A_short_study_of_Renyi_entropy.ipynb

{
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    "toc": "true"
   },
   "source": [
    "# Table of Contents\n",
    " <p><div class=\"lev1 toc-item\"><a href=\"#A-short-study-of-Rényi-entropy\" data-toc-modified-id=\"A-short-study-of-Rényi-entropy-1\"><span class=\"toc-item-num\">1&nbsp;&nbsp;</span>A short study of Rényi entropy</a></div><div class=\"lev2 toc-item\"><a href=\"#Requirements\" data-toc-modified-id=\"Requirements-11\"><span class=\"toc-item-num\">1.1&nbsp;&nbsp;</span>Requirements</a></div><div class=\"lev2 toc-item\"><a href=\"#Utility-functions\" data-toc-modified-id=\"Utility-functions-12\"><span class=\"toc-item-num\">1.2&nbsp;&nbsp;</span>Utility functions</a></div><div class=\"lev2 toc-item\"><a href=\"#Definition,-common-and-special-cases\" data-toc-modified-id=\"Definition,-common-and-special-cases-13\"><span class=\"toc-item-num\">1.3&nbsp;&nbsp;</span>Definition, common and special cases</a></div><div class=\"lev2 toc-item\"><a href=\"#Plotting-some-values\" data-toc-modified-id=\"Plotting-some-values-14\"><span class=\"toc-item-num\">1.4&nbsp;&nbsp;</span>Plotting some values</a></div><div class=\"lev2 toc-item\"><a href=\"#Conclusion\" data-toc-modified-id=\"Conclusion-15\"><span class=\"toc-item-num\">1.5&nbsp;&nbsp;</span>Conclusion</a></div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# A short study of Rényi entropy\n",
    "\n",
    "I want to study here the Rényi entropy, using [Python](https://www.python.org/).\n",
    "I will define a function implementing $H_{\\alpha}(X)$, from the given formula, for discrete random variables, and check the influence of the parameter $\\alpha$,\n",
    "$$ H_{\\alpha}(X) := \\frac{1}{1-\\alpha} \\log_2(\\sum_i^n p_i^{\\alpha}),$$\n",
    "where $X$ has $n$ possible values, and the $i$-th outcome has probability $p_i\\in[0,1]$.\n",
    "\n",
    "- *Reference*: [this blog post by John D. Cook](https://www.johndcook.com/blog/2018/11/21/renyi-entropy/), [this Wikipédia page](https://en.wikipedia.org/wiki/R%C3%A9nyi_entropy) and [this page on MathWorld](http://mathworld.wolfram.com/RenyiEntropy.html),\n",
    "- *Author*: [Lilian Besson](https://perso.crans.org/besson/)\n",
    "- *License*: [MIT License](https://lbesson.mit-license.org/)\n",
    "- *Date*: 22th of November, 2018"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Requirements"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "code_folding": [],
    "scrolled": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
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     ]
    }
   ],
   "source": [
    "!pip install watermark matplotlib numpy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The watermark extension is already loaded. To reload it, use:\n",
      "  %reload_ext watermark\n",
      "Lilian Besson \n",
      "\n",
      "CPython 3.6.6\n",
      "IPython 7.0.1\n",
      "\n",
      "matplotlib 3.0.2\n",
      "numpy 1.14.5\n",
      "\n",
      "compiler   : GCC 8.0.1 20180414 (experimental) [trunk revision 259383\n",
      "system     : Linux\n",
      "release    : 4.15.0-38-generic\n",
      "machine    : x86_64\n",
      "processor  : x86_64\n",
      "CPU cores  : 4\n",
      "interpreter: 64bit\n",
      "Git hash   : a119f96f2de5b449131a73b6c9861f26b2c0d3f8\n"
     ]
    }
   ],
   "source": [
    "%load_ext watermark\n",
    "%watermark -v -m -a \"Lilian Besson\" -g -p matplotlib,numpy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "----\n",
    "## Utility functions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We start by giving three examples of such vectors $X=(p_i)_{1\\leq i \\leq n}$, a discrete probability distributions on $n$ values."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [],
   "source": [
    "X1 = [0.25, 0.5, 0.25]\n",
    "X2 = [0.1, 0.25, 0.3, 0.45]\n",
    "X3 = [0, 0.5, 0.5]\n",
    "\n",
    "X4 = np.full(100, 1/100)\n",
    "X5 = np.full(1000, 1/1000)\n",
    "\n",
    "X6 = np.arange(100, dtype=float)\n",
    "X6 /= np.sum(X6)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We need a function to safely compute $x \\mapsto x \\log_2(x)$, with special care in case $x=0$. This one will accept a numpy array or a single value as argument:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{'divide': 'ignore', 'over': 'ignore', 'under': 'ignore', 'invalid': 'ignore'}"
      ]
     },
     "execution_count": 50,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.seterr(all=\"ignore\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [],
   "source": [
    "def x_log2_x(x):\n",
    "    \"\"\" Return x * log2(x) and 0 if x is 0.\"\"\"\n",
    "    results = x * np.log2(x)\n",
    "    if np.size(x) == 1:\n",
    "        if np.isclose(x, 0.0):\n",
    "            results = 0.0\n",
    "    else:\n",
    "        results[np.isclose(x, 0.0)] = 0.0\n",
    "    return results"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For examples:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [
    {
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      ]
     },
     "execution_count": 52,
     "metadata": {},
     "output_type": "execute_result"
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    {
     "data": {
      "text/plain": [
       "-0.5"
      ]
     },
     "execution_count": 52,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "0.0"
      ]
     },
     "execution_count": 52,
     "metadata": {},
     "output_type": "execute_result"
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    {
     "data": {
      "text/plain": [
       "2.0"
      ]
     },
     "execution_count": 52,
     "metadata": {},
     "output_type": "execute_result"
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    {
     "data": {
      "text/plain": [
       "33.219280948873624"
      ]
     },
     "execution_count": 52,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x_log2_x(0)\n",
    "x_log2_x(0.5)\n",
    "x_log2_x(1)\n",
    "x_log2_x(2)\n",
    "x_log2_x(10)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "and with vectors, slots with $p_i=0$ are handled without error:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([-0.5, -0.5, -0.5])"
      ]
     },
     "execution_count": 54,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "array([-0.33219281, -0.5       , -0.52108968, -0.51840139])"
      ]
     },
     "execution_count": 54,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "array([ 0. , -0.5, -0.5])"
      ]
     },
     "execution_count": 54,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "array([-0.06643856, -0.06643856, -0.06643856, -0.06643856, -0.06643856,\n",
       "       -0.06643856, -0.06643856, -0.06643856, -0.06643856, -0.06643856])"
      ]
     },
     "execution_count": 54,
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     "output_type": "execute_result"
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     "data": {
      "text/plain": [
       "array([-0.00996578, -0.00996578, -0.00996578, -0.00996578, -0.00996578,\n",
       "       -0.00996578, -0.00996578, -0.00996578, -0.00996578, -0.00996578])"
      ]
     },
     "execution_count": 54,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "array([ 0.        , -0.00247944, -0.00455483, -0.00647773, -0.00830159,\n",
       "       -0.0100518 , -0.01174333, -0.01338606, -0.01498701, -0.01655143])"
      ]
     },
     "execution_count": 54,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x_log2_x(X1)\n",
    "x_log2_x(X2)\n",
    "x_log2_x(X3)\n",
    "\n",
    "x_log2_x(X4)[:10]\n",
    "x_log2_x(X5)[:10]\n",
    "x_log2_x(X6)[:10]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "----\n",
    "## Definition, common and special cases"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "From the mathematical definition, an issue will happen if $\\alpha=1$ or $\\alpha=\\inf$, so we deal with the special cases manually.\n",
    "$X$ is here given as the vector of $(p_i)_{1\\leq i \\leq n}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [],
   "source": [
    "def renyi_entropy(alpha, X):\n",
    "    assert alpha >= 0, \"Error: renyi_entropy only accepts values of alpha >= 0, but alpha = {}.\".format(alpha)  # DEBUG\n",
    "    if np.isinf(alpha):\n",
    "        # XXX Min entropy!\n",
    "        return - np.log2(np.max(X))\n",
    "    elif np.isclose(alpha, 0):\n",
    "        # XXX Max entropy!\n",
    "        return np.log2(len(X))\n",
    "    elif np.isclose(alpha, 1):\n",
    "        # XXX Shannon entropy!\n",
    "        return - np.sum(x_log2_x(X))\n",
    "    else:\n",
    "        return (1.0 / (1.0 - alpha)) * np.log2(np.sum(X ** alpha))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Curryfied version\n",
    "def renyi_entropy_2(alpha):\n",
    "    def re(X):\n",
    "        return renyi_entropy(alpha, X)\n",
    "    return re"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Curryfied version\n",
    "def renyi_entropy_3(alphas, X):\n",
    "    res = np.zeros_like(alphas)\n",
    "    for i, alpha in enumerate(alphas):\n",
    "        res[i] = renyi_entropy(alpha, X)\n",
    "    return res"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "----\n",
    "## Plotting some values"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [],
   "source": [
    "alphas = np.linspace(0, 10, 1000)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([1.5849625 , 1.58414417, 1.58332491, 1.58250473, 1.58168363,\n",
       "       1.58086162, 1.58003871, 1.5792149 , 1.57839021, 1.57756464])"
      ]
     },
     "execution_count": 59,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "renyi_entropy_3(alphas, X1)[:10]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {},
   "outputs": [],
   "source": [
    "def plot_renyi_entropy(alphas, X):\n",
    "    fig = plt.figure()\n",
    "    plt.plot(alphas, renyi_entropy_3(alphas, X))\n",
    "    plt.xlabel(r\"Value for $\\alpha$\")\n",
    "    plt.ylabel(r\"Value for $H_{\\alpha}(X)$\")\n",
    "    plt.title(r\"Réniy entropy for $X={}$\".format(X[:10]))\n",
    "    plt.show()\n",
    "    # return fig"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_renyi_entropy(alphas, X1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_renyi_entropy(alphas, X2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_renyi_entropy(alphas, X3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_renyi_entropy(alphas, X4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_renyi_entropy(alphas, X5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_renyi_entropy(alphas, X6)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "----\n",
    "## Conclusion"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It is not surprising that $H_{\\alpha}(X)$ appears to be continuous as a function of $\\alpha$, as one can easily verify that it is."
   ]
  }
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