ELOB (evidence lower bound) for diffusion I am trying to understand the loss definition for diffusion, but lots of questions already arise during the first step of the derivation. (I have nearly zero statistic knowledge, but I am good at linear algebra and calculus, so please point me to the right resource if possible). I am currently learning from https://www.assemblyai.com/blog/diffusion-models-for-machine-learning-introduction/

$\text{L}=\mathbb{E}_q\left[-\text{log}\frac{p_\theta(x_{0:T})}{q(x_{1:T}|x_0)}\right]$
using Markov assumption, we get
$\mathbb{E}_q\left[-\text{log}p(x_T)-\sum_{t\geq1}\text{log}\frac{p_\theta(x_{t-1}|x_t)}{q(x_{t}|x_{t-1})}\right]$


*

*What is the meaning of $x_{0:T}$? Is it equal to $p_\theta(x_{0:T}) = p_\theta(x_1,x_2,\dots,x_T)$? I haven't seen anywhere explaining it.

*From these two steps, does it mean that $p_\theta(x_{0:T})=\prod_{t\geq1} p_\theta(x_{t-1}|x_t)$? Is it the Markov assumption mentioned above? But when I check WIKI, the definition of Markov property is $p(x_t|x_{t-1},\dots,x_{0})=p(x_t|x_{t-1})$, which is quite different.

*I have the same doubt for $q(x_{1:T}|x_0)$

*As mentioned in another post, $\mathbb{E}_q(X)$ basically means $\int Xdq$. Does it mean that $\mathbb{E}_q\left[-\text{log}\frac{p_\theta(x_{0:T})}{q(x_{1:T}|x_0)}\right]=\int -\text{log}\frac{p_\theta(x_{0:T})}{q(x_{1:T}|x_0)}dq$ here?


*

*Yes. It should have been defined but that is the usual notation.

*Its not,

$$  P(x_0, ..., x_t) = P(x_T|x_0, ... x_{t-1})P(x_0, ..., x_{T-1})$$
Using Markov property
$$ P(x_t|x_{t-1})P(x_{t-1}|x_{t-2})...P(x_1|x_0)P(x_0) = P(x_0) \prod_{t\geq1} P(x_t|x_{t-1}).$$


*Similar to 3, but notice that we already condition on $x_0$.

