migrated invariants.md to docusaurus

2022-04-13 14:46:01 -04:00 · 2022-04-13 14:46:01 -04:00 · b0cf8adb9f
commit b0cf8adb9f
parent 7251f5864a
5 changed files with 225 additions and 6 deletions
--- a/packages/website/docs/Internal/Invariants.md
+++ b/packages/website/docs/Internal/Invariants.md
@ -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
--- a/packages/website/docusaurus.config.js
+++ b/packages/website/docusaurus.config.js
@ -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"),
@ -111,6 +115,15 @@ const config = {
        darkTheme: darkCodeTheme,
      },
    }),
+  stylesheets: [
+    {
+      href: "https://cdn.jsdelivr.net/npm/katex@0.13.24/dist/katex.min.css",
+      type: "text/css",
+      integrity:
+        "sha384-odtC+0UGzzFL/6PNoE8rX/SPcQDXBJ+uRepguP4QkPCm2LBxH3FA3y+fKSiJ+AmM",
+      crossorigin: "anonymous",
+    },
+  ],
 };

 module.exports = config;
--- a/packages/website/package.json
+++ b/packages/website/package.json
@ -17,7 +17,10 @@
    "clsx": "^1.1.1",
    "prism-react-renderer": "^1.2.1",
    "react": "^18.0.0",
-    "react-dom": "^18.0.0"
+    "react-dom": "^18.0.0",
+    "remark-math": "^3",
+    "rehype-katex": "^5",
+    "hast-util-is-element": "1.1.0"
  },
  "browserslist": {
    "production": [
--- a/packages/website/sidebars.js
+++ b/packages/website/sidebars.js
@ -40,6 +40,16 @@ const sidebars = {
        },
      ],
    },
+    {
+      type: "category",
+      label: "Internal",
+      items: [
+        {
+          type: "autogenerated",
+          dirName: "Internal",
+        },
+      ],
+    },
  ],

  // But you can create a sidebar manually
--- a/yarn.lock
+++ b/yarn.lock
@ -3890,6 +3890,11 @@
  resolved "https://registry.yarnpkg.com/@types/json5/-/json5-0.0.29.tgz#ee28707ae94e11d2b827bcbe5270bcea7f3e71ee"
  integrity sha1-7ihweulOEdK4J7y+UnC86n8+ce4=

+"@types/katex@^0.11.0":
+  version "0.11.1"
+  resolved "https://registry.yarnpkg.com/@types/katex/-/katex-0.11.1.tgz#34de04477dcf79e2ef6c8d23b41a3d81f9ebeaf5"
+  integrity sha512-DUlIj2nk0YnJdlWgsFuVKcX27MLW0KbKmGVoUHmFr+74FYYNUDAaj9ZqTADvsbE8rfxuVmSFc7KczYn5Y09ozg==
+
 "@types/lodash@^4.14.181":
  version "4.14.181"
  resolved "https://registry.yarnpkg.com/@types/lodash/-/lodash-4.14.181.tgz#d1d3740c379fda17ab175165ba04e2d03389385d"
@ -6394,7 +6399,7 @@ commander@^6.2.1:
  resolved "https://registry.yarnpkg.com/commander/-/commander-6.2.1.tgz#0792eb682dfbc325999bb2b84fddddba110ac73c"
  integrity sha512-U7VdrJFnJgo4xjrHpTzu0yrHPGImdsmD95ZlgYSEajAn2JKzDhDTPG9kBTefmObL2w/ngeZnilk+OV9CG3d7UA==

-commander@^8.3.0:
+commander@^8.0.0, commander@^8.3.0:
  version "8.3.0"
  resolved "https://registry.yarnpkg.com/commander/-/commander-8.3.0.tgz#4837ea1b2da67b9c616a67afbb0fafee567bca66"
  integrity sha512-OkTL9umf+He2DZkUq8f8J9of7yL6RJKI24dVITBmNfZBmri9zYZQrKkuXiKhyfPSu8tUhnVBB1iKXevvnlR4Ww==
@ -9278,6 +9283,11 @@ hast-util-from-parse5@^6.0.0:
    vfile-location "^3.2.0"
    web-namespaces "^1.0.0"

+hast-util-is-element@1.1.0, hast-util-is-element@^1.0.0:
+  version "1.1.0"
+  resolved "https://registry.yarnpkg.com/hast-util-is-element/-/hast-util-is-element-1.1.0.tgz#3b3ed5159a2707c6137b48637fbfe068e175a425"
+  integrity sha512-oUmNua0bFbdrD/ELDSSEadRVtWZOf3iF6Lbv81naqsIV99RnSCieTbWuWCY8BAeEfKJTKl0gRdokv+dELutHGQ==
+
 hast-util-parse-selector@^2.0.0:
  version "2.2.5"
  resolved "https://registry.yarnpkg.com/hast-util-parse-selector/-/hast-util-parse-selector-2.2.5.tgz#d57c23f4da16ae3c63b3b6ca4616683313499c3a"
@ -9310,6 +9320,15 @@ hast-util-to-parse5@^6.0.0:
    xtend "^4.0.0"
    zwitch "^1.0.0"

+hast-util-to-text@^2.0.0:
+  version "2.0.1"
+  resolved "https://registry.yarnpkg.com/hast-util-to-text/-/hast-util-to-text-2.0.1.tgz#04f2e065642a0edb08341976084aa217624a0f8b"
+  integrity sha512-8nsgCARfs6VkwH2jJU9b8LNTuR4700na+0h3PqCaEk4MAnMDeu5P0tP8mjk9LLNGxIeQRLbiDbZVw6rku+pYsQ==
+  dependencies:
+    hast-util-is-element "^1.0.0"
+    repeat-string "^1.0.0"
+    unist-util-find-after "^3.0.0"
+
 hastscript@^5.0.0:
  version "5.1.2"
  resolved "https://registry.yarnpkg.com/hastscript/-/hastscript-5.1.2.tgz#bde2c2e56d04c62dd24e8c5df288d050a355fb8a"
@ -10968,6 +10987,13 @@ junk@^3.1.0:
  resolved "https://registry.yarnpkg.com/junk/-/junk-3.1.0.tgz#31499098d902b7e98c5d9b9c80f43457a88abfa1"
  integrity sha512-pBxcB3LFc8QVgdggvZWyeys+hnrNWg4OcZIU/1X59k5jQdLBlCsYGRQaz234SqoRLTCgMH00fY0xRJH+F9METQ==

+katex@^0.13.0:
+  version "0.13.24"
+  resolved "https://registry.yarnpkg.com/katex/-/katex-0.13.24.tgz#fe55455eb455698cb24b911a353d16a3c855d905"
+  integrity sha512-jZxYuKCma3VS5UuxOx/rFV1QyGSl3Uy/i0kTJF3HgQ5xMinCQVF8Zd4bMY/9aI9b9A2pjIBOsjSSm68ykTAr8w==
+  dependencies:
+    commander "^8.0.0"
+
 keyv@^3.0.0:
  version "3.1.0"
  resolved "https://registry.yarnpkg.com/keyv/-/keyv-3.1.0.tgz#ecc228486f69991e49e9476485a5be1e8fc5c4d9"
@ -14541,6 +14567,18 @@ regjsparser@^0.8.2:
  dependencies:
    jsesc "~0.5.0"

+rehype-katex@^5:
+  version "5.0.0"
+  resolved "https://registry.yarnpkg.com/rehype-katex/-/rehype-katex-5.0.0.tgz#b556f24fde918f28ba1cb642ea71c7e82f3373d7"
+  integrity sha512-ksSuEKCql/IiIadOHiKRMjypva9BLhuwQNascMqaoGLDVd0k2NlE2wMvgZ3rpItzRKCd6vs8s7MFbb8pcR0AEg==
+  dependencies:
+    "@types/katex" "^0.11.0"
+    hast-util-to-text "^2.0.0"
+    katex "^0.13.0"
+    rehype-parse "^7.0.0"
+    unified "^9.0.0"
+    unist-util-visit "^2.0.0"
+
 rehype-parse@^6.0.2:
  version "6.0.2"
  resolved "https://registry.yarnpkg.com/rehype-parse/-/rehype-parse-6.0.2.tgz#aeb3fdd68085f9f796f1d3137ae2b85a98406964"
@ -14550,6 +14588,14 @@ rehype-parse@^6.0.2:
    parse5 "^5.0.0"
    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"
+  integrity sha512-fOiR9a9xH+Le19i4fGzIEowAbwG7idy2Jzs4mOrFWBSJ0sNUgy0ev871dwWnbOo371SjgjG4pwzrbgSVrKxecw==
+  dependencies:
+    hast-util-from-parse5 "^6.0.0"
+    parse5 "^6.0.0"
+
 relateurl@^0.2.7:
  version "0.2.7"
  resolved "https://registry.yarnpkg.com/relateurl/-/relateurl-0.2.7.tgz#54dbf377e51440aca90a4cd274600d3ff2d888a9"
@ -14589,6 +14635,11 @@ remark-footnotes@2.0.0:
  resolved "https://registry.yarnpkg.com/remark-footnotes/-/remark-footnotes-2.0.0.tgz#9001c4c2ffebba55695d2dd80ffb8b82f7e6303f"
  integrity sha512-3Clt8ZMH75Ayjp9q4CorNeyjwIxHFcTkaektplKGl2A1jNGEUey8cKL0ZC5vJwfcD5GFGsNLImLG/NGzWIzoMQ==

+remark-math@^3:
+  version "3.0.1"
+  resolved "https://registry.yarnpkg.com/remark-math/-/remark-math-3.0.1.tgz#85a02a15b15cad34b89a27244d4887b3a95185bb"
+  integrity sha512-epT77R/HK0x7NqrWHdSV75uNLwn8g9qTyMqCRCDujL0vj/6T6+yhdrR7mjELWtkse+Fw02kijAaBuVcHBor1+Q==
+
 remark-mdx@1.6.22:
  version "1.6.22"
  resolved "https://registry.yarnpkg.com/remark-mdx/-/remark-mdx-1.6.22.tgz#06a8dab07dcfdd57f3373af7f86bd0e992108bbd"
@ -14673,7 +14724,7 @@ repeat-element@^1.1.2:
  resolved "https://registry.yarnpkg.com/repeat-element/-/repeat-element-1.1.4.tgz#be681520847ab58c7568ac75fbfad28ed42d39e9"
  integrity sha512-LFiNfRcSu7KK3evMyYOuCzv3L10TW7yC1G2/+StMjK8Y6Vqd2MG7r/Qjw4ghtuCOjFvlnms/iMmLqpvW/ES/WQ==

-repeat-string@^1.5.4, repeat-string@^1.6.1:
+repeat-string@^1.0.0, repeat-string@^1.5.4, repeat-string@^1.6.1:
  version "1.6.1"
  resolved "https://registry.yarnpkg.com/repeat-string/-/repeat-string-1.6.1.tgz#8dcae470e1c88abc2d600fff4a776286da75e637"
  integrity sha1-jcrkcOHIirwtYA//Sndihtp15jc=
@ -16509,6 +16560,18 @@ unified@^8.4.2:
    trough "^1.0.0"
    vfile "^4.0.0"

+unified@^9.0.0:
+  version "9.2.2"
+  resolved "https://registry.yarnpkg.com/unified/-/unified-9.2.2.tgz#67649a1abfc3ab85d2969502902775eb03146975"
+  integrity sha512-Sg7j110mtefBD+qunSLO1lqOEKdrwBFBrR6Qd8f4uwkhWNlbkaqwHse6e7QvD3AP/MNoJdEDLaf8OxYyoWgorQ==
+  dependencies:
+    bail "^1.0.0"
+    extend "^3.0.0"
+    is-buffer "^2.0.0"
+    is-plain-obj "^2.0.0"
+    trough "^1.0.0"
+    vfile "^4.0.0"
+
 union-value@^1.0.0:
  version "1.0.1"
  resolved "https://registry.yarnpkg.com/union-value/-/union-value-1.0.1.tgz#0b6fe7b835aecda61c6ea4d4f02c14221e109847"
@ -16545,6 +16608,13 @@ unist-builder@2.0.3, unist-builder@^2.0.0:
  resolved "https://registry.yarnpkg.com/unist-builder/-/unist-builder-2.0.3.tgz#77648711b5d86af0942f334397a33c5e91516436"
  integrity sha512-f98yt5pnlMWlzP539tPc4grGMsFaQQlP/vM396b00jngsiINumNmsY8rkXjfoi1c6QaM8nQ3vaGDuoKWbe/1Uw==

+unist-util-find-after@^3.0.0:
+  version "3.0.0"
+  resolved "https://registry.yarnpkg.com/unist-util-find-after/-/unist-util-find-after-3.0.0.tgz#5c65fcebf64d4f8f496db46fa8fd0fbf354b43e6"
+  integrity sha512-ojlBqfsBftYXExNu3+hHLfJQ/X1jYY/9vdm4yZWjIbf0VuWF6CRufci1ZyoD/wV2TYMKxXUoNuoqwy+CkgzAiQ==
+  dependencies:
+    unist-util-is "^4.0.0"
+
 unist-util-generated@^1.0.0:
  version "1.1.6"
  resolved "https://registry.yarnpkg.com/unist-util-generated/-/unist-util-generated-1.1.6.tgz#5ab51f689e2992a472beb1b35f2ce7ff2f324d4b"