From bd3f2c99d1165ae509d07d859ae25fa795f1e735 Mon Sep 17 00:00:00 2001
From: NunoSempere <nuno.sempere@protonmail.com>
Date: Thu, 14 Apr 2022 16:22:21 -0400
Subject: [PATCH] tweak: add explanation for magic number

---
 .../SymbolicDist/SymbolicDist.res             |  7 +++--
 .../Internal/ProcessingConfidenceIntervals.md | 31 +++++++++++++++++++
 2 files changed, 36 insertions(+), 2 deletions(-)
 create mode 100644 packages/website/docs/Internal/ProcessingConfidenceIntervals.md

diff --git a/packages/squiggle-lang/src/rescript/Distributions/SymbolicDist/SymbolicDist.res b/packages/squiggle-lang/src/rescript/Distributions/SymbolicDist/SymbolicDist.res
index 28119e57..eb6ea136 100644
--- a/packages/squiggle-lang/src/rescript/Distributions/SymbolicDist/SymbolicDist.res
+++ b/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. *. 1.6448536269514722)
+    let stdev = (high -. low) /. (2. *. normal95confidencePoint)
     #Normal({mean: mean, stdev: stdev})
   }
   let inv = (p, t: t) => Jstat.Normal.inv(p, t.mean, t.stdev)
@@ -120,7 +123,7 @@ module Lognormal = {
     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 *. 1.6448536269514722)
+    let sigma = (logHigh -. logLow) /. (2.0 *. normal95confidencePoint)
     #Lognormal({mu: mu, sigma: sigma})
   }
   let fromMeanAndStdev = (mean, stdev) => {
diff --git a/packages/website/docs/Internal/ProcessingConfidenceIntervals.md b/packages/website/docs/Internal/ProcessingConfidenceIntervals.md
new file mode 100644
index 00000000..202c291c
--- /dev/null
+++ b/packages/website/docs/Internal/ProcessingConfidenceIntervals.md
@@ -0,0 +1,31 @@
+# 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
\ No newline at end of file