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Calculus

Basics of probability

Probability distributions

Bayes' theorem

Frequentist techniques

Pitfalls

External links

The following gives a list of blog articles and short glossary entries covering some of the basic mathematics used on this blog. At the bottom is a list of web-based material authored by people who typically know more maths than I.

This round-up is mainly intended to serve as an easy entry point to interested readers of the blog who find the mathematical content difficult. It may also serve as a quick reference for readers comfortable with maths, but who, like me, don't see the need to memorize every useful formula. This page is actually a very slow response to a useful suggestion made by Richard Carrier, in summer 2012.

Other, more esoteric topics in probability have been covered on the blog, but are not included here, in an effort to restrict this list to matters of the most direct practical significance.

If you have any questions about the material here, or other mathematical procedures I've used, you can ask a question in the comments section, below, or contact me directly by email. My email address is in my blogger profile, linked on the right hand side of the page.

I have tried to arrange the list in a logical order. If you are new to probability and statistics, the following can be treated as a handy short course, that will hopefully bring you up to the level of being able to begin performing real-life statistical analyses yourself. One suggestion, if you want to work through the material in this way: several blog posts start with the statement of some technical problem - get the most out of the offered solution by first spending a bit of time thinking about it and trying to map out a route to the solution yourself.

N.B.: The tables are for reference only, do not be intimidated by the amount of information in them!

Specific distribution functions:

Pitfalls in Inference

A tiny selection from an almost limitless list of human intellectual frailties. The first three items occur both in quantitative analysis and in informal, daily reasoning, while the last two are usually limited to numerical, statistical analyses.

Resources Elsewhere on the Web

Other, more esoteric topics in probability have been covered on the blog, but are not included here, in an effort to restrict this list to matters of the most direct practical significance.

If you have any questions about the material here, or other mathematical procedures I've used, you can ask a question in the comments section, below, or contact me directly by email. My email address is in my blogger profile, linked on the right hand side of the page.

I have tried to arrange the list in a logical order. If you are new to probability and statistics, the following can be treated as a handy short course, that will hopefully bring you up to the level of being able to begin performing real-life statistical analyses yourself. One suggestion, if you want to work through the material in this way: several blog posts start with the statement of some technical problem - get the most out of the offered solution by first spending a bit of time thinking about it and trying to map out a route to the solution yourself.

## Calculus

Short, easy articles covering the very basic concepts (including tables of standard formulae), for those unfamiliar, or in need of reminding:- Derivatives (jump directly to table of standard derivatives)
- Integrals (jump to table of standard integrals)

N.B.: The tables are for reference only, do not be intimidated by the amount of information in them!

## Basics of Probability

3 Basic Laws of Probability:

Simple application examples:

## Probability Distributions

Specific distribution functions:

## Bayes' Theorem

- Bayes's theorem, basic description in the glossary
- The Base Rate Fallacy, application example - testing binary hypotheses
- How to make
*ad-hominem*arguments, application example - an infinite number of hypotheses - Error Bars for Binary Parameters gives further analysis of an equivalent problem
- Signal and Noise, another application of infinite hypotheses, using a Gaussian noise model
- Parameter estimation, introduced in the glossary
- Model comparison in the glossary
- Ockham's razor, 1st blog post on model comparison
- The Ockham factor, 2nd blog post on model comparison

Frequentist Techniques

- The frequency interpretation of probability, introduced in the glossary
- Null-hypothesis significance tests, in the glossary
- Null-hypothesis significance tests, critical blog article
- p-values
- Maximum likelihood
- Fly papers and photon detectors, blog article about maximum likelihood

Pitfalls in Inference

A tiny selection from an almost limitless list of human intellectual frailties. The first three items occur both in quantitative analysis and in informal, daily reasoning, while the last two are usually limited to numerical, statistical analyses.

- Base-Rate Fallacy
- Mind-Projection Fallacy (discussed in the glossary also)
- Regression Fallacy
- Simpson's Paradox
- Berkson's Paradox

Resources Elsewhere on the Web

- Wikipedia has countless good-quality technical articles on mathematical topics, with a good mix of technical and not-too-technical info. It's one of the first places I look, when I need to find a formula or research some new topic. Many useful external links are also provided. Occasional technical errors or inconsistencies can be found.

- Mathworld also covers an enormous breadth of topics, but the articles tend to dive straight in at a highly technical level. I don't recall spotting any errors - certainly a good place to verify other research.

- Wolfram Alpha is a fantastic free web resource. You can type in mathematical questions (requests for analytical formulae or specific calculations), then a search engine tries its best (with reasonable success) to interpret your question precisely, then return an answer, with as much additional info (plots, series expansions, etc.) as you are likely to want.

- Mathsisfun.com is a website providing friendly introductions to several important mathematical topics (e.g. algebra, geometry, statistics, calculus). The information content is of a high standard, and the explanations simple and clear enough for young school students. Some advanced topics are also covered, and the site serves as a goldmine for maturer people, looking for a quick refresher.

- Green Tea Press has several free good-quality eBooks on computing and statistics, authored by Allen Downey. A forthcoming book at Green Tea Press (as of 7-8-2013, in draft form and not on their main page), Think Bayes, is worth noting.

- Paul's Online Math Notes is a set of excellent lecture notes, mostly on differential and integral calculus, and the solution of differential equations.

- Richard Carrier has two wonderful talks on Bayes' theorem. They are: (1) Lust For Glory (introduced here) and (2) Miracles and Historical Method. They're highly accessible, with no prior mathematical skill required, and they demonstrate beautifully the applicability of probability theory to all fields of inquiry, including history and supernatural claims.

- On the topic of common errors of reasoning, Fallacy Files is an excellent website.

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