scvi equations

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the scVI generative model

\[\begin{aligned} &w_{n g} \sim \operatorname{Gamma}\left(\rho_{n}^{g}, \theta\right)\\ &y_{n g} \sim \operatorname{Poisson}\left(\ell_{n} w_{n g}\right)\\ &h_{n g} \sim \text { Bernoulli }\left(f_{h}^{g}\left(z_{n}, s_{n}\right)\right)\\ &x_{n g}=\left\{\begin{array}{l} {y_{n g} \text { if } h_{n g}=0} \\ {0 \text { otherwise }} \end{array}\right. \end{aligned}\]

derivation of ELBO

1.

\(\begin{align} \log P_\theta(x) =& \log P_\theta(x)\int q_\phi(x|z)dz \\ =& E_{q_\phi(z|x)}\log P_\theta(x) \end{align}\) because the integration is not related with x

2.

\(\begin{align} \log P_\theta(x)=&\log \frac{P_\theta(x,z)}{P_\theta(z|x)}\\ =&\log \frac{P_\theta(x,z)}{q_\phi(z|x)}\frac{q_\phi(z|x)}{P_\theta(z|x)}\\ =&\log P_\theta(x|z)-\log \frac{q_\phi(z|x)}{P_\theta(z)} + \log \frac{q_\phi(z|x)}{P_\theta(z|x)} \end{align}\)

3.

if we combine the 2 above equations: \(\begin{align} \log P_\theta(x) =& E_q\log P_\theta(x|z) - D_{KL}(q_\phi(z|x)||P_\theta(z)) + D_{KL}(q_\phi(z|x)||P_\theta(z|x))\\\\ \log P_\theta(x) \geq& E_q\log P_\theta(x|z) - D_{KL}(q_\phi(z|x)||P_\theta(z)) \end{align}\)

4. meanfield assumption

\(D_{KL}(q_\phi(z,l|x,s)||P_\theta(z,l)) = D_{KL}(q_\phi(z|x,s)||P_\theta(z))+D_{KL}(q_\phi(l|x,s)||P_\theta(l))\)