{
"metadata": {
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 3
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.7.7-final"
},
"orig_nbformat": 2,
"kernelspec": {
"name": "Python 3.7.7 64-bit ('blog': conda)",
"display_name": "Python 3.7.7 64-bit ('blog': conda)",
"metadata": {
"interpreter": {
"hash": "a791f4b9424e3975ed64217961a14e58af61de2cbd0b16cd6f0b3a8e9eaf4389"
}
}
}
},
"nbformat": 4,
"nbformat_minor": 2,
"cells": [
{
"source": [
"# Beta distribution"
],
"cell_type": "markdown",
"metadata": {}
},
{
"source": [
"When trying to to Bayesian interference to estimate probability, we can use Beta distribution as a prior. Below are some steps to calculate it.\n",
"\n",
"From the [Data Science from Scratch book](https://www.oreilly.com/library/view/data-science-from/9781492041122/)."
],
"cell_type": "markdown",
"metadata": {}
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [],
"source": [
"import math as m"
]
},
{
"source": [
"$ f(x, \\alpha, \\beta) = x^{\\alpha - 1} (1 - x)^{\\beta - 1} \\frac {1} {\\text{B}(\\alpha, \\beta)} $\n",
"\n",
"where\n",
"\n",
"$ \\text{B} = \\Gamma(\\alpha) \\Gamma(\\beta) \\frac{1} {\\Gamma(\\alpha + \\beta)} $\n",
"\n",
"where\n",
"\n",
"$\\Gamma(n)$ is the Gamma function which, for positive integers is\n",
"\n",
"$ \\Gamma(n) = (n - 1)!$"
],
"cell_type": "markdown",
"metadata": {}
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [],
"source": [
"def B(alpha: float, beta: float) -> float:\n",
" \"This scales the parameters between 0 and 1\"\n",
" return m.gamma(alpha) * m.gamma(beta) / m.gamma(alpha + beta)"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [],
"source": [
"def beta_pdf(x: float, alpha: float, beta:float) -> float:\n",
" if x <= 0 or x >=1: return 0\n",
"\n",
" return x ** (alpha - 1) * (1 - x) ** (beta - 1) / B(alpha, beta)"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [],
"source": [
"import numpy as np\n",
"import pandas as pd\n",
"import altair as alt"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [],
"source": [
"df = pd.DataFrame()\n",
"\n",
"Beta_combinations = [(1, 1), (10, 10), (4, 16), (16, 4), (30, 30)]\n",
"\n",
"for Beta in Beta_combinations:\n",
" alpha, beta = Beta\n",
" df_B = pd.DataFrame()\n",
" df_B['x'] = pd.Series(np.arange(0.01, 1, .01)) \n",
" df_B['y'] = df_B['x'].apply(lambda x: beta_pdf(x, alpha, beta))\n",
" df_B['Beta'] = f'({alpha}, {beta})'\n",
"\n",
" df = pd.concat([df, df_B])\n"
]
},
{
"source": [
"The distribution centers around $ \\alpha \\frac{1} {\\alpha + \\beta} $\n",
"\n",
"- Beta(1, 1) is the uniform distribution in [1, 1]\n",
"- When alpha is greater than beta the distribution is skewed to the left (and respectively, in the opposite case)\n",
"- The greater alpha and beta are the 'tighter' is the distribution"
],
"cell_type": "markdown",
"metadata": {}
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {},
"outputs": [
{
"output_type": "execute_result",
"data": {
"text/html": "\n\n",
"text/plain": [
"alt.Chart(...)"
]
},
"metadata": {},
"execution_count": 6
}
],
"source": [
"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)"
]
}
]
}