11.4隐变量的视角

文档摘要

11.4 隐变量的视角 11.4.1 生成过程和概率模型 $$ p(\boldsymbol{x}|z{k} = 1) = \mathcal{N}(\boldsymbol{x}|\boldsymbol{\mu}{k}, \boldsymbol{\Sigma}{k}). \tag{11.58} $$ $$ p(z) = \boldsymbol{\pi} = [\pi{1}, \dots, \pi{K}]^{\top}, \quad \sum\limits{k=1}^{K} \pi{k} = 1, \tag{11.59} $$ $$ \pi{k} = p(z{k} = 1) \tag{11.

11.4 隐变量的视角

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11.4.1 生成过程和概率模型

p(\boldsymbol{x}|z_{k} = 1) = \mathcal{N}(\boldsymbol{x}|\boldsymbol{\mu}_{k}, \boldsymbol{\Sigma}_{k}). \tag{11.58}

p(z) = \boldsymbol{\pi} = [\pi_{1}, \dots, \pi_{K}]^{\top}, \quad \sum\limits_{k=1}^{K} \pi_{k} = 1, \tag{11.59}

\pi_{k} = p(z_{k} = 1) \tag{11.60}

p(\boldsymbol{x}, z_{k} = 1) = p(\boldsymbol{x}|z_{k} = 1)p(z_{k} =1) = \pi_{k}\,\mathcal{N}(\boldsymbol{x}|\boldsymbol{\mu}_{k}, \boldsymbol{\Sigma}_{k})\tag{11.61}

p(\boldsymbol{x}, \boldsymbol{z}) = \begin{bmatrix} p(\boldsymbol{x}, z_{1} = 1)\\ \vdots \\ p(\boldsymbol{x}, z_{K} = 1) \end{bmatrix} = \begin{bmatrix} \pi_{1}\,\mathcal{N}(\boldsymbol{x}|\boldsymbol{\mu}_{1}, \boldsymbol{\Sigma}_{1})\\ \vdots \\ \pi_{K}\,\mathcal{N}(\boldsymbol{x}|\boldsymbol{\mu}_{K}, \boldsymbol{\Sigma}_{K}) \end{bmatrix},

11.4.2 似然函数

p(\boldsymbol{x}|\boldsymbol{\theta}) = \sum\limits_{z} p(\boldsymbol{x}|\boldsymbol{\theta}, z)p(z|\boldsymbol{\theta}), \quad \boldsymbol{\theta} \coloneqq \{ \boldsymbol{\mu}_{k}, \boldsymbol{\Sigma}_{k}, \pi_{k}: k, 1, \dots, K \}. \tag{11.63}

\begin{bmatrix} 1\\0\\0 \end{bmatrix}, \quad \begin{bmatrix} 0\\1\\0 \end{bmatrix}, \quad \begin{bmatrix} 0\\0\\1 \end{bmatrix}. \tag{11.64}

\begin{align} p(\boldsymbol{x}|\boldsymbol{\theta}) &= \sum\limits_{z} p(\boldsymbol{x}|\boldsymbol{\theta}, z)p(z|\boldsymbol{\theta})\tag{11.65a}\\ &= \sum\limits_{k=1}^{K} p(\boldsymbol{x}|\boldsymbol{\theta}, z_{k} =1)p(z_{k} = 1|\boldsymbol{\theta})\tag{11.65b} \end{align}

\begin{align} p(\boldsymbol{x}|\boldsymbol{\theta}) &\xlongequal{\text{(11.65b)}} \sum\limits_{k=1}^{K} p(\boldsymbol{x}|\boldsymbol{\theta}, z_{k} =1)p(z_{k} = 1|\boldsymbol{\theta})\tag{11.66a}\\ &= \sum\limits_{k=1}^{K} \pi_{k}\,\mathcal{N}(\boldsymbol{x}|\boldsymbol{\mu}_{k}, \boldsymbol{\Sigma}_{k}), \tag{11.66b} \end{align}

p(\mathcal{X}|\boldsymbol{\theta}) = \prod\limits_{n=1}^{N} p(\boldsymbol{x}_{n}|\boldsymbol{\theta}) \xlongequal{\text{(11.66b)}} \prod\limits_{n=1}^{N} \sum\limits_{k=1}^{K} \pi_{k}\,\mathcal{N}(\boldsymbol{x}_{n}|\boldsymbol{\mu}_{k}, \boldsymbol{\Sigma}_{k}) \tag{11.67}

11.4.3 后验分布

p(z_{k} = 1|\boldsymbol{x}) = \frac{p(z_{k} = 1)p(\boldsymbol{x}|z_{k} = 1)}{p(\boldsymbol{x})} \tag{11.68}

\tag{11.69} p(z_{k} = 1|\boldsymbol{x}) = \frac{p(z_{k} = 1)p(\boldsymbol{x}|z_{k} = 1)}{\sum_{j=1}^{K} p(z_{j} = 1) p(\boldsymbol{x}|z_{j} = 1)} = \frac{\pi_{k}\,\mathcal{N}(\boldsymbol{x}|\boldsymbol{\mu}_{k}, \boldsymbol{\Sigma}_{k})}{\sum_{j=1}^{K} \pi_{j}\,\mathcal{N}(\boldsymbol{x}|\boldsymbol{\mu}_{j}, \boldsymbol{\Sigma}_{j})}

11.4.4 延拓至整个数据集

\boldsymbol{z}_{n} = [z_{n, 1}, \dots, z_{n, K}]^{\top} \in \mathbb{R}^{K}. \tag{11.70}

p(\boldsymbol{x}_{1}, \dots, \boldsymbol{x}_{N}|\boldsymbol{z}_{1}, \dots, \boldsymbol{z}_{N}) = \prod\limits_{n=1}^{N} p(\boldsymbol{x}_{n}|\boldsymbol{z}_{n}). \tag{11.71}

\begin{align} p(z_{n, k} = 1|\boldsymbol{x}_{n}) &= \frac{p(z_{n,k} = 1)p(\boldsymbol{x}_{n}|z_{n,k} = 1)}{\sum_{j=1}^{K} p(z_{n,j} = 1) p(\boldsymbol{x}_{n}|z_{n,j} = 1)} \tag{11.72a}\\ &= \frac{\pi_{k}\,\mathcal{N}(\boldsymbol{x}_{n}|\boldsymbol{\mu}_{k}, \boldsymbol{\Sigma}_{k})}{\sum_{j=1}^{K} \pi_{j}\,\mathcal{N}(\boldsymbol{x}_{n}|\boldsymbol{\mu}_{j}, \boldsymbol{\Sigma}_{j})} =r_{n, k}. \tag{11.72b} \end{align}

11.4.5 重访 EM 算法

\begin{align} Q(\boldsymbol{\theta}|\boldsymbol{\theta}^{(t)}) &= \mathbb{E}_{\boldsymbol{z}|\boldsymbol{x}, \boldsymbol{\theta}^{(t)}}[\log p(\boldsymbol{x}, \boldsymbol{z}|\boldsymbol{\theta})] \tag{11.73a}\\ &= \int p(\boldsymbol{z}|\boldsymbol{x}, \boldsymbol{\theta}^{(t)}) \cdot \log p(\boldsymbol{x}, \boldsymbol{z}|\boldsymbol{\theta}) \, \mathrm d{\boldsymbol{z}} \tag{11.73b} \end{align}