# Compute the shortest length binomial confidence interval

## Problem description

The problem is pretty simple. Suppose we have a coin, toss it $n$ times, and observe $x$ number of heads. Now, we want to estimate the possibility $p$ of head. The best estimator is $\frac{x}{n}$. But we are also interested in providing a confidence interval(CI) of $p$, $L(x)$ and $R(x)$.

Suppose that the confidence level is $C$. And a CI of confidence level $C$ is that $\forall p \in [0, 1], P(p \in [L(x), R(x)] | p) \ge C$.

## Background

I didn’t find out an existing algorithm to compute out an optimal confidence interval yet. https://en.wikipedia.org/wiki/Binomial_proportion_confidence_interval

One possible reason I guess is that all the existing solutions are focusing on working out an analytical solution which is in the form of math expressions.

The optimal CI here means that the its maximum length interval is the shortest one among all the maximum lengths of all possible CIs.

In this post, the goal is to compute out an optimal CI for given $N$ and $C$, which $\underset{(R,L) \in \text{CIs}}{\text{minimize}} \underset{x \in \{0, \dots, N\}}{\max} (R(x) - L(x))$.

## Solution

### Main problem - binary search on the maximum length.

Let $t = \max_{x} R(x) - L(x)$, the high level idea is doing a binary search on $t$. The subproblem is that for a given $T$ and $N, C$, whether a valid $L(x)$ and $R(x)$ exists, where $\forall x \in \{0, \dots, N\}, R(x) - L(x) \ge T$ and $\forall p \in [0, 1], P(p \in [L(x), R(x)] | p) \ge C$.

And also, we can relax the problem by assuming that all $R(x)-L(x)$ are the same length, let $T = R(x) - L(x), \forall x \in \{0, \dots, N\}$. Now, we are ready to move to the subprobem.

### Sub problem - A proof by induction.

With this same length assumption, to solve the subproblem, we could construct $L(x)$ by induction. If the construction succeeds, then the result is a valid CL with the length $T$. If the inductive construction fails, then it proves that no valid $L(x)$ exists for the given $T$.

the base case: $k = 0$, $L(k) = 0.0$ and $R(k) = T$.

the induction step: $L(k+1)$ is $p’$ such that

$$\forall p \le p', \sum_{0 \le x \le k} 1(x) \times P(x | p') \ge C$$

and

$$\forall p > p', \sum_{0 \le x \le k} 1(x) \times P(x | p') < C$$

where

$1(x) = 1 \text{ if } L(x) \le p' \le L(x) + T, \text{else }0$

Clearly, this $p’$ could get found by binary search again, this inducative construction will give us the optimal candidate $L(x)$. In order to validate if it’s a valid L(x), we can check if $L(N) + T \ge 1$ or not.

if $L(N) + T < 1$, then it proves that no valid solution exists.

if $L(N) + T \ge 1$, then it’s a valid CI with length T.

### Summary

Now, we get an optimal binomial confidence interval, :)

Input: N, C

binary_search(T, left=0.0, right=1.0)
t = (left + right) / 2
L(x) = proof_by_induction(t,N,C)
if L(N) + T < 1.0
left = t
else:
right = t
L(x) = proof_by_induction(right,N,C)
R(x) = L(x) + t
return L, R


## Code

Jupyter Notebook

Example:

n = 20, confidence level = 90%

minimum interval length = 0.347

n: 20 , confidence: 0.9 , interval: 0.34707111480793
0 [0.0, 0.34707111480793] 0.0
1 [0.0052541740694689865, 0.352325288877399] 0.05
2 [0.026914132614011044, 0.37398524742194106] 0.1
3 [0.05641789624311604, 0.40348901105104606] 0.15
4 [0.09021345530747042, 0.43728457011540045] 0.2
5 [0.1269260599358087, 0.4739971747437387] 0.25
6 [0.16587238278892374, 0.5129434975968538] 0.3
7 [0.20666403328284658, 0.5537351480907766] 0.35
8 [0.2490648213798084, 0.5961359361877384] 0.4
9 [0.29292888519207044, 0.6400000000000005] 0.45
10 [0.33817090338965594, 0.685242018197586] 0.5
11 [0.3811895005158469, 0.728260615323777] 0.55
12 [0.4271718075480454, 0.7742429223559755] 0.6
13 [0.4739972939527387, 0.8210684087606688] 0.65
14 [0.5129436168058538, 0.8600147316137838] 0.7
15 [0.5537352672997766, 0.9008063821077066] 0.75
16 [0.5961360553967384, 0.9432071702046685] 0.8
17 [0.6720098441544733, 1.0] 0.85
18 [0.728260734532777, 1.0] 0.9
19 [0.79683990144431, 1.0] 0.95
20 [0.874186048296206, 1.0] 1.0