Merge pull request #278 from quantified-uncertainty/audit-2022-04-14-SymbolicDist.res

feat: Audit SymbolicDist.res

packages/squiggle-lang/__tests__/ReducerInterface/ReducerInterface_Distribution_test.res

@@ -23,8 +23,8 @@ describe("eval on distribution functions", () => {
   })
   describe("to", () => {
     testEval("5 to 2", "Error(TODO: Low value must be less than high value.)")
-    testEval("to(2,5)", "Ok(Lognormal(1.1512925464970227,0.278507821238345))")
-    testEval("to(-2,2)", "Ok(Normal(0,1.215913388057542))")
+    testEval("to(2,5)", "Ok(Lognormal(1.1512925464970227,0.27853260523016377))")
+    testEval("to(-2,2)", "Ok(Normal(0,1.2159136638235384))")
   })
   describe("mean", () => {
     testEval("mean(normal(5,2))", "Ok(5)")
@@ -57,8 +57,8 @@ describe("eval on distribution functions", () => {
   describe("multiply", () => {
     testEval("normal(10, 2) * 2", "Ok(Normal(20,4))")
     testEval("2 * normal(10, 2)", "Ok(Normal(20,4))")
-    testEval("lognormal(5,2) * lognormal(10,2)", "Ok(Lognormal(15,4))")
-    testEval("lognormal(10, 2) * lognormal(5, 2)", "Ok(Lognormal(15,4))")
+    testEval("lognormal(5,2) * lognormal(10,2)", "Ok(Lognormal(15,2.8284271247461903))")
+    testEval("lognormal(10, 2) * lognormal(5, 2)", "Ok(Lognormal(15,2.8284271247461903))")
     testEval("2 * lognormal(5, 2)", "Ok(Lognormal(5.693147180559945,2))")
     testEval("lognormal(5, 2) * 2", "Ok(Lognormal(5.693147180559945,2))")
   })

packages/squiggle-lang/src/rescript/Distributions/SymbolicDist/SymbolicDist.res

@@ -1,5 +1,8 @@
 open SymbolicDistTypes
 
+let normal95confidencePoint = 1.6448536269514722
+// explained in website/docs/internal/ProcessingConfidenceIntervals
+
 module Normal = {
   type t = normal
   let make = (mean: float, stdev: float): result<symbolicDist, string> =>
@@ -11,7 +14,7 @@ module Normal = {
 
   let from90PercentCI = (low, high) => {
     let mean = E.A.Floats.mean([low, high])
-    let stdev = (high -. low) /. (2. *. 1.644854)
+    let stdev = (high -. low) /. (2. *. normal95confidencePoint)
     #Normal({mean: mean, stdev: stdev})
   }
   let inv = (p, t: t) => Jstat.Normal.inv(p, t.mean, t.stdev)
@@ -21,12 +24,12 @@ module Normal = {
 
   let add = (n1: t, n2: t) => {
     let mean = n1.mean +. n2.mean
-    let stdev = sqrt(n1.stdev ** 2. +. n2.stdev ** 2.)
+    let stdev = Js.Math.sqrt(n1.stdev ** 2. +. n2.stdev ** 2.)
     #Normal({mean: mean, stdev: stdev})
   }
   let subtract = (n1: t, n2: t) => {
     let mean = n1.mean -. n2.mean
-    let stdev = sqrt(n1.stdev ** 2. +. n2.stdev ** 2.)
+    let stdev = Js.Math.sqrt(n1.stdev ** 2. +. n2.stdev ** 2.)
     #Normal({mean: mean, stdev: stdev})
   }
 
@@ -132,19 +135,22 @@ module Lognormal = {
   let mean = (t: t) => Ok(Jstat.Lognormal.mean(t.mu, t.sigma))
   let sample = (t: t) => Jstat.Lognormal.sample(t.mu, t.sigma)
   let toString = ({mu, sigma}: t) => j`Lognormal($mu,$sigma)`
+
   let from90PercentCI = (low, high) => {
     let logLow = Js.Math.log(low)
     let logHigh = Js.Math.log(high)
     let mu = E.A.Floats.mean([logLow, logHigh])
-    let sigma = (logHigh -. logLow) /. (2.0 *. 1.645)
+    let sigma = (logHigh -. logLow) /. (2.0 *. normal95confidencePoint)
     #Lognormal({mu: mu, sigma: sigma})
   }
   let fromMeanAndStdev = (mean, stdev) => {
+    // https://math.stackexchange.com/questions/2501783/parameters-of-a-lognormal-distribution
+    // https://wikiless.org/wiki/Log-normal_distribution?lang=en#Generation_and_parameters
     if stdev > 0.0 {
-      let variance = Js.Math.pow_float(~base=stdev, ~exp=2.0)
-      let meanSquared = Js.Math.pow_float(~base=mean, ~exp=2.0)
-      let mu = Js.Math.log(mean) -. 0.5 *. Js.Math.log(variance /. meanSquared +. 1.0)
-      let sigma = Js.Math.pow_float(~base=Js.Math.log(variance /. meanSquared +. 1.0), ~exp=0.5)
+      let variance = stdev ** 2.
+      let meanSquared = mean ** 2.
+      let mu = 2. *. Js.Math.log(mean) -. 0.5 *. Js.Math.log(variance +. meanSquared)
+      let sigma = Js.Math.sqrt(Js.Math.log(variance /. meanSquared +. 1.))
       Ok(#Lognormal({mu: mu, sigma: sigma}))
     } else {
       Error("Lognormal standard deviation must be larger than 0")
@@ -152,8 +158,9 @@ module Lognormal = {
   }
 
   let multiply = (l1, l2) => {
+    // https://wikiless.org/wiki/Log-normal_distribution?lang=en#Multiplication_and_division_of_independent,_log-normal_random_variables
     let mu = l1.mu +. l2.mu
-    let sigma = l1.sigma +. l2.sigma
+    let sigma = Js.Math.sqrt(l1.sigma ** 2. +. l2.sigma ** 2.) // m
     #Lognormal({mu: mu, sigma: sigma})
   }
   let divide = (l1, l2) => {

packages/website/docs/Internal/ProcessingConfidenceIntervals.md

@@ -0,0 +1,32 @@
+# Processing confidence intervals
+
+This page explains what we are doing when we take a 95% confidence interval, and we get a mean and a standard deviation from it
+
+## For normals
+
+```js
+module Normal = {
+  //...
+  let from90PercentCI = (low, high) => {
+    let mean = E.A.Floats.mean([low, high])
+    let stdev = (high -. low) /. (2. *. 1.6448536269514722)
+    #Normal({mean: mean, stdev: stdev})
+  }
+  //...
+}
+```
+
+We know that for a normal with mean $\mu$ and standard deviation $\sigma$,
+
+$$
+
+a \cdot Normal(\mu, \sigma) = Normal(a\cdot \mu, |a|\cdot \sigma)
+
+
+$$
+
+We can now look at the inverse cdf of a $Normal(0,1)$. We find that the 95% point is reached at $1.6448536269514722$. ([source](https://stackoverflow.com/questions/20626994/how-to-calculate-the-inverse-of-the-normal-cumulative-distribution-function-in-p)) This means that the 90% confidence interval is $[-1.6448536269514722, 1.6448536269514722]$, which has a width of $2 \cdot 1.6448536269514722$.
+
+So then, if we take a $Normal(0,1)$ and we multiply it by $\frac{(high -. low)}{(2. *. 1.6448536269514722)}$, it's 90% confidence interval will be multiplied by the same amount. Then we just have to shift it by the mean to get our target normal.
+
+## For lognormals