Variational AutoEncoder VAE
Tags: #machine learning #VAEEquation
$$\log p_{\theta}(x)=\mathbb{E}_{q_{\phi}(z|x)}[\log p_{\theta}(x)] \\ =\mathbb{E}_{q_{\phi}(z|x)}[\log \frac{p_{\theta}(x,z)}{p_{\theta}(z|x)}] \\ =\mathbb{E}_{q_{\phi}(z|x)}[\log [\frac{p_{\theta}(x,z)}{q_{\phi}(z|x)} \times \frac{q_{\phi}(z|x)}{p_{\theta}(z|x)}]] \\ =\mathbb{E}_{q_{\phi}(z|x)}[\log [\frac{p_{\theta}(x,z)}{q_{\phi}(z|x)} ]] +D_{KL}(q_{\phi}(z|x) || p_{\theta}(z|x))\\$$Latex Code
\log p_{\theta}(x)=\mathbb{E}_{q_{\phi}(z|x)}[\log p_{\theta}(x)] \\
=\mathbb{E}_{q_{\phi}(z|x)}[\log \frac{p_{\theta}(x,z)}{p_{\theta}(z|x)}] \\
=\mathbb{E}_{q_{\phi}(z|x)}[\log [\frac{p_{\theta}(x,z)}{q_{\phi}(z|x)} \times \frac{q_{\phi}(z|x)}{p_{\theta}(z|x)}]] \\
=\mathbb{E}_{q_{\phi}(z|x)}[\log [\frac{p_{\theta}(x,z)}{q_{\phi}(z|x)} ]] +D_{KL}(q_{\phi}(z|x) || p_{\theta}(z|x))\\
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Introduction
Estimating the Log-likelihood and Posterior
Equation
Latex Code
\log p_{\theta}(x)=\mathbb{E}_{q_{\phi}(z|x)}[\log p_{\theta}(x)] \\
=\mathbb{E}_{q_{\phi}(z|x)}[\log \frac{p_{\theta}(x,z)}{p_{\theta}(z|x)}] \\
=\mathbb{E}_{q_{\phi}(z|x)}[\log [\frac{p_{\theta}(x,z)}{q_{\phi}(z|x)} \times \frac{q_{\phi}(z|x)}{p_{\theta}(z|x)}]] \\
=\mathbb{E}_{q_{\phi}(z|x)}[\log [\frac{p_{\theta}(x,z)}{q_{\phi}(z|x)} ]] +D_{KL}(q_{\phi}(z|x) || p_{\theta}(z|x))\\
Explanation
Evidence Lower Bound
Equation
Latex Code
\mathbb{L}_{\theta,\phi}(\mathbf{x})=\mathbb{E}_{q_{\phi}(\mathbf{z}|\mathbf{x})}[\log p_{\theta}(\mathbf{x},\mathbf{z})-\log q_{\phi}(\mathbf{z}|\mathbf{x}) ]
Explanation
Reparameterization trick
Equation
Latex Code
z = \mu + \epsilon \cdot \sigma
Explanation
VAE latex code is illustrated above. See paper for more details Auto-Encoding Variational Bayes
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