migrated invariants.md to docusaurus

This commit is contained in:
Quinn Dougherty 2022-04-13 14:46:01 -04:00
parent 7251f5864a
commit b0cf8adb9f
5 changed files with 225 additions and 6 deletions

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@ -0,0 +1,123 @@
---
title: Statistical properties of algebraic combinations of distributions for property testing.
urlcolor: blue
author:
- Nuño Sempere
- Quinn Dougherty
abstract: This document outlines some properties about algebraic combinations of distributions. It is meant to facilitate property tests for [Squiggle](https://squiggle-language.com/), an estimation language for forecasters. So far, we are focusing on the means, the standard deviation and the shape of the pdfs.
---
The academic keyword to search for in relation to this document is "[algebra of random variables](https://wikiless.org/wiki/Algebra_of_random_variables?lang=en)". Squiggle doesn't yet support getting the standard deviation, denoted by $\sigma$, but such support could yet be added.
## Means and standard deviations
### Sums
$$
mean(f+g) = mean(f) + mean(g)
$$
$$\sigma(f+g) = \sqrt{\sigma(f)^2 + \sigma(g)^2}
$$
In the case of normal distributions,
$$
mean(normal(a,b) + normal(c,d)) = mean(normal(a+c, \sqrt{b^2 + d^2}))
$$
### Subtractions
$$
mean(f-g) = mean(f) - mean(g)
$$
$$
\sigma(f-g) = \sqrt{\sigma(f)^2 + \sigma(g)^2}
$$
### Multiplications
$$
mean(f \cdot g) = mean(f) \cdot mean(g)
$$
$$
\sigma(f \cdot g) = \sqrt{ (\sigma(f)^2 + mean(f)) \cdot (\sigma(g)^2 + mean(g)) - (mean(f) \cdot mean(g))^2}
$$
### Divisions
Divisions are tricky, and in general we don't have good expressions to characterize properties of ratios. In particular, the ratio of two normals is a Cauchy distribution, which doesn't have to have a mean.
## Probability density functions (pdfs)
Specifying the pdf of the sum/multiplication/... of distributions as a function of the pdfs of the individual arguments can still be done. But it requires integration. My sense is that this is still doable, and I (Nuño) provide some _pseudocode_ to do this.
### Sums
Let $f, g$ be two independently distributed functions. Then, the pdf of their sum, evaluated at a point $z$, expressed as $(f + g)(z)$, is given by:
$$
(f + g)(z)= \int_{-\infty}^{\infty} f(x)\cdot g(z-x) \,dx
$$
See a proof sketch [here](https://www.milefoot.com/math/stat/rv-sums.htm)
Here is some pseudocode to approximate this:
```js
// pdf1 and pdf2 are pdfs,
// and cdf1 and cdf2 are their corresponding cdfs
let epsilonForBounds = 2 ** -16;
let getBounds = (cdf) => {
let cdf_min = -1;
let cdf_max = 1;
let n = 0;
while (
(cdf(cdf_min) > epsilonForBounds || 1 - cdf(cdf_max) > epsilonForBounds) &&
n < 10
) {
if (cdf(cdf_min) > epsilonForBounds) {
cdf_min = cdf_min * 2;
}
if (1 - cdf(cdf_max) > epsilonForBounds) {
cdf_max = cdf_max * 2;
}
}
return [cdf_min, cdf_max];
};
let epsilonForIntegrals = 2 ** -16;
let pdfOfSum = (pdf1, pdf2, cdf1, cdf2, z) => {
let bounds1 = getBounds(cdf1);
let bounds2 = getBounds(cdf2);
let bounds = [
Math.min(bounds1[0], bounds2[0]),
Math.max(bounds1[1], bounds2[1]),
];
let result = 0;
for (let x = bounds[0]; (x = x + epsilonForIntegrals); x < bounds[1]) {
let delta = pdf1(x) * pdf2(z - x);
result = result + delta * epsilonForIntegrals;
}
return result;
};
```
## Cumulative density functions
TODO
## Inverse cumulative density functions
TODO
# To do:
- Provide sources or derivations, useful as this document becomes more complicated
- Provide definitions for the probability density function, exponential, inverse, log, etc.
- Provide at least some tests for division
- See if playing around with characteristic functions turns out anything useful

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@ -1,5 +1,7 @@
// @ts-check
// Note: type annotations allow type checking and IDEs autocompletion
const math = require("remark-math");
const katex = require("rehype-katex");
const lightCodeTheme = require("prism-react-renderer/themes/github");
const darkCodeTheme = require("prism-react-renderer/themes/dracula");
@ -14,7 +16,7 @@ const config = {
onBrokenLinks: "throw",
onBrokenMarkdownLinks: "warn",
favicon: "img/favicon.ico",
organizationName: "QURIResearch", // Usually your GitHub org/user name.
organizationName: "quantified-uncertainty", // Usually your GitHub org/user name.
projectName: "squiggle", // Usually your repo name.
plugins: [
@ -47,13 +49,15 @@ const config = {
sidebarPath: require.resolve("./sidebars.js"),
// Please change this to your repo.
editUrl:
"https://github.com/foretold-app/squiggle/tree/master/packages/website/",
"https://github.com/quantified-uncertainty/squiggle/tree/master/packages/website/",
remarkPlugins: [math],
rehypePlugins: [katex],
},
blog: {
showReadingTime: true,
// Please change this to your repo.
editUrl:
"https://github.com/foretold-app/squiggle/tree/master/packages/website/",
"https://github.com/quantified-uncertainty/squiggle/tree/master/packages/website/",
},
theme: {
customCss: require.resolve("./src/css/custom.css"),
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darkTheme: darkCodeTheme,
},
}),
stylesheets: [
{
href: "https://cdn.jsdelivr.net/npm/katex@0.13.24/dist/katex.min.css",
type: "text/css",
integrity:
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crossorigin: "anonymous",
},
],
};
module.exports = config;

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@ -17,7 +17,10 @@
"clsx": "^1.1.1",
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},
"browserslist": {
"production": [

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@ -40,6 +40,16 @@ const sidebars = {
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},
{
type: "category",
label: "Internal",
items: [
{
type: "autogenerated",
dirName: "Internal",
},
],
},
],
// But you can create a sidebar manually

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hast-util-to-text@^2.0.0:
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rehype-katex@^5:
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resolved "https://registry.yarnpkg.com/rehype-katex/-/rehype-katex-5.0.0.tgz#b556f24fde918f28ba1cb642ea71c7e82f3373d7"
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unified "^9.0.0"
unist-util-visit "^2.0.0"
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xtend "^4.0.0"
rehype-parse@^7.0.0:
version "7.0.1"
resolved "https://registry.yarnpkg.com/rehype-parse/-/rehype-parse-7.0.1.tgz#58900f6702b56767814afc2a9efa2d42b1c90c57"
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