3.2.6 Bayes’ theorem figure3.5

In Sections 3.2.4 and 3.2.5 we considered a Gaussian p(x) in which we parti

tioned the vector x into two subvectors x = (xa, xb) and then found expressions

for the conditional distribution p(xa|xb) and the marginal distribution p(xa). We

noted that the mean of the conditional distribution p(xa|xb) was a linear function of

xb. Here we will suppose that we are given a Gaussian marginal distribution p(x)

and a Gaussian conditional distribution p(y|x) in which p(y|x) has a mean that is a

linear function of x and a covariance that is independent of x. This is an example

of a linear-Gaussian model (Roweis and Ghahramani, 1999). We wish to find the

marginal distribution p(y) and the conditional distribution p(x|y). This is a struc

ture that arises in several types of generative model and it will prove convenient to

derive the general results here.

在3.2.4和3.2.5节中，我们考虑了一个高斯分布 $p(x)$，其中我们将向量 $x$ 划分为两个子向量 $x = (x_a, x_b)$，然后找到了条件分布 $p(x_a | x_b)$ 和边际分布 $p(x_a)$ 的表达式。我们注意到条件分布 $p(x_a | x_b)$ 的均值是 $x_b$ 的线性函数。在这里，我们假设给定一个高斯边际分布 $p(x)$ 和一个高斯条件分布 $p(y | x)$，其中 $p(y | x)$ 的均值是 $x$ 的线性函数，协方差与 $x$ 无关。这是一个线性高斯模型的例子（Roweis 和 Ghahramani, 1999）。我们希望找到边际分布 $p(y)$ 和条件分布 $p(x | y)$。

我们将边际分布和条件分布表示为：

import numpy as np
import matplotlib.pyplot as plt
from scipy.stats import multivariate_normal

# 设置均值和协方差矩阵
mean = [0.5, 0.5]
cov = [[0.01, 0.007], [0.007, 0.01]]

# 创建网格
x, y = np.meshgrid(np.linspace(0, 1, 100), np.linspace(0, 1, 100))
pos = np.dstack((x, y))

# 创建多元高斯分布
rv = multivariate_normal(mean, cov)

# 绘制高斯分布轮廓
plt.figure(figsize=(10, 5))

# 子图 (a): 高斯分布轮廓
plt.subplot(1, 2, 1)
plt.contourf(x, y, rv.pdf(pos), levels=10, cmap='viridis')
plt.colorbar()
plt.title('(a) Contours of a Gaussian distribution p(xa, xb)')
plt.xlabel('xa')
plt.ylabel('xb')

# 子图 (b): 边际分布和条件分布
plt.subplot(1, 2, 2)

# 边际分布 p(xa)
xa = np.linspace(0, 1, 100)
marginal_p_xa = multivariate_normal(mean[0], cov[0][0]).pdf(xa)
plt.plot(xa, marginal_p_xa, 'b-', label='p(xa)')

# 条件分布 p(xa | xb = 0.7)
xb = 0.7
conditional_mean = mean[0] + cov[0][1] / cov[1][1] * (xb - mean[1])
conditional_cov = cov[0][0] - cov[0][1] ** 2 / cov[1][1]
conditional_p_xa_given_xb = multivariate_normal(conditional_mean, conditional_cov).pdf(xa)
plt.plot(xa, conditional_p_xa_given_xb, 'r-', label='p(xa | xb = 0.7)')

plt.title('(b) Marginal and Conditional Distributions')
plt.xlabel('xa')
plt.ylabel('Density')
plt.legend()

plt.tight_layout()
plt.show()

书上的图

上述代码结果 colab 运行