finalize putting opitimization equations

Author: olivier-bernard-creatis

collections/_posts/2025-10-15-TreeVAE.md

@@ -194,13 +194,22 @@ The full presentation, including links to the diffusion model (DDPM), is availab
   ➔ <span style="color:darkred">Use of Monte Carlo framework to estimate some distributions !</span>  
 
 
-### Data fidelity
+### Data fidelity term
 * The data fidelity term can be expressed as:
-  ➔ $$\mathcal{L}_{rec} = \mathbb{E}_{q(z_{\mathcal{P}_l},\mathcal{P}_l \mid x)}[ \text{log} \, p(x \mid z_{\mathcal{P}_l},\mathcal{P}_l) ]$$
-  ➔ <span style="color:darkred">$$\mathcal{L}_{rec} \approx \frac{1}{M} \sum_{m=1}^{M} \sum_{l\in \mathbb{L}}{P(l;c)\log \mathcal{N(x \mid \mu_{x,l}(z_l^{(m)}),\sigma_{x,l}^2(z_l^{(m)}))}}$$</span> <br><br>   
+  ➔ $$\mathcal{L}_{rec} = \mathbb{E}_{q(z_{\mathcal{P}_l},\mathcal{P}_l \mid x)}[ \text{log} \, p(x \mid z_{\mathcal{P}_l},\mathcal{P}_l) ]$$ <br>
+  ➔ <span style="color:darkred">$$\mathcal{L}_{rec} \approx \frac{1}{M} \sum_{m=1}^{M} \sum_{l\in \mathbb{L}}{P(l;c)\log (\mathcal{N(\mu_{x,l}(z_l^{(m)}),\sigma_{x,l}^2(z_l^{(m)})))}}$$</span> <br><br>   
   - with $$P(i;c)=\prod_{j\in P_{i \backslash \{0\}}}q(c_{pa(j) \rightarrow j} \mid x)$$ for $$i \in \mathbb{V}$$ the probability of reaching node $$i$$ which is the product over the probabilities of the decisions in the path until $$i$$
   - with $$z_l^{(m)}$$ the Monte Carlo (MC) samples, and $$M$$ the number of the MC samples.
 
+### Distributions fit term
+* The distributions fit term can be expressed as:
+  ➔ <span style="color:darkred">$$\text{KL}(q(z_{\mathcal{P}_l},\mathcal{P}_l \mid x) \parallel p(z_{\mathcal{P}_l},\mathcal{P}_l) ) = \text{KL}_{root} + \text{KL}_{nodes} + \text{KL}_{decisions}$$</span><br>
+  ➔ $$\text{KL}_{root} = \text{KL}(q(z_0 \mid x) \parallel p(z_0))$$ <br>
+  ➔ $$\text{KL}_{nodes} \approx \frac{1}{M} \sum_{m=1}^{M} \sum_{i \in \mathbb{V} \backslash \{0\}} P(i;c) \, \text{KL}(q(z_i^{(m)} \mid pa(z_i^{(m)})) \parallel p(z_i^{(m)} \mid pa(z_i^{(m)})))$$ <br>
+  ➔ $$\text{KL}_{decisions} \approx \frac{1}{M} \sum_{m=1}^{M} \sum_{i \in \mathbb{V} \backslash \{\mathbb{L}\}} P(i;c) \, \text{KL}(q(c_i \mid x) \parallel p(c_i \mid z_i))$$ <br>
+
+
+
 # Experiments
 
 * Evaluation on 8 public datasets (small images): MNIST, Fashion-MNIST, 20Newsgroups, Omniglot, Omniglot-5, CIFAR-10 and CIFAR-100, CelebA