如何从多元正态分布创建多个观察？

Question

I'm attempting to create n = 100 observations x from multivariate normal distribution (mean, I), while x is a (2,) vector, and it is known that mean follows another multivariate distribution (([0,0],sigma)). How can I achieve these? I was stuck with the size part in np.random.multivariate_normal function and it seems that I never set the dimensions I want.

mean_mean = np.array([0,0])
mean_sigma = np.array([[0.1,0],[0,0.1]])
mu = np.random.multivariate_normal(m0,S0,1).reshape(2,)
u = np.array([0,0])
I = np.array([[1,0],[0,1]])
x = np.random.multivariate_normal(u,I,100)

Answer 1

Since we don't know the value of the mean and instead only know that the mean follows some distribution, we first need to make n=100 samples of the mean, which we can do as so:

import numpy as np

samples = 100

mean_mean = np.array([0, 0])
mean_sigma = np.array([[0.1, 0], [0, 0.1]])
mu = np.random.multivariate_normal(mean_mean, mean_sigma, samples)

With this we can then sample the x values we desired. Unfortunately, the mean we pass to multivariate_normal must be 1 dimensional, so we can't vectorise the draws over the different means. Here, instead, I loop over the means to give our x values.

I = np.array([[1, 0], [0, 1]])

x = np.zeros([samples, 2])
for i in range(samples):
    x[2*i:2*(i+1)] = np.random.multivariate_normal(mu[i], I, 1)

Edit

After a little thought it's easy to vectorise the draws of x . To do so we will draw from the standard multivariate normal and transform the output.

x = np.random.multivariate_normal(np.zeros(2), I, samples)
x += mu