论文标题
数据分析食谱：贝叶斯推论中多元高斯人的产品
Data Analysis Recipes: Products of multivariate Gaussians in Bayesian inferences
论文作者
论文摘要
两个高斯（或正常分布）的产物是另一种高斯。这是一个有价值且有用的事实！在这里，我们使用它来得出多元高斯人的共同产物的重构：高斯可能性时代的高斯先验的产物，其中某些或全部这些参数仅在平均值中输入可能性，并且仅在线性上输入可能性。也就是说，线性，高斯，贝叶斯模型。可能会将以前的PDF重构为边缘化的可能性（或贝叶斯证据）的产物，乘以PDF的后pDF，在这种情况下（在这种情况下）这两种都是高斯。重构高斯人的平均值和方差张量是直接获得闭合形式表达式的。在这里，我们进行了讨论。封闭形式的表达式可用于加快和提高包含高斯先验线性参数的推论精度。我们将这些方法与在物理和天文学中经常产生的推论联系起来。 如果您想要的只是答案，那么在第3节开始时就提出和回答了问题。我们在第4节中以工作练习的形式展示了两个玩具示例。本说明中的解决方案，讨论和练习的目的是针对已经熟悉贝叶斯推理和可能性基本思想的人。
A product of two Gaussians (or normal distributions) is another Gaussian. That's a valuable and useful fact! Here we use it to derive a refactoring of a common product of multivariate Gaussians: The product of a Gaussian likelihood times a Gaussian prior, where some or all of those parameters enter the likelihood only in the mean and only linearly. That is, a linear, Gaussian, Bayesian model. This product of a likelihood times a prior pdf can be refactored into a product of a marginalized likelihood (or a Bayesian evidence) times a posterior pdf, where (in this case) both of these are also Gaussian. The means and variance tensors of the refactored Gaussians are straightforward to obtain as closed-form expressions; here we deliver these expressions, with discussion. The closed-form expressions can be used to speed up and improve the precision of inferences that contain linear parameters with Gaussian priors. We connect these methods to inferences that arise frequently in physics and astronomy. If all you want is the answer, the question is posed and answered at the beginning of Section 3. We show two toy examples, in the form of worked exercises, in Section 4. The solutions, discussion, and exercises in this Note are aimed at someone who is already familiar with the basic ideas of Bayesian inference and probability.
