Put your code inside a raw tag. This helps to stop Liquid templating e.g. {{ }} is Moustache.js syntax that must not be evaluated by Jekyll.
I donâ€™t know if this will prevent _foo_ from becoming <em>foo</em>

This code will be evaluated.
{{ site.description }}
{% raw %}
This code will stay as plain text and will not be evaluated by Jekyll
{{ site.description }}
{% endraw %}
This code will also be evaluated.
{{ site.description }}

Using this {{ â€¦ }} syntax to inject math formula in will produce:

The page build failed for the `master` branch with the following error:
The variable `{{ $\begin{aligned}` on line 7 in `_posts/2021-01-17-Expectation.md` was not properly closed with `}}`.

Speaking of that attempt is there a way to disable liquid somehow?

I used at first the span and div tags but it failed disabling the liquid to interpret the _.

The other solution you proposed looks like a solution but is complicated.

{% raw %} seams it is not working inside the .md file, it would work from .html, but for articles I use .md files. I wonder if I can use {% raw %} from .md somehow.

Instead of putting your script tag on your page, you put it in a separate file and import or â€śincludeâ€ť it so it gets added to the current page. This also makes it more reusuable.

Another approach. If you are happy to have your script tags on every page, then put them in a layout - which is HTML and will not try and evaluate underscores as markdown.

Note that type defaults to text/javascript, so you can leave it out.

For performance reasons, it doesnâ€™t matter whether you actually use the script on a page after it is loaded. The browser will cache the external .js script so that on a navigation click, the next page will load faster because that asset is already in the browser.

Hi, it looks like using div tag is working now. The only trick is that div tag must not be close to the formula, there should be the space in between. When using the below code it wonâ€™t work.

---
published: true
layout: post
title: Expectation
permalink: /expectation
---
In here I will set some notation of the mathematical expectation of discrete and continuous random variable (RV).
### Discrete RV expectation
In case of the discrete variable the expectation, or expected value, of some function $f(x)$ with respect to a probability distribution $P(x)$ is the average, or mean value, that $f$ takes on when $x$ is drawn from $P$:
{% raw %}
$\begin{aligned} \mathbb{E}_{\mathrm{x} \sim P} [ f(x) ]= \sum_{x} P(x) f(x) \end{aligned}$
{% endraw %}
$\mathrm{x} \sim P$ means $\mathrm{x}$ is drawn from distribution $P(x)$ or just from $P$. Inside the $[\ldots]$ brackets we should have some function $f(x)$, or in special case just $x$.
### Continuous RV expectation
For continuous variables it is computed with the integral:
$\mathbb{E}_{\mathrm{x} \sim p}[f(x)]=\int p(x) f(x) d x$
If the identity of the distribution is clear from the context we may write simple:
$\mathbb{E}_{\mathrm{x} }[f(x)]$
If the random variable is clear from the context we may write:
$\mathbb{E}[f(x)]$
By default, we can assume that
$\mathbb{E}[\cdot]$
averages over the values of all the random variables inside the brackets.
Likewise, when there is no ambiguity, we may omit the square brackets:
$\mathbb{E}$
Expectations are linear:
$\mathbb{E} \_ {\mathrm{x}}[\alpha f(x)+\beta g(x)]=\alpha \mathbb{E}_{\mathrm{x}}[f(x)]+\beta \mathbb{E}_{\mathrm{x}}[g(x)]$
We define the <span> $\mathbb X = \{{\pmb x^{(1)}}, \ldots ,{\pmb x^{(m)}}\}$ </span>
$p_{data}(\mathrm x)$
$p_{model}(\pmb {\mathrm x}; \pmb \theta)$
$p_{model}(\pmb {x}; \pmb \theta)$ maps any concrete configuration to $p_{data}(\pmb {x})$
$\theta_{ML} = arg max$
$\begin{aligned} \boldsymbol{\theta}_{\mathrm{ML}} &=\underset{\boldsymbol{\theta}}{\arg \max } p_{\text {model }}(\mathbb{X} ; \boldsymbol{\theta}) \\ &=\underset{\boldsymbol{\theta}}{\arg \max } \prod_{i=1}^{m} p_{\text {model }}\left(\boldsymbol{x}^{(i)} ; \boldsymbol{\theta}\right) \end{aligned}$
For numeric stability:
$\boldsymbol{\theta}_{\mathrm{ML}}=\underset{\boldsymbol{\theta}}{\arg \max } \sum_{i=1}^{m} \log p_{\text {model }}\left(\boldsymbol{x}^{(i)} ; \boldsymbol{\theta}\right)$
Defined by train data:
$\boldsymbol{\theta}_{\mathrm{ML}}=\underset{\boldsymbol{\theta}}{\arg \max } \mathbb{E}_{\mathbf{x} \sim \hat{p}_{\text {data }}} \log p_{\text {model }}(\boldsymbol{x} ; \boldsymbol{\theta})$
Final:
$D_{\mathrm{KL}}\left(\hat{p}_{\text {data }} \| p_{\text {model }}\right)=
\mathbb{E}_{\mathbf{x} \sim \hat{p}_{\text {data }}}
\left[\overbrace{\log \hat{p}_{\text {data }}(\pmb{x})}^{\ data \ generating \ process}-\log p_{\text {model }}(\pmb{x})\right]$
Important part:
$\mathbb{E}_{\mathbf{x} \sim \hat{p}_{\text {data }}}\left[-\log p_{\text {model }}(\pmb{x})\right]$
where:
$\hat{p}_{\text {data }}$ is empirical distribution, ${p}_{\text {data }}$ is true distribution
Once we have the expectation we can define the variance and covariance.
## Variance
Variance give us the measure how much values of a function of a random variable $\text x$ vary as we sample different values of $\mathrm x$ from it's probability distribution.
$\operatorname{Var}(f(x))=\mathbb{E}\left[(f(x)-\mathbb{E}[f(x)])^{2}\right]$
## Covariance
The covariance gives some sense of how much two values are linearly related to each other, as well as the scale of these variables:
$\operatorname{Cov}\left(f(x), g(y)\right)=\mathbb{E}[(f(x)-\mathbb{E}[f(x)])(g(y)-\mathbb{E}[g(y)])]$

While div tag works I wonder, why I may not use div tag with $$ â€¦ $$ and why I need to use div tag with $ â€¦ $. Is there something like a setting I may use.

I wonder if there is some explanation possible checking the Jekyll source.