Cum sociis natoque penatibus et magnis dis parturient montes, nascetur ridiculus mus. Aenean eu leo quam. Pellentesque ornare sem lacinia quam venenatis vestibulum. Sed posuere consectetur est at lobortis. Cras mattis consectetur purus sit amet fermentum.

Curabitur blandit tempus porttitor. Nullam quis risus eget urna mollis ornare vel eu leo. Nullam id dolor id nibh ultricies vehicula ut id elit.

Etiam porta sem malesuada magna mollis euismod. Cras mattis consectetur purus sit amet fermentum. Aenean lacinia bibendum nulla sed consectetur.

Inline HTML elements

HTML defines a long list of available inline tags, a complete list of which can be found on the Mozilla Developer Network.

• To bold text, use **To bold text**.
• To italicize text, use *To italicize text*.
• Abbreviations, like HTML should be defined like this *[HTML]: HyperText Markup Language.
• Citations, like — Mark otto, should use <cite>.
• Deleted text should use ~~deleted~~ and inserted text should use <ins>.
• Superscript ^text uses <sup> and subscript _text uses <sub>.

Most of these elements are styled by browsers with few modifications on our part.

Heading 2

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Heading 3

Vivamus sagittis lacus vel augue rutrum faucibus dolor auctor.

Heading 4

Vivamus sagittis lacus vel augue rutrum faucibus dolor auctor.

Heading 5

Vivamus sagittis lacus vel augue rutrum faucibus dolor auctor.

Heading 6

Vivamus sagittis lacus vel augue rutrum faucibus dolor auctor.

Code

Cum sociis natoque penatibus et magnis dis code element montes, nascetur ridiculus mus.

// Example can be run directly in your JavaScript console

// Create a function that takes two arguments and returns the sum of those
// arguments
var adder = new Function("a", "b", "return a + b");

// Call the function
adder(2, 6);
// > 8

Lists

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• Praesent commodo cursus magna, vel scelerisque nisl consectetur et.
• Donec id elit non mi porta gravida at eget metus.
• Nulla vitae elit libero, a pharetra augue.

Donec ullamcorper nulla non metus auctor fringilla. Nulla vitae elit libero, a pharetra augue.

1. Vestibulum id ligula porta felis euismod semper.
2. Cum sociis natoque penatibus et magnis dis parturient montes, nascetur ridiculus mus.
3. Maecenas sed diam eget risus varius blandit sit amet non magna.

Cras mattis consectetur purus sit amet fermentum. Sed posuere consectetur est at lobortis.

HyperText Markup Language (HTML)

The language used to describe and define the content of a Web page

Cascading Style Sheets (CSS)

Used to describe the appearance of Web content

JavaScript (JS)

The programming language used to build advanced Web sites and applications

Integer posuere erat a ante venenatis dapibus posuere velit aliquet. Morbi leo risus, porta ac consectetur ac, vestibulum at eros. Nullam quis risus eget urna mollis ornare vel eu leo.

Images

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Center image with caption

This is caption for image above

Tables

Aenean lacinia bibendum nulla sed consectetur. Lorem ipsum dolor sit amet, consectetur adipiscing elit.

But what I want/need is to have one row per student and the periods across the top as. For example:

Nullam id dolor id nibh ultricies vehicula ut id elit. Sed posuere consectetur est at lobortis. Nullam quis risus eget urna mollis ornare vel eu leo.

MathJax

Let’s formalize. For finite sample $\mathcal{X}$ of size $N$, we define an unnormalized mass function $\tilde{p} : \mathcal{X} \to [0, \infty)$ and let $Z:= \sum_{x \in \mathcal{X}} \tilde{p}(x)$ be the normalizing partition function. We then define $\phi(x) := \ln \tilde{p}(x)$ as the log-unnormalized probabilities or the potential function.

Our algorithm is then:

1. Add Gumbel distributed noise to our potential functions
2. Find the MAP of this perturbed value over all $x \in \mathcal{X}$. Call this value $z_i$
3. Repeat steps 1 and 2 multiple times and then collect the mean $\hat{Z} \approx Z$

But why?

We want to prove the supposedly useful Gumbel trick then using the Perturb-and-MAP method, specifically

where $\phi(x)$ has been defined as the potentials and $\gamma \sim \text{Gumbel}(-c)$ where $c$ is the Euler-Mascheroni constant.

Because the mean of $\text{Gumbel}(\mu)$ is $\mu + c$, we can show that $\ln Z$ and then $Z$ are recoverable.

A brief Gumbal interlude

The Gumbel distribution is traditionally used to model the maxima of already extreme events. For example, what will be worst earthquake next year given the measurements of the worst earthquakes in the past 10 years in San Francisco?

A variable $X$ drawn from $\text{Gumbel}(\mu)$ has the probability distribution

where $z = x - \mu$. For this problem, we set the scale parameter $\beta = 1$ whereas the location parameter $\mu$ remains free.

Additionally, the cumulative distribution function of a $\text{Gumbel}(\mu)$ is

This will become useful!

The actual proof

We want to find the value of $x$ that maximizes $\phi(x) + \gamma(x)$. Thinking in terms of the CDF, we want all values of $x \in \mathcal{X}$ to produce smaller or equal values

The first equality follows from multiplying the Gumbel CDF $F(t)$ of $\gamma(x)$ of all possible values to capture the maximum. The second equality comes from expanding out the Gumbel CDF. The third equality consolidates the potential functions such that $Z = \sum_{x \in \mathcal{X}} \phi(x)$. The fourth equality sticks the $\ln Z$ back into the $\exp$ function. The last equality compresses the probability back into the Gumbel CDF, except set at a different location. $\blacksquare$

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