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Welcome to the stats map, a (growing) collection of notes on various subtopics in statistics, probability, and theoretical computer science.

Unlike a traditional textbook, these notes prioritize the connections between subjects instead of giving a full treatment of each subject on its own. I don’t provide many proofs or formal definitions. This is an attempt to understand how the world of statistics hangs together; to chart the outlines of what we know and the techniques we use. It is an attempt at a Zettelkasten—or second brain—for statistics and probability.

These are, obviously, my notes. They are not unbiased. They reflect my opinions on what’s interesting and what’s not. There will be significant detail on some topics and a total absence of many topics. That said, this is still in it’s early stages (it’s missing even basic material from a standard first year grad course in theoretical statistics), and it will continue to be expanded in the coming months and years.

How to navigate

If you click on a link you will follow it. You can see a list of all notes on the left hand side of the page, and backlinks to the current note on the right hand side (or on the bottom if you’re on a small screen).

What’s in here

Possible places to start:

• game-theoretic statistics and game-theoretic probability
• e-values and e-processes
• concentration inequalities or techniques for multivariate concentration
• maximal inequalities
• central limit theorems
• hypothesis testing

Or if you’re interested in the philosophical foundations of statistics:

• The foundations of statistics or arguments therein, like Fisher’s paradigm versus the NP paradigm or why current statistical practice combines the Fisherian and Neyman-Pearson perspectives.
• Various statistical schools of thoughts and their methods (non-contradictory): Bayesian statistics, frequentist statistics, and game-theoretic statistics
• Various philosophies of probability (very much contradictory): Bayesian interpretation of probability, frequentist interpretation of probability, instrumentalist theory of probability
• More than you wanted to know about some philosophical issues with p-values

Why make this?

1. Writing helps me digest and aggregate information. You haven’t truly understood something until you can teach it, and you can’t teach it until you can write it down coherently.
2. You can know a lot of material but not know how it’s connected. The best researchers don’t simply have detailed knowledge of many ideas, they have a mental map of how all these areas fit together. This is my exported mental map.
3. Emphasizing the connections between topics helps you spot patterns. Similar techniques are often used in different areas.
4. It helps you spot holes in the literature. Writing down what is known helps carve out what is unknown.

More on the philosophy

This is inspired by the Zettelkasten method, an approach to note-taking and personal knowledge management.

Most note taking methods are based on hierarchies. You keep folders on various topics, subfolders on subtopics, and so on. In statistics, you might have a folders on concentration inequalities, information theory, and hypothesis testing. But this makes it hard to see the connections between these areas. For instance, there is an explicit connection between concentration and hypothesis testing (see duality between hypothesis tests and CIs), and information theoretic methods are often used to construct concentration inequalities (see techniques for multivariate concentration). Where should you put the notes on these connections?

Zettelkasten systems solve this problem by abolishing hierarchical note-taking. They instead adopt a network-like architecture, prioritizing the connections between ideas just as much as the ideas themselves. Concretely, it consists of small files (zettels) that are linked to each other. Here this is done via hyperlinks between markdown files, but historically it was done with index cards. Zettelkasten systems have a long history:

Born out of the commonplace tradition with modifications by Conrad Gessner (1516-1565) and descriptions by Johann Jacob Moser (1701–1785), the Zettelkasten, a German word translated as “slip box”, is generally a collection of highly curated atomic notes collected on slips of paper or index cards. Zettelkasten were made simpler to create and maintain with the introduction of the mass manufacture of index cards (and card boxes and furniture) in the early 20th century.

• Chris Aldrich on the history of note-taking tools.

There has been a renewed interest in Zettelkasten and related note-taking systems in recent years. Common keywords in this space are evergreen notes, second-brain, digital garden, and personal knowledge management. Some public examples of these kind of note taking systems come from Andy Matushack, Nitin Pai, and Joschua.

Zettelkasten systems are typically used to construct and grow an inventory of your own thoughts and insights. Seeing as I did not invent the vast majority of the maths discussed here, this note set deviates from this philosophy. But I think network-style note taking systems for technical subjects are an under-explored medium (for both research and pedagogy), so here we are.

Who am I?

I am Ben. I’m a PhD student in the machine learning and statistics program at CMU. I study things like sequential statistics, e-values, and concentration of measure. Occasionally I also read, write, and yell about philosophy.

Here is my face, cartoonified.

If you spot any mistakes, you can email me at benchugg $\alpha \tau$ cmu.edu.