaussian
摘要：
当均值已知时，Wishart分布是多元高斯似然精度的标准共轭先验，而当均值也未知时，可以使用正态Wishart分布。然而，如何为这些分布的超参数赋值并不那么明显。特别是，当形成这些分布的非信息限制时，Wishart的形状（或自由度）参数必须小心处理。一些作者提出的直接将形状解释为伪计数并使其归零的直观解决方案违反了对形状参数的限制。我们展示了如何使用多变量高斯之间的缩放库尔贝克-莱布勒散度作为能量函数来构造Wishart和正常Wishart共轭先验。当用作信息先验时，这些分布的显着特征是模式，而KL缩放因子用作伪计数。比例因子可以下降到零的极限，以形成不违反Wishart形状参数限制的非信息先验。这个限制是无信息的意义上的后验模式是相同的最大似然估计的高斯参数。
英文摘要：
The Wishart distribution is the standard conjugate prior for the precision of the multivariate Gaussian likelihood, when the mean is known -- while the normal-Wishart can be used when the mean is also unknown. It is however not so obvious how to assign values to the hyperparameters of these distributions. In particular, when forming non-informative limits of these distributions, the shape (or degrees of freedom) parameter of the Wishart must be handled with care. The intuitive solution of directly interpreting the shape as a pseudocount and letting it go to zero, as proposed by some authors, violates the restrictions on the shape parameter. We show how to use the scaled KL-divergence between multivariate Gaussians as an energy function to construct Wishart and normal-Wishart conjugate priors. When used as informative priors, the salient feature of these distributions is the mode, while the KL scaling factor serves as the pseudocount. The scale factor can be taken down to the limit at zero, to form non-informative priors that do not violate the restrictions on the Wishart shape parameter. This limit is non-informative in the sense that the posterior mode is identical to the maximum likelihood estimate of the parameters of the Gaussian.
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