feat: Added notes over pdfs

as well as cleaned up the doc a bit

note: I don't really like the term "invariants"

packages/squiggle-lang/.gitignore

@@ -19,3 +19,4 @@ yarn-error.log
 dist
 *.coverage
 _coverage
+result

packages/squiggle-lang/__tests__/docs/README.md

@@ -0,0 +1,16 @@
+# Properties
+
+Using `nix`. Where `o` is `open` on OSX and `xdg-open` on linux, 
+
+```sh
+nix-build
+o result/property-tests.pdf
+```
+
+Without `nix`, you can install `pandoc` and `pdflatex` yourself and see `make.sh` for the rendering command. 
+
+## _Details_
+
+The `invariants.pdf` document is _normative and aspirational_. It does not document tests as they exist in the codebase, but represents how we think squiggle ought to be tested. 
+
+We are partially bottlenecked by the rescript ecosystem's maturity with respect to property-based testing. 

packages/squiggle-lang/__tests__/docs/default.nix

@@ -0,0 +1,26 @@
+{ chan ? "a7ecde854aee5c4c7cd6177f54a99d2c1ff28a31"  # 21.11
+, pkgs ? import (builtins.fetchTarball { url = "https://github.com/NixOS/nixpkgs/archive/${chan}.tar.gz"; }) {}
+}:
+# Style sheets https://github.com/citation-style-language/styles/
+with pkgs;
+
+let deps = [
+  pandoc 
+  (texlive.combine 
+    { inherit (texlive) scheme-small datetime; }
+  )
+]; in
+stdenv.mkDerivation {
+  name = "render_squiggle_properties";
+  src = ./.;
+  buildInputs = deps;
+  buildPhase = ''
+    echo rendering...
+    pandoc -s invariants.md -o invariants.pdf 
+    echo rendered.
+  '';
+  installPhase = ''
+    mkdir -p $out
+    cp invariants.pdf $out/invariants.pdf
+    '';
+}

packages/squiggle-lang/__tests__/docs/invariants.md

@@ -0,0 +1,115 @@
+---
+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
+

packages/squiggle-lang/__tests__/docs/make.sh

@@ -0,0 +1 @@
+ pandoc -s invariants.md -o invariants.pdf