Beta distribution¶

When trying to to Bayesian interference to estimate probability, we can use Beta distribution as a prior. Below are some steps to calculate it.

From the Data Science from Scratch book.

In [1]:

import math as m

$ f(x, \alpha, \beta) = x^{\alpha - 1} (1 - x)^{\beta - 1} \frac {1} {\text{B}(\alpha, \beta)} $

where

$ \text{B} = \Gamma(\alpha) \Gamma(\beta) \frac{1} {\Gamma(\alpha + \beta)} $

where

$\Gamma(n)$ is the Gamma function which, for positive integers is

$ \Gamma(n) = (n - 1)!$

In [2]:

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)

In [3]:

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)

In [4]:

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

In [5]:

df = pd.DataFrame()

Beta_combinations = [(1, 1), (10, 10), (4, 16), (16, 4), (30, 30)]

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])

The distribution centers around $ \alpha \frac{1} {\alpha + \beta} $

Beta(1, 1) is the uniform distribution in [1, 1]
When alpha is greater than beta the distribution is skewed to the left (and respectively, in the opposite case)
The greater alpha and beta are the 'tighter' is the distribution

In [6]:

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)

Out[6]: