从python中的多元概率密度函数采样 [英] Sampling from a multivariate probability density function in python
问题描述
我有一个多元概率密度函数P(x，y，z)，我想从中取样.通常，我将使用numpy.random.choice()进行此类任务，但是此函数仅适用于一维概率密度.多元pdf有等效功能吗?
这里可以遵循一些不同的路径.
(1)如果P(x，y，z)的因子为P(x，y，z)= P(x)P(y)P(z)(即x，y和z独立)那么您可以分别对每个样品进行采样.
(2)如果P(x，y，z)具有更一般的因式分解，则可以将需要采样的变量数量减少到其他变量的条件.例如.如果P(x，y，z)= P(z | x，y)P(y | x)P(x)，则可以依次对x，y(给定x)和z(分别给定y和x)进行采样.
(3)对于某些特定分布，有已知的采样方法.例如.对于多元高斯，如果x是来自均值0和恒等方差高斯的样本(即，仅将每个x_i采样为N(0，1))，则y = L x + m的均值为m，协方差S = LL'，其中L是S的下三角Cholesky分解，必须为正定.
(4)对于许多多元分布，以上都不适用，并且应用了更复杂的方案，例如马尔可夫链蒙特卡罗.
也许您对这个问题说的更多，可以给出更具体的建议.
I have a multivariate probability density function P(x,y,z), and I want to sample from it. Normally, I would use numpy.random.choice() for this sort of task, but this function only works for 1-dimensional probability densities. Is there an equivalent function for multivariate pdfs?
There a few different paths one can follow here.
(1) If P(x,y,z) factors as P(x,y,z) = P(x) P(y) P(z) (i.e., x, y, and z are independent) then you can sample each one separately.
(2) If P(x,y,z) has a more general factorization, you can reduce the number of variables that need to be sampled to whatever's conditional on the others. E.g. if P(x,y,z) = P(z|x, y) P(y | x) P(x), then you can sample x, y given x, and z given y and x in turn.
(3) For some particular distributions, there are known ways to sample. E.g. for multivariate Gaussian, if x is a sample from a mean 0 and identity covariance Gaussian (i.e. just sample each x_i as N(0, 1)), then y = L x + m has mean m and covariance S = L L' where L is the lower-triangular Cholesky decomposition of S, which must be positive definite.
(4) For many multivariate distributions, none of the above apply, and a more complicated scheme such as Markov chain Monte Carlo is applied.
Maybe if you say more about the problem, more specific advice can be given.
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