Variational EM via mean-field

Problem

Marginal Log-Likelihood

Session information

Last updated: 2018-08-20

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Problem

We need to make MLE of $\theta$ for $P(\mathbf{y}|\theta)$ . The marginal $\log P(\mathbf{y}|\theta) = \log \int P(\mathbf{y}, Z|\theta) dZ$ is difficule to compute. However $\log P(\mathbf{y}|\theta) = \log \int P(\mathbf{y}| Z, \theta)P(Z|\theta) dZ$ is easier. $P(\mathbf{y}| Z, \theta)$ is likelihood, $P(Z|\theta)$ is prior.

Marginal Log-Likelihood

For any normalized distribution

$q(Z)$ ,

$\begin{align*} \log P(\mathbf{y}|\theta) &= \log P(\mathbf{y}|\theta) \int_{Z} q(Z) dZ \\ &= \int_{Z} \left(\log P(\mathbf{y}|\theta) \right)q(Z) dZ \\ &= \int_{Z} \left( \log \frac{P(\mathbf{y}, Z|\theta)}{P(Z|\mathbf{y}, \theta)}\right) q(Z) dZ \\ &= \int_{Z} \left(\log \frac{P(\mathbf{y}, Z|\theta) q(Z)}{P(Z|\mathbf{y}, \theta)q(Z)}\right) q(Z) dZ \\ &= \int_{Z} \left(\log \frac{P(\mathbf{y}|Z,\theta)P(Z|\theta)}{q(Z)} \frac{q(Z)}{P(Z|\mathbf{y}, \theta)}\right) q(Z) dZ \\ &= \int_{Z} \left( \log P(\mathbf{y}|Z,\theta) \right) q(Z)dZ + \int_{Z} \log \frac{p(Z|\theta)}{q(Z)} q(Z) dZ + \int_{Z} \log \frac{q(Z)}{P(Z|\mathbf{y}, \theta)} q(Z)dZ \\ &= \mathbb{E}_{q}\left[ \log P(\mathbf{y}|Z,\theta) \right] + \mathbb{E}_{q}\left[ \log \frac{p(Z|\theta)}{q(Z)} \right] + \mathbb{E}_{q}\left[ \log \frac{q(Z)}{P(Z|\mathbf{y}, \theta)} \right] \\ &= \mathbb{E}_{q}\left[ \log P(\mathbf{y}|Z,\theta) \right] + \mathbb{E}_{q}\left[ \log \frac{p(Z|\theta)}{q(Z)} \right] + D_{KL}(q||p(Z|\mathbf{y},\theta)) \\ &\geq \mathbb{E}_{q}\left[ \log P(\mathbf{y}|Z,\theta) \right] + \mathbb{E}_{q}\left[ \log \frac{p(Z|\theta)}{q(Z)} \right] = \mathcal{L}(q,\theta) \end{align*}$

The term $D_{KL}(q||p(Z|\mathbf{y},\theta)) \geq 0$ , so $\mathcal{L}(q,\theta)$ is a lower bound.

Set $q(Z) = \prod_{i} q_{i}(Z_{i})$

$q_{j}^{\star}(Z_{j}) = \frac{\exp\left( \mathbb{E}_{i\neq j}\left[ \log P(Z, \mathbf{y}) \right] \right)}{\int \exp\left( \mathbb{E}_{i\neq j}\left[ \log P(Z, \mathbf{y}) \right] \right) dZ_{j}}$

$\max_{q \in \Omega} \log P(\mathbf{y}|\theta)$ : Set $q(Z) = \prod_{i} q_{i}(Z_{i})$
$\max_{\theta} \mathbb{E}_{q} \left[ \log P(\mathbf{y}|Z,\theta) \right] + \sum_{i} \mathbb{E}_{q}\left[ \log \frac{P(Z_{i}|\theta)}{q_{i}(Z_{i})} \right]$

Session information

sessionInfo()

R version 3.5.1 (2018-07-02)
Platform: x86_64-apple-darwin15.6.0 (64-bit)
Running under: macOS High Sierra 10.13.6

Matrix products: default
BLAS: /Library/Frameworks/R.framework/Versions/3.5/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/3.5/Resources/lib/libRlapack.dylib

locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

loaded via a namespace (and not attached):
 [1] workflowr_1.1.1   Rcpp_0.12.18      digest_0.6.15    
 [4] rprojroot_1.3-2   R.methodsS3_1.7.1 backports_1.1.2  
 [7] git2r_0.23.0      magrittr_1.5      evaluate_0.11    
[10] stringi_1.2.4     whisker_0.3-2     R.oo_1.22.0      
[13] R.utils_2.6.0     rmarkdown_1.10    tools_3.5.1      
[16] stringr_1.3.1     yaml_2.2.0        compiler_3.5.1   
[19] htmltools_0.3.6   knitr_1.20

This reproducible R Markdown analysis was created with workflowr 1.1.1

Variational EM via mean-field

Yuxin Zou

2018-5-29

Problem

Marginal Log-Likelihood

Session information