universality-of-the-uniform.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Universality of the Uniform\n",
    "The standard uniform distribution $U$ has two very interesting property. \n",
    "1. You can create random-variables of any distribution with it, given that you have its inverse CDF (Quantile function). $F^{-1}(U)=X$\n",
    "2. You can do the inverse of the above and create the standard uniform distribution, given that you have the CDF and the rvs. of that CDF."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 1. Create the rvs., given the inverse CDF\n",
    "$F^{-1}(U)=X$\n",
    "#### For illustration we will use logistic rvs.\n",
    "Logistic CDF:\n",
    "$F(x)=\\frac{e^x}{1+e^x}$\n",
    "\n",
    "Logistic Inverse CDF: $F^-1(p)=log(\\frac{p}{1-p})$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Now let p be U\n",
    "$F^-1(U)=log(\\frac{U}{1-U})=X$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Run simulation:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 86,
   "metadata": {},
   "outputs": [],
   "source": [
    "n_simulations = int(10e4)\n",
    "rv_counts = {}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 87,
   "metadata": {},
   "outputs": [],
   "source": [
    "for i in range(n_simulations):\n",
    "    r = np.random.rand()\n",
    "    x_float = np.log(r/(1-r))\n",
    "    x_int = int(np.round(x_float))\n",
    "    try:\n",
    "        rv_counts[x_int] += 1\n",
    "    except KeyError:\n",
    "        rv_counts[x_int] = 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 89,
   "metadata": {},
   "outputs": [],
   "source": [
    "bar_x = []\n",
    "bar_y = []\n",
    "\n",
    "for x in rv_counts:\n",
    "    bar_x.append(x)\n",
    "    bar_y.append(rv_counts[x])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 94,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.bar(bar_x, bar_y, align='center')\n",
    "plt.title('Logistic PDF approximation')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 2. Create standard uniform rvs., given the CDF and its rvs.\n",
    "$F(X)=U$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 185,
   "metadata": {},
   "outputs": [],
   "source": [
    "n_simulations = int(10e4)\n",
    "rv_counts = {}\n",
    "\n",
    "for i in range(n_simulations):\n",
    "    X = np.random.logistic()\n",
    "    F = np.exp(X) / (1 + np.exp(X))\n",
    "    # scale to easily convert to categorical\n",
    "    F *= 10\n",
    "    F = np.ceil(F)\n",
    "    F /= 10\n",
    "    try:\n",
    "        rv_counts[F] += 1\n",
    "    except KeyError:\n",
    "        rv_counts[F] = 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 186,
   "metadata": {},
   "outputs": [],
   "source": [
    "bar_x = []\n",
    "bar_y = []\n",
    "\n",
    "for x in rv_counts:\n",
    "    bar_x.append(x)\n",
    "    bar_y.append(rv_counts[x])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 187,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.bar(bar_x, bar_y, align='center')\n",
    "plt.title('Uniformly distributed rvs.')\n",
    "plt.xlim(0.0, 1.0)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Why does this work?\n",
    "There are two ways to prove this:\n",
    "1. Mathematically\n",
    "2. By using a story (formally)\n",
    "\n",
    "I'll be presenting both, but it is much more important to understand the latter story-proof.\n",
    "\n",
    "### 1. Mathematical Proof\n",
    "We are assuming $X$ to be equal to $F^{-1}(U)$, so lets see if the laws of probability are still valid and the same then:\n",
    "\n",
    "$ P(X\\leq x) = P(F^{-1}(U)\\leq x) = P(U\\leq F(x)) = F(x) $\n",
    "\n",
    "$ P(X\\leq x) = F(x)$\n",
    "\n",
    "For the proof of the second statement $F(X)=U$ we let $Y=F(X)$ and bare in mind that $ P(U\\leq p)=p $.\n",
    "\n",
    "$ P(Y\\leq y) = P(F(X)\\leq y) = P(X\\leq F^{-1}(y)) = F(F^{-1}(y)) = y $\n",
    "\n",
    "### 2. Story Proof"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Imagine students taking an exam and let $X$ be the score (lets say between 0 and 100) of a random student. Also assume that the scores are continous (which we have to do).\n",
    "\n",
    "\n",
    "##### $F(X)=U$  Proof\n",
    "The percentile function $F(x)$ says how much better a student with score $x$ is better than the rest (in the top 0.xx %). Now it may often be the case that X has a very strange looking distribution, but its percentiles are all uniform (for example: there are as many students at the bottom 10% as in the top 10%, or between [0.4;0.5] and [0.5;0.6]). Therefore the percentiles of the students are uniformly distributed.\n",
    "\n",
    "##### $F^{-1}(U)=X$  Proof\n",
    "The quantile function $F^{-1}(p)$ give you the points that are needed for a student to be in the top 0.xx percent. Since the percentiles of the students (top 5%, bottom 5%) are uniformly distributed, sampling a random uniform percentile (between 0.0 and 1.0) is like picking a random studend, which is the definition of $X$. So $F^{-1}(U)=X$"
   ]
  }
 ],
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