feat: Added notes over pdfs
as well as cleaned up the doc a bit note: I don't really like the term "invariants"
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# Squiggle invariants
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---
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title: Statistical properties of algebraic combinations of distributions for property testing.
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urlcolor: blue
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author:
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- Nuño Sempere
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- Quinn Dougherty
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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.
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Here are some property tests for squiggle. I am testing mostly for the mean and the standard deviation. I know that squiggle doesn't yet have functions for the standard deviation, but they could be added.
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---
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The keywords to search for are "[algebra of random variables](https://wikiless.org/wiki/Algebra_of_random_variables?lang=en)".
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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.
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## Means and standard deviations
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## Means and standard deviations
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### Sums
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### Sums
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$$ mean(f+g) = mean(f) + mean(g) $$
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$$ mean(f+g) = mean(f) + mean(g) $$
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$$ std(f+g) = \sqrt{std(f)^2 + std(g)^2} $$
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$$ \sigma(f+g) = \sqrt{\sigma(f)^2 + \sigma(g)^2} $$
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In the case of normal distributions,
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In the case of normal distributions,
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@ -19,28 +25,77 @@ $$ mean(normal(a,b) + normal(c,d)) = mean(normal(a+c, \sqrt{b^2 + d^2})) $$
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$$ mean(f-g) = mean(f) - mean(g) $$
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$$ mean(f-g) = mean(f) - mean(g) $$
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$$ std(f-g) = \sqrt{std(f)^2 + std(g)^2} $$
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$$ \sigma(f-g) = \sqrt{\sigma(f)^2 + \sigma(g)^2} $$
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### Multiplications
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### Multiplications
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$$ mean(f \cdot g) = mean(f) \cdot mean(g) $$
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$$ mean(f \cdot g) = mean(f) \cdot mean(g) $$
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$$ std(f \cdot g) = \sqrt{ (std(f)^2 + mean(f)) \cdot (std(g)^2 + mean(g)) - (mean(f) \cdot mean(g))^2} $$
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$$ \sigma(f \cdot g) = \sqrt{ (\sigma(f)^2 + mean(f)) \cdot (\sigma(g)^2 + mean(g)) - (mean(f) \cdot mean(g))^2} $$
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### Divisions
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### Divisions
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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.
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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.
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# To do:
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## Probability density functions (pdfs)
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- Provide sources or derivations, useful as this document becomes more complicated
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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.
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- Provide definitions for the probability density function, exponential, inverse, log, etc.
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- Provide at least some tests for division
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- See if playing around with characteristic functions turns out anything useful
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## Probability density functions
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### Sums
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TODO
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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:
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$$ (f + g)(z)= \int_{-\infty}^{\infty} f(x)\cdot g(z-x) \,dx $$
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See a proof sketch [here](https://www.milefoot.com/math/stat/rv-sums.htm)
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Here is some pseudocode to approximate this:
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```js
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// pdf1 and pdf2 are pdfs,
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// and cdf1 and cdf2 are their corresponding cdfs
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let epsilonForBounds = 2**(-16)
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let getBounds = cdf => {
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let cdf_min = -1
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let cdf_max = 1
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let n=0
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while(
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(
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cdf(cdf_min) > epsilonForBounds ||
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( 1 - cdf(cdf_max) ) > epsilonForBounds
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) &&
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n < 10
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){
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if(cdf(cdf_min) > epsilonForBounds){
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cdf_min = cdf_min * 2
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}
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if((1-cdf(cdf_max)) > epsilonForBounds){
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cdf_max = cdf_max * 2
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}
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}
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return [cdf_min, cdf_max]
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}
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let epsilonForIntegrals = 2**(-16)
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let pdfOfSum = (pdf1, pdf2, cdf1, cdf2, z) => {
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let bounds1 = getBounds(cdf1)
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let bounds2 = getBounds(cdf2)
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let bounds = [
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Math.min(bounds1[0], bounds2[0]),
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Math.max(bounds1[1], bounds2[1])
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]
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let result = 0
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for(let x = bounds[0]; x=x+epsilonForIntegrals; x<bounds[1]){
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let delta = pdf1(x) * pdf2(z-x)
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result = result + delta * epsilonForIntegrals
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}
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return result
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}
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```
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## Cumulative density functions
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## Cumulative density functions
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@ -49,3 +104,12 @@ TODO
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## Inverse cumulative density functions
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## Inverse cumulative density functions
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TODO
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TODO
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# To do:
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- Provide sources or derivations, useful as this document becomes more complicated
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- Provide definitions for the probability density function, exponential, inverse, log, etc.
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- Provide at least some tests for division
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- See if playing around with characteristic functions turns out anything useful
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BIN
packages/squiggle-lang/__tests__/docs/invariants.pdf
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BIN
packages/squiggle-lang/__tests__/docs/invariants.pdf
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