050-parameter-estimation.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Bayesian vs Maximum Likelihood Parameter Estimation\n",
    "\n",
    "Thomas Breuel\n",
    "\n",
    "Deep Learning Summer School 2018, Genoa, Italy\n",
    "\n",
    "_This is a low-quality PDF conversion; to see the original notebook, please go to: [github.com/tmbdev/dl-2018](github.com/tmbdev/dl-2018)_\n",
    "\n",
    "The notebooks are directly executable."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Populating the interactive namespace from numpy and matplotlib\n"
     ]
    }
   ],
   "source": [
    "%pylab inline\n",
    "from scipy import stats\n",
    "from scipy.stats import norm"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Tiny Review"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Notation for Classification Problems\n",
    "\n",
    "- $\\omega$: class\n",
    "- $p(\\omega, x)$: joint distribution defining the classification problem\n",
    "- $D=\\{(\\omega_1, x_1),...\\}$: i.i.d. sample from $p$\n",
    "- $\\arg\\max_\\omega P(\\omega | x)$: optimal zero-one loss function classifier\n",
    "- $\\hat{\\theta}$: a maximum likelihood estimate\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Bayes Rule \n",
    "\n",
    "$$ P( \\omega | x ) = \\frac{ p(x | \\omega) P(\\omega) } { p(x) } $$\n",
    "\n",
    "- $P(\\omega | x)$ posterior distribution at input $x$\n",
    "- $p(x | \\omega)$ class conditional density (\"generative model\")\n",
    "- $p(x)$ sample distribution or \"evidence\"\n",
    "- $P(\\omega)$ prior class probabilities\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Maximum Likelihood Estimation\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Neural Networks and Maximum Likelihood Estimation\n",
    "\n",
    "A DNN is ultimate a parameterized function $f_\\theta(x)$, with $\\theta\\in\\mathbb{R}^{1000...}$.\n",
    "\n",
    "We're training the DNN on some random training sample $D$ drawn according to some joint distribution $p(y, x)$, and we try to maximize $\\theta$ so that the model matches the dataset best under some loss function."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## \"One Parameter Generative Network\"\n",
    "\n",
    "Let's drop all the complexity of classification etc. and just focus on parameter estimation. Let's use a one-parameter model:\n",
    "\n",
    "$$p_\\theta(x) = (2\\pi)^{-1/2} e^{-\\frac{(x-\\theta)^2}{2}}$$\n",
    "\n",
    "Our training set $D$ consists of $N$ samples from $p_\\theta(x)$ and we need to find $\\theta$.\n",
    "\n",
    "You might call this a \"one parameter generative network\"."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Maximum Likelihood Estimate\n",
    "\n",
    "How do we find $\\theta$? You already know a formula for finding $\\theta$:\n",
    "\n",
    "$$ \\hat{\\theta} = \\frac{1}{N} \\sum_{x\\in D} x $$\n",
    "\n",
    "This is the _maximum likelihood estimate_ of $\\theta$.\n",
    "\n",
    "Let's derive this."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Univariate Normal Density\n",
    "\n",
    "For the univariate normal density, $\\theta = (\\mu,\\sigma)$, but we assume $\\sigma=1$ is known. We write:\n",
    "\n",
    "$$p(x|\\theta) = p_\\theta(x) = (2\\pi)^{-1/2} e^{-\\frac{(x-\\theta)^2}{2}}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Univariate Normal Density (2)\n",
    "\n",
    "Now we can write a density for the entire dataset:\n",
    "\n",
    "$$ p(D|\\theta) = p(x_1,\\ldots,x_n|\\theta) $$\n",
    "$$ = p(x_1|\\theta)\\cdot\\ldots\\cdot p(x_n|\\theta)$$\n",
    "$$ = \\prod_{i=1}^n p(x_i|\\theta)$$\n",
    "\n",
    "We call this the _likelihood_ of the data.\n",
    "\n",
    "Note that likelihoods are parameterized densities viewed as functions of the parameters."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Maximum Likelihood Estimate\n",
    "\n",
    "The _maximum likelihood estimate_ is given by:\n",
    "\n",
    "$$\\hat{\\theta} = \\arg\\max_\\theta ~ p(D|\\theta)$$\n",
    "\n",
    "This seems like a reasonable thing to do: choose the parameter that was\n",
    "most likely to produce the data set."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Maximum Likelihood Instability\n",
    "\n",
    "However, in general, there is little reason why the maximum of the likelihood function\n",
    "should mean anything.\n",
    "\n",
    "For example, we can easily modify the likelihood function to put a tiny spike in it\n",
    "that moves the maximum somewhere arbitrary without actually changing the problem much at all.\n",
    "\n",
    "For bimodal densities, the Maximum Likelihood solution can jump between the two peaks with tiny changes in the data."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Maximum Likelihood Estimation of the Mean of a Normal Density\n",
    "\n",
    "Let's derive the ML estimator explicitly.\n",
    "\n",
    "We're trying to maximize:\n",
    "\n",
    "$$p(D|\\theta) = \\prod_i (2\\pi)^{-1/2} e^{-\\frac{(x_i-\\theta)^2}{2}}$$\n",
    "\n",
    "Let's take logarithms\n",
    "\n",
    "$$l(\\mu) = \\sum_i \\log p(x_i|\\theta) = \\hbox{const} - {1\\over 2} (x_i-\\theta)\\cdot(x_i-\\theta)$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Maximizing the Likelihood\n",
    "\n",
    "If $l$ is sufficiently well behaved (it is), then a necessary condition for a local maximum is that the gradient in the parameter to be estimated is 0. The gradient is:\n",
    "\n",
    "$$\\nabla_\\theta l(\\theta) = \\nabla_\\theta(\\hbox{const} - {1\\over 2} (x_i-\\theta)\\cdot(x_i-\\theta)) = \\sum_i (x_i-\\theta)$$\n",
    "\n",
    "$$ \\nabla_\\theta l(\\theta)= \\sum_i (x_i-\\theta) = 0 \\Rightarrow \\sum_i x_i = n \\theta $$\n",
    "\n",
    "$$ \\hat{\\theta} = \\frac{1}{n}\\sum x_i $$\n",
    "\n",
    "Therefore: the arithmetic mean is the maximum likelihood estimator for the parameter of the normal density with known variance."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Connection to Classification\n",
    "\n",
    "Let's say you have a classification problem in which you know that both classes are distributed according to $p_\\theta(x)$ for two different parameters, $\\theta_1$ and $\\theta_2$.\n",
    "\n",
    "Recipe for classification:\n",
    "- compute $\\theta_1$ and $\\theta_2$ from the training samples\n",
    "- apply Bayes formula to derive $P(\\omega | x; \\theta_1, \\theta_2)$ using the parametric densities\n",
    "\n",
    "Other recipe:\n",
    "- write down $P(\\omega | x; \\theta_1, \\theta_2)$\n",
    "- use gradient descent to optimize training set error"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Bayesian vs Maximum Likelihood\n",
    "\n",
    "Let's say that the $\\theta$ are the parameters of a DNN classifier $P_\\theta(\\omega|x)$. In typical DL, we estimate a single parameter vector $\\hat{\\theta}$ and use that for classification:\n",
    "\n",
    "$$\\hat{\\theta} = \\arg\\max_\\theta p(D | \\theta)$$\n",
    "\n",
    "In Bayesian methods, we average out our classifications over all possible parameter vectors:\n",
    "\n",
    "$$P(\\omega | x) \\propto \\int P_\\theta(\\omega | x) p(\\theta | D) d\\theta$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [],
   "source": [
    "# parameter estimate for nornmal density, unknown mean, sigma=1\n",
    "\n",
    "def p_x_theta(x, theta, sigma=1):\n",
    "    return exp(-(x-theta)**2/(2*sigma**2))/sqrt(2*pi*sigma**2)\n",
    "def p_theta_D(theta, D):\n",
    "    return prod([p_x_theta(x, theta) for x in D], axis=0)\n",
    "\n",
    "D = [0.7, 3.3]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "code_folding": [],
    "slideshow": {
     "slide_type": "slide"
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   },
   "outputs": [
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     "execution_count": 3,
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P//LrRCJBIpHIZHhSINzDU++4cbEjyR/NmsEFF4T/Ll27xo5GYiovL6e8vLzB19W7qYyZ9QCGu3tZ8nwo4O4+IqXMaGCOu09Nni8DTnT39WbWAngceNLdR6ZcsxRIJMu0TV5/aA3316YyAsBLL4WllFesCGvoRJfjTWVq8+qroZN49Wpo3jx2NJIvMrmpzHygk5l1NLOWQH9gerUy04GByRv3ADZWNfUA9wCvpSaAlGsuTX59CfBYGrFICZs4MTz15kUCyCOHHw577QVz58aORApRWttLmlkZMJKQNMa6+y1mNohQI7grWWYUUAZsBi5194VmdhwwF1hMaO5x4AZ3f8rM9gAeAPYDVgPnufvGGu6tmoBQUQH77QezZ0OXLvWXz4k8qQkA3HprqCHdfXfsSCRfpFsT0B7DUhDmzIFrr4UFC2JHkiKPksBbb0G3bvDuu9CyZexoJB9oj2EpKpMnl+5iceno0AEOOwxmzowdiRQaJQHJe59/Dg8/HDqFpXYDBoRkKdIQSgKS92bODE+5HTrEjiS/nXsuPPFEWGZbJF1KApL31BSUnr33DpPHplcfuydSByUByWubN4en23POiR1JYVCTkDSUkoDktenTw+Yxe+8dO5LCcMYZYbOZDz6IHYkUCiUByWtqCmqY3XaDU06BadNiRyKFQklA8taHH4an2jO0vmyDqElIGkJJQPLWtGnhqXbXXWNHUlj69IGFC2Ht2tiRSCFQEpC8paagxtlxR+jXDx54IHYkUgiUBCQvrV0blojo0yd2JIVJTUKSLiUByUsPPBCeZnfcMXYkhalXL1i1KuzEJlIXJQHJS2oKapoWLcLciilTYkci+U5JQPLOG2+Ep9hevWJHUtguuEBNQlI/JQHJO1OmhKdY7SPcND17wscfw+LFsSORfKYkIHlHTUGZ0awZ9O+v2oDUTUlA8srixbBpU1gqQppuwIBQs8qTvW8kDykJSF6ZPDk8vTbTb2ZGfOtb0KoVvPBC7EgkX+l/Nckb7jBpUujQlMwwUwex1E1JQPLG88/DzjuHp1fJnAEDYOpU+OKL2JFIPlISkLwxaRJceGF4epXM6dQJDjgAnn02diSSj5QEJC9s2wYPPqhRQdlywQUhyYpUpyQgeeGZZ+Cgg+DAA2NHUpzOOy9s0PPpp7EjkXyTVhIwszIzW2Zmy81sSC1lbjezFWa2yMy6pbw+1szWm9kr1coPM7M1ZrYgeZQ17aNIIVOHcHa1bQvf+Q48/njsSCTf1JsEzKwZMAo4FTgcGGBmXaqV6Q0c5O4HA4OAP6e8fW/y2prc5u5HJ4+nGvMBpPBt2QIzZoSnVckeNQlJTdKpCXQHVrj7anffBkwB+lUr0w+YAODu84DWZtYmef4c8GEt31tdgMKMGXDMMdCmTexIituZZ8Ls2WHHNpEq6SSBdsDbKedrkq/VVeadGsrUZHCy+WiMmbVOo7wUoYkT1RSUC61bw8kna/9h+aqYS3TdCdzk7m5mNwO3AVfUVHD48OFffp1IJEgkErmIT3Jgw4awj/D998eOpDRceCHcfjtceWXsSCTTysvLKS8vb/B15vUsKmJmPYDh7l6WPB8KuLuPSCkzGpjj7lOT58uAE919ffK8IzDD3Y+q5R61vm9mXl+MUrjuvBPmzi3Qde/NCm5Rnq1boV27sGtbhw6xo5FsMjPcvd4m93Sag+YDncyso5m1BPoD06uVmQ4MTN64B7CxKgFUxUO19n8za5tyehawJI1YpMjcfz9cfHHsKEpHq1ZhmW51EEuVepOAu1cAg4FZwKvAFHdfamaDzOz7yTJPAG+a2UrgL8BVVdeb2STgeeAQM3vLzC5LvnWrmb1iZouAE4H/yOQHk/z3xhuwciWcckrsSErLRRfBffcVXCVGsqTe5qDY1BxUvG68Ed5/P7RRF6QCbA4CqKwME/Mefhi6dau/vBSmTDYHiWSce2gKuuii2JGUnmbNwn93dcYLKAlIJC++GP4Yfec7sSMpTRddFJaXrqiIHYnEpiQgUdx3X/hDpBVD4+jcGdq318qioiQgEWzbFta3v/DC2JGUNjUJCSgJSARPPhmeRLViaFz9+4eVRT/+OHYkEpOSgOTcuHFw6aWxo5B99oETTtAyEqVOSUByasOGsIjZuefGjkQgJOPx42NHITEpCUhOTZ4MffuGxcwkvr59YckSePPN2JFILEoCklNqCsovLVuGLT0nTIgdicSiJCA588or8N57cNJJsSORVFVNQpWVsSORGJQEJGfGj4eBA6F589iRSKpu3WCXXeDvf48dicSgJCA5sW1b2DzmkktiRyLVmYXawLhxsSORGJQEJCeeego6dYKDD44didTkwgvh0Ufhk09iRyK5piQgOXHPPeoQzmdt2sDxx8ODD8aORHJNS0lL1r37Lhx2GLz1Fuy6a+xoMqhAl5KuzfTpcMst8PzzsSORTNBS0pI3xo8Pu1kVVQIoQn36wOrV8OqrsSORXFISkKyqrIQxY+Df/z12JFKfFi3gssvCz0tKh5KAZFV5Oey8s/YNKBSXXx5WFv3ss9iRSK4oCUhW3X13qAVo34DCcOCB0LUrPPJI7EgkV9QxLFmzYUMYFvrmm7D77rGjyYIi6xiu8sADMHp0WOhPCpc6hiW6+++H732vSBNAEevXLywqt3Jl7EgkF5QEJCvcQ1PQlVfGjkQaqlUruPhidRCXCiUByYq5c8PIoBNOiB2JNMagQXDvvbB1a+xIJNvSSgJmVmZmy8xsuZkNqaXM7Wa2wswWmVm3lNfHmtl6M3ulWvndzWyWmb1uZjPNTCvMF5E//QmuukodwoXqkEPgW9/SDOJSUG8SMLNmwCjgVOBwYICZdalWpjdwkLsfDAwC/pzy9r3Ja6sbCjzj7p2B2cD1jfoEknfWroWnnw4rhkrhuvrqkMyluKVTE+gOrHD31e6+DZgC9KtWph8wAcDd5wGtzaxN8vw54MMavm8/oGpju/HAGQ0PX/LRXXeFTcy1e1hh69s3JPQFC2JHItmUThJoB7ydcr4m+VpdZd6poUx1+7j7egB3Xwfsk0Yskue2bQtJ4OqrY0ciTdW8OfzgB6oNFLsWsQNIUeuA6+HDh3/5dSKRIJFI5CAcaYxHHgnLRR9xROxIJBOuvDL0D/zud7DHHrGjkbqUl5dTXl7e4OvqnSxmZj2A4e5eljwfCri7j0gpMxqY4+5Tk+fLgBOrnvTNrCMww92PSrlmKZBw9/Vm1jZ5/aE13F+TxQrIiSeGWsB558WOJAeKdLJYdRdfHDqJr7sudiTSEJmcLDYf6GRmHc2sJdAfmF6tzHRgYPLGPYCNVQmgKp7kUf2aS5NfXwI8lkYskseWLIEVK+DMM2NHIpl09dXw5z9DRUXsSCQb6k0C7l4BDAZmAa8CU9x9qZkNMrPvJ8s8AbxpZiuBvwBXVV1vZpOA54FDzOwtM7ss+dYI4GQzex3oBdySwc8lEfzxj/DDH8IOO8SORDLpmGNgzz3h8cdjRyLZoLWDJCPWr4dDD4Xly2GvvWJHkyMl0hwEMGUK3HlnmAQohUFrB0lO/elPcP75JZQASsw554QNZ+bPjx2JZJpqAtJkW7bA/vvDc8+FkSQlo4RqAgC33QYvvhhqBZL/VBOQnJkwAXr2LLEEUIKuvDLMBF+9OnYkkkmqCUiTVFZCly5hxcmSWyyuxGoCEIaJVlaGWoHkN9UEJCdmzAjLQxx/fOxIJBeuuQbGjYOPPoodiWSKkoA0mnuYSXrttVottFR06AC9e8Nf/hI7EskUNQdJo5WXw/e/D0uXhnVmSk4JNgcBLF4Mp5wCb7wBO+8cOxqpjZqDJOtuvhmuv75EE0AJO/JI6NEj7BwnhU81AWmUf/wDBgwIy0SU7AzhEq0JALz0UtiLeOVK2HHH2NFITVQTkKz61a9g6NASTgAl7tvfDovKjRsXOxJpKtUEpMGqngLfeCNsSl6ySrgmAKoN5jvVBCRrbr4ZfvazEk8AQs+e0KkT3Hdf7EikKVQTkAZZuBD69NHIEKDkawIQFpS77LIwQqxly9jRSCrVBCQrrr8efvELJQAJTjgh7CQ3ZkzsSKSxVBOQtM2ZE9aP0VNfkmoCQKgdnnZaWEZ8l11iRyNVVBOQjHIPo4F+9SslAPmqbt3CtqJ//GPsSKQxVBOQtDz8cEgAL70EzfToEKgm8KWVK8MEsmXLtKdEvki3JqAkIPX64gs44ojwpFdWFjuaPKIk8BVXXQU77QR/+EPsSASUBCSD7r4bJk2C2bO1UNxXKAl8xbp1cPjhoba4//6xoxElAcmIDz8Mewf/9a9hlqikUBL4mptugpdfhmnTYkciSgKSEddcA59/DqNHx44kDykJfM2nn4bawF13wXe/Gzua0qYkIE22eDH06hWGhO65Z+xo8pCSQI0eeyyMJHv5ZY0ki0lDRKVJ3OFHP4Lhw5UApGFOPz30CdxxR+xIJB1pJQEzKzOzZWa23MyG1FLmdjNbYWaLzKxrfdea2TAzW2NmC5KHxp3kkQcegI0bYdCg2JFIoTGDkSPht7+Fd9+NHY3Up97mIDNrBiwHegFrgflAf3dfllKmNzDY3U8zs2OAke7eo65rzWwY8LG717lltZqDcu+jj8KQ0EmTtHdwndQcVKehQ2H1apg8OXYkpSmTzUHdgRXuvtrdtwFTgH7VyvQDJgC4+zygtZm1SeNaDTjMQ9deC337KgFI0/zyl2G46KOPxo5E6pJOEmgHvJ1yvib5Wjpl6rt2cLL5aIyZtU47asmamTPhmWfg1ltjRyKFbuedYexYuPpq+OCD2NFIbVpk6fum84R/J3CTu7uZ3QzcBlxRU8Hhw4d/+XUikSCRSGQgRKlu06awcfyYMbDrrrGjkWJw/PFw9tnwk5/AhAmxoylu5eXllJeXN/i6dPoEegDD3b0seT4UcHcfkVJmNDDH3acmz5cBJwIH1Hdt8vWOwAx3P6qG+6tPIEd+8AOoqNAG4mlTn0BaNm8Om9PfcUdYbVRyI5N9AvOBTmbW0cxaAv2B6dXKTAcGJm/cA9jo7uvrutbM2qZcfxawJI1YJEtmzoQnnoDf/z52JFJsvvGN0Cw0aBBs2BA7Gqmu3uYgd68ws8HALELSGOvuS81sUHjb73L3J8ysj5mtBDYDl9V1bfJb35ocSloJrAI0GDGStWvh0kthyhRorZ4ZyYKTToILLoCBA+Hxx7USbT7RjOES98UXYXp/r17wX/8VO5oCo+agBtm2Lew7cOaZ8NOfxo6m+GnZCEnLL38Jzz8fmoOaN48dTYFREmiwt96C73wnDBvt2TN2NMVNy0ZIvZ55JrTVTpyoBCC50aFDGHjQvz+8/37saASUBErWihVw0UVw333Qpk3saKSUnH46nHcenHtuWKFW4lJzUAn64INQFb/22jAvQBpJzUGNVlER+gb22SfUDLRZUeapOUhq9PnnYfJO375KABJP8+ZhbaqXXoLf/S52NKVNNYES4g5XXhnGaj/8sPoBmkw1gSZbsybUSkeOhLPOih1NcUm3JpCtZSMkz7jDddfBK6/AnDlKAJIf2rcPm9CUlYVJZaeeGjui0qPmoBLgDjfcEDaKnzkTdtkldkQi2x19dBgyevHF4XdUcktJoATceGOYpfn007DHHrGjEfm6Y4+Fhx4KQ0fnzo0dTWlREihi7vDzn4ddwp59FvbaK3ZEIrU74YSwAc0554QHFskNJYEi9fnnYT2gZ56Bv/0tDMUTyXe9esG0aWEOi5aezg11DBehTZvCMNCddgqdwDvvHDsikfQdfzyUl0OfPvD226E/S/MIskc1gSLz+utw3HHQqVMYBqoEIIXo0EPDmlbTpoUa7ebNsSMqXkoCRWTyZPh//w8GD4Y774QWqudJAdt3X/j730PfVvfusHRp/ddIw2myWBHYsiXMAXj66dAJ3K1b7IhKhCaL5YQ73HMPDB0Kf/hDGEqq5qH6admIEvHss3DUUfDRR/DPfyoBSPExgyuuCL/rt94K3/teWJJaMkNJoEB98AFcfnk4Ro4My0FrVzApZkcdBQvRubM3AAAHPElEQVQWQI8eYYLZqFFhITppGiWBAvPpp2HBrc6dw8zfJUu0ebeUjpYt4Re/gOeegwcfhK5dw0RItco1nvoECsTWrWHt/xtvDJ1kv/41dOkSO6oSpz6BqNxDAhg6NMyE//Wvw/BS9RcE2l6ySHzwAYweHaq+RxwBN90UqsOSB5QE8kJFRZhY9pvfwO67h30yzj5bo+OUBApYZWVYP2X8+LDCYr9+8B//EdpEJY8oCeSViopQM/jDH2DVqjC/YODAMGemFCkJFJjKSnjxRZg+PWy20bo1XHIJXHABtG0bOzqpkZJA3lq0KDxETZoUksD554dRRQccEDuy3MloEjCzMuCPhI7kse4+ooYytwO9gc3Ape6+qK5rzWx3YCrQEVgFnOfuH9XwfYs2CaxZE9b1mT0b/vrXsMDb974XfmG7do0dndRLSSDvbdsWlk9/5JFQS9hnnzCQIpEIM+t33TV2hNmTsSRgZs2A5UAvYC0wH+jv7stSyvQGBrv7aWZ2DDDS3XvUda2ZjQDed/dbzWwIsLu7D63h/kWRBDZtCk8nCxaELfWefz6M7T/00HLOPTdB375w4IGxo8y88vJyEolE7DCyotyMRBH8btam2H52FRWhtv3kk+Hh68UXyznyyATdu8O3vx2GnR52GOywQ+xIMyOTO4t1B1a4++rkN54C9AOWpZTpB0wAcPd5ZtbazNoAB9RxbT/gxOT144Fy4GtJoFC4w4cfhgWvVq0KxxtvhKnuS5eG9448MvyyJRIwZEj4hbvppnKuuSYRN/gsKrY/JKnKgUTkGLKp2H52zZuHrSx79gznv/hFOaeckuCf/9w+Ee3NN0OTUZcu4TjgANh//3C0axd2Pys26SSBdsDbKedrCImhvjLt6rm2jbuvB3D3dWYWbbFj97D08tatYRx+1fHJJ9uPTZtg48bw9L5xY9ind8MGeO89WLcO3n0XWrWCDh22/9IceCD07h0Ww+rQAZppVoZI3mjRIuxhcMIJ21/77DNYuTI8uL3+eqg5TJ0aksPateH/8X33Df10e+0Vjj33hG9+M/TjtW4Nu+0W5vDssktIGjvuGFb03WmncH2LFvk1jDVbg6ga8xFrrVeffHL4Q13bUVm5/d+Kiq//W1EBX3yx/di2bfuxdWtIADvsEH5AVT+snXb66g+y6gf8zW+GH3rnztt/Cdq2Db8YWrFTpLDtuGMYin3EEV9/zz08AL77bnjwe//98CD4/vuwfj0sXx7e//jjsOpp1QPkp5+G5PLpp+HvTWVl+FvTsmX4u1N1tGix/WjePBzNmm3/N/Uw2/4vhH+rH2lz9zoPoAfwVMr5UGBItTKjgfNTzpcBbeq6FlhKqA0AtAWW1nJ/16FDhw4dDT/q+/vu7mnVBOYDncysI/Au0B8YUK3MdOBqYKqZ9QA2uvt6M9tQx7XTgUuBEcAlwGM13Tydjg0REWmcepOAu1eY2WBgFtuHeS41s0Hhbb/L3Z8wsz5mtpIwRPSyuq5NfusRwANmdjmwGjgv459ORETqlPeTxUREJHsKYryKmZ1jZkvMrMLMjo4dTyaYWZmZLTOz5cl5EkXDzMaa2XozeyV2LNlgZu3NbLaZvWpmi83smtgxZZKZtTKzeWa2MPn5hsWOKdPMrJmZLTCz6bFjyTQzW2VmLyd/fi/WV74gkgCwGDgT+FvsQDIhOYluFHAqcDgwwMyKaU3QewmfrVh9Afynux8O9ASuLqafn7tvBU5y925AV6C3mVUfFl7ofgy8FjuILKkEEu7ezd3r/bkVRBJw99fdfQWNG3qaj76cgOfu24CqSXRFwd2fAz6MHUe2uPu6qmVR3P0Twki3dnGjyix335L8shWh77Bo2o3NrD3QBxgTO5YsMRrwt70gkkARqm1ynRQYM9uf8LQ8L24kmZVsLlkIrAOedvf5sWPKoP8GfkoRJbZqHHjazOab2b/XVzhvVtw2s6cJcwu+fInwYX7u7jPiRCVSOzPbBXgI+HGyRlA03L0S6GZmuwGPmtlh7l7wzSdmdhqw3t0XmVmC4mldSHWcu79rZnsTksHSZO28RnmTBNz95Ngx5NA7QIeU8/bJ16RAmFkLQgK4z91rnONSDNx9k5nNAcoojjb044DTzawPsBOwq5lNcPeBkePKGHd/N/nve2b2CKH5udYkUIjNQcWQub+cgGdmLQmT6IptlIJRHD+r2twDvObuI2MHkmlmtpeZtU5+vRNwMl9dMLJgufsN7t7B3Q8k/H83u5gSgJntnKyhYmbfAE4BltR1TUEkATM7w8zeJixD8biZPRk7pqZw9wqgahLdq8CUlEl0Bc/MJgHPA4eY2VtmdlnsmDLJzI4DLgT+LTkMb0Fy34xisS8wx8wWEfo6Zrr7E5FjkvS0AZ5L9ue8AMxw91l1XaDJYiIiJawgagIiIpIdSgIiIiVMSUBEpIQpCYiIlDAlARGREqYkICJSwpQERERKmJKAiEgJ+/9gsOy5WEP6dQAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f8e52a97150>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# plot\n",
    "xs = linspace(-2, 6, 1000)\n",
    "thetas = linspace(-1, 5, 100)\n",
    "# p(theta|D)\n",
    "plot(thetas, p_theta_D(thetas, D), color=\"blue\")\n",
    "# arg max p(theta|D)... or mean(D)\n",
    "plot([mean(D)]*2, [0, 0.03], color=\"red\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "code_folding": [],
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f8e52a8dc10>]"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f8e52a8dbd0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# p(x|D) using Bayesian vs ML\n",
    "\n",
    "def bayesian_integral(x, D):\n",
    "    thetas = linspace(-5, 10, 1000)\n",
    "    return sum(p_x_theta(x, thetas) * p_theta_D(thetas, D)) / sum(p_theta_D(thetas, D)) \n",
    "\n",
    "ps = [(x, bayesian_integral(x, D)) for x in xs]\n",
    "\n",
    "ps = array(ps); plot(ps[:,0], ps[:,1])\n",
    "plot(xs, p_x_theta(xs, mean(D)), color=\"red\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f8e525c9490>]"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f8e526ab050>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# overparameterizing the distribution\n",
    "plot(ps[:,0], ps[:,1])\n",
    "plot(xs, p_x_theta(xs, mean(D)), color=\"black\", alpha=0.3)\n",
    "plot(xs, p_x_theta(xs, mean(D), var(D)**.5), color=\"red\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Maximum Likelihood vs Bayesian Estimates\n",
    "\n",
    "The Bayesian estimate $p(x|D)$ usually is quite different from the ML estimate $p(x|\\hat{\\theta}(D))$. It often doesn't even have the same form.\n",
    "\n",
    "ML estimates tend to be \"overtrained\" / underestimate variance. Bayesian estimates can never be \"overtrained\".\n",
    "\n",
    "Using an overparameterization accidentally can compensate for this to some degree (that probably makes DNNs work better)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Maximum A-Posteriori Estimate"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## Maximum A Posteriori Estimate\n",
    "\n",
    "By analogy to Bayesian methods, we can also ``multiply in'' a prior:\n",
    "\n",
    "$$\\hat{\\theta} = \\arg\\max_\\theta ~ p(D|\\theta) p(\\theta)$$\n",
    "\n",
    "This is called the _maximum a-posterior estimate_ (MAP) for the parameter $\\theta$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "## MAP vs Bayesian Methods\n",
    "\n",
    "Although MAP looks like the derivation may have involved _Bayes rule_, it is\n",
    "not a Bayesian method at all.\n",
    "Bayesian methods are *not* methods that involve Bayes rule somewhere.\n",
    "\n",
    "**Bayesian methods are methods that result in decisions that minimize expected loss.**\n",
    "\n",
    "**Bayesian methods are NOT methods that happen to use Bayes rule somewhere.**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Summary\n",
    "\n",
    "We know how to do classification / estimation optimally and correctly: Bayesian methods.\n",
    "\n",
    "Why aren't we using them with DNNs? Because they are computationally prohibitive.\n",
    "\n",
    "However, several methods attempt to approximate Bayesian solutions:\n",
    "- dropout methods\n",
    "- ensemble methods\n",
    "- some forms of variational methods"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Uniform Distribution Example"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Bayesian vs Maximum Likelihood Example\n",
    "==============================\n",
    "\n",
    "Let's apply this to another, simple example and compare maximum likelihood and bayesian approaches\n",
    "~\n",
    "Assume that the samples $x$ come from a uniform density over the interval $[0,\\theta]$\n",
    "\n",
    "$$p(x|\\theta) = U(x;0,\\theta) = 1/\\theta \\cdot \\lfloor x\\in[0,\\theta]\\rfloor$$\n",
    "\n",
    "We also assume a prior\n",
    "\n",
    "$$p(\\theta) = U(\\theta;0,10)$$\n",
    "\n",
    "That is, $\\theta$ is distributed uniformly over the interval $[0,10]$\n",
    "\n",
    "Let's assume we see a sequence of training examples $D = \\\\{4,7,2\\\\}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f8e523c5a90>]"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f8e524a7710>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "xs = linspace(-1,11,1000)\n",
    "mus = xs\n",
    "C = (amax(xs)-amin(xs))/len(xs)\n",
    "def pxt(x,mu): return (xs>=0)*(x<=mu)*1.0/maximum(mu,1e-6)\n",
    "plot(xs,pxt(xs,5.5))\n",
    "pmu = (xs>=0)*(xs<=10)*0.1\n",
    "plot(xs,pmu)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Now assume we draw the sample $x_1=4$.  How should we update our estimate?\n",
    "\n",
    "$$p(\\mu|x) = \\frac{p(x|\\mu) p(\\mu)}{p(x)}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f8e52238050>]"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f8e526376d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pmu1 = pxt(4,mus)*pmu\n",
    "pmu1 /= C*sum(pmu1)\n",
    "plot(xs,pmu1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "The maximum likelihood estimate is clearly at $\\mu=4$.  If we now plug this\n",
    "into $p(x|\\mu)$ we get..."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f8e52269110>]"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f8e52396c50>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "ylim(0,1)\n",
    "plot(xs,pxt(xs,4))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "This is odd because it predicts that only values between 0 and 4 can occur.\n",
    "But the training sample $x_1=4$ only excludes that $\\mu\\lt4$; it doesn't \n",
    "exclude any values greater than $4$.\n",
    "\n",
    "What's the Bayesian estimate?\n",
    "\n",
    "$$p(x|D) = \\int p(x|\\theta) p(\\theta|D) d\\theta \\propto \\int_0^{10} 1/\\theta \\cdot \\lfloor x\\in[0,\\theta]\\rfloor \\cdot 1/\\theta \\cdot \\lfloor \\theta\\in[4,10]\\rfloor d\\theta\n",
    "= \\int_{x_1}^{10} 1/\\theta^2 \\cdot \\lfloor x\\in[0,\\theta]\\rfloor d\\theta$$\n",
    "\n",
    "You can either think about it, or we can simply perform this integral numerically."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f8e521ffad0>]"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f8e52567190>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "result = zeros(xs.shape)\n",
    "for i,mu in enumerate(mus):\n",
    "    weight = pmu1[i]\n",
    "    result += weight * pxt(xs,mu)\n",
    "result /= C*sum(result)\n",
    "plot(xs,result)\n",
    "plot(xs,pxt(xs,4))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Now assume we get another sample, $x_1=7$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f8e5203de10>]"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f8e5221b490>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pmu2 = pxt(7,mus)*pmu1\n",
    "pmu2 /= C*sum(pmu2)\n",
    "plot(xs,pmu2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f8e51f87c10>]"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f8e5200b810>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "result = zeros(xs.shape)\n",
    "for i,mu in enumerate(mus):\n",
    "    weight = pmu2[i]\n",
    "    result += weight * pxt(xs,mu)\n",
    "result /= C*sum(result)\n",
    "plot(xs,result)\n",
    "plot(xs,pxt(xs,7))\n",
    "plot(xs,pxt(xs,4))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "This is even weirder.  After seeing the first sample, the maximum likelihood\n",
    "estimator predicts only values between 0 and 4 occurring, but after seeing\n",
    "another training sample, it is changing its mind and now predicts that values\n",
    "between 0 and 7 can occur.\n",
    "\n",
    "The Bayesian estimator, in contrast, \"knows\" that the parameter must be greater than 7,\n",
    "so it predicts a uniform distribution for the interval [0...7] and then a tradeoff\n",
    "between the parameter distribution and the uniform distribution of the parameters."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "The last sample illustrates this further.\n",
    "\n",
    "A sample of $x_3=2$ doesn't cause any update to the maximum likelihood estimator,\n",
    "but it does cause an update to posterior distribution."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f8e520865d0>]"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f8e52086690>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "pmu3 = pxt(2,mus)*pmu2\n",
    "pmu3 /= C*sum(pmu3)\n",
    "plot(xs,pmu3)\n",
    "plot(xs,pmu2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f8e51f54dd0>]"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f8e51f54750>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "result = zeros(xs.shape)\n",
    "total = 0\n",
    "for i,mu in enumerate(mus):\n",
    "    weight = pmu3[i]\n",
    "    result += weight * pxt(xs,mu)\n",
    "    total += weight\n",
    "result /= total\n",
    "plot(xs,result)\n",
    "plot(xs,pxt(xs,7))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "In fact, if we repeat the same process with a lot of samples (in this case\n",
    "the true parameter is 7), we see that the Bayesian parameter estimate\n",
    "becomes an increasingly peaked distribution close to the true value.\n",
    "\n",
    "I.e., if, out of 100 samples, we haven't seen a value greater than 7,\n",
    "then the probability that the mean is significantly greater than 7 must\n",
    "be very small."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f8e52410550>]"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f8e5254d310>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "p = pmu.copy()\n",
    "for i in range(100):\n",
    "    p = pxt(rand()*7,mus)*p\n",
    "    p /= C*sum(p)\n",
    "plot(mus,p)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Loss Functions for Parameter Estimation\n",
    "========================================\n",
    "\n",
    "Consider $p(\\theta|x)$ from the previous example again."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f8e51d46950>]"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f8e51e4b110>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot(mus,pmu2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Assume now that we are supposed to return a \"best estimate\" of the parameter.\n",
    "\n",
    "By itself, that isn't sufficient.\n",
    "\n",
    "But now assume that we are given a loss function: if our estimate is with in $\\pm 0.5$ of the true value,\n",
    "we don't need to pay a penalty, otherwise, we need to pay a penalty of 1.  What value should we return?\n",
    "\n",
    "The most likely value is 7, but no values less than 7 can occur."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Therefore, it is better to return 7.5.  That way, not only do we have the most likely value, but we also\n",
    "get all the probability mass between 7 and 8 as well and our expected loss is about half of what it would be\n",
    "if we had returned 7."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "Now assume we are penalized if we are outside the range $\\pm 1$ from the true value.\n",
    "By the same reasoning, our parameter estimate should now be 8."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "As you can see from this simple example, there is not \"best\" answer to the parameter estimation\n",
    "problem; our answer depends on the loss function.\n",
    "\n",
    "But we can see that for any symmetric loss function, the value 7 (the maximum likelihood estimate)\n",
    "is never the optimal answer."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "celltoolbar": "Slideshow",
  "kernelspec": {
   "display_name": "Python 2",
   "language": "python",
   "name": "python2"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 2
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython2",
   "version": "2.7.15"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 1
}