From 09f6f9951a5707475d3b0ff59eed4c5077453a58 Mon Sep 17 00:00:00 2001
From: chris-admin <chricchio.risi@icloud.com>
Date: Wed, 3 Sep 2025 19:28:05 +0200
Subject: [PATCH] Added Probabilistic View in chapter 10

---
 Chapters/10-Autoencoders/INDEX.md | 50 +++++++++++++++++++++++++++++++
 1 file changed, 50 insertions(+)

diff --git a/Chapters/10-Autoencoders/INDEX.md b/Chapters/10-Autoencoders/INDEX.md
index 3474685..0b79c4f 100644
--- a/Chapters/10-Autoencoders/INDEX.md
+++ b/Chapters/10-Autoencoders/INDEX.md
@@ -111,3 +111,53 @@ similar to a point, nor too far apart from each other***
 $$
 L(x) = ||x - \hat{x}||^{2}_{2} + KL[N(\mu_{x}, \Sigma_{x}), N(0, 1)]
 $$
+
+### Probabilistic View
+
+- $\mathcal{X}$: Set of our data
+- $\mathcal{Y}$: Latent variable set
+- $p(x|y)$: Probabilistic encoder, tells us the distribution of $x$ given $y$
+- $p(y|x)$: Probabilistic decoder, tells us the distribution of $y$ given $x$
+
+> [!NOTE]
+> Bayesian a Posteriori Probability
+> $$
+>   \underbrace{p(A|B)}_{\text{Posterior}} = \frac{
+>       \overbrace{p(B|A)}^{\text{Likelihood}}
+>       \overbrace{\cdot p(A)}^{\text{Prior}}
+>   }{
+>       \underbrace{p(B)}_{\text{Marginalization}}
+>   }
+>   = \frac{p(B|A) \cdot p(A)}{\int{p(B|u)p(u)du}}
+> $$
+>
+>   - **Posterior**: Probability of A being true given B
+>   - **Likelihood**: Probability of B being true
+given A
+>   - **Prior**: Probability of A being true (knowledge)
+>   - **Marginalization**: Probability of B being true
+
+By making the assumption of the probability of
+$y$ of being a gaussian with 0 mean and identity
+deviation, and assuming $x$ and $y$ independent
+and identically distributed:
+
+- $p(y) = \mathcal{N}(0, I) \rightarrow p(x|y) = \mathcal{N}(f(y), cI)$
+
+Since we technically need an integral over the denominator, we use
+**approximate techniques** such as **Variational INference**
+
+### Variational Inference
+
+<!-- TODO: See PDF 10 pgs 59 to 65-->
+
+### Reparametrization Trick
+
+Since $y$ is **technically sampled**, this makes it impossible
+to backpropagate the `mean` and `std-dev`, thus we add another
+variable, sampled from a *standard gaussian* $\zeta$, so that
+we have
+
+$$
+y = \sigma_x \cdot \zeta + \mu_x
+$$