We try to use beta distributions to estimate if our coin is fair.

From the Data Science from Scratch book.

Libraries and helper functions 

import math as m

import numpy as np
import altair as alt
import pandas as pd

def B(alpha: float, beta: float) -> float:
    "This scales the parameters between 0 and 1"
    return m.gamma(alpha) * m.gamma(beta) / m.gamma(alpha + beta)

def beta_pdf(x: float, alpha: float, beta:float) -> float:
    if x <= 0 or x >=1: return 0

    return x ** (alpha - 1) * (1 - x) ** (beta - 1) / B(alpha, beta)

Example 1

  1. We do not want to make assumptions beforehand, so we choose both $\alpha$ and $\beta$ to be 1: $B(1, 1)$
  2. We flip the coin 10 times and get 3 heads
  3. Our new posterior distribution becomes $B(4, 8)$ centered around 0.33
  4. We have to assume that the observed probabilty is the real

Example 2

  1. We have a strong assumption that the coin is fair so we choose a $B(20, 20)
  2. Again, we got 3 heads out of 10
  3. Our new Beta is $B(23, 27)$ centered around 0.46
  4. It suggest that the coin is slightly biased toward tails

Example 3

  1. We believe that the coin is biased toward head by 75% of the time, so we choose $B(30, 10)$
  2. Again, we got 3 heads out of 10
  3. Our poseterior distribution is $B(33, 17)$ centered around 0.66
  4. It suggest that the coin is biased toward the head, although less strongly as we believed
df = pd.DataFrame()

Beta_combinations = [(1, 1), (4, 8), (20, 20), (23, 27), (30, 10), (33, 17)]

for Beta in Beta_combinations:
    alpha, beta = Beta
    df_B = pd.DataFrame()
    df_B['x'] = pd.Series(np.arange(0.01, 1, .01))    
    df_B['y'] = df_B['x'].apply(lambda x: beta_pdf(x, alpha, beta))
    df_B['Beta'] = f'({alpha}, {beta})'

    df = pd.concat([df, df_B])

alt.Chart(df).mark_line().encode(
    alt.X('x:Q'), alt.Y('y:Q'), alt.Color('Beta'), tooltip=['x', 'y', 'Beta'], strokeDash='Beta'
).properties(width=600, title='Beta distributions')