{
 "metadata": {
  "name": "",
  "signature": "sha256:4ecdae6210f95e36d3af59848f494f74e2f29cdacd65867ecf2aa7690cd9e1b4"
 },
 "nbformat": 3,
 "nbformat_minor": 0,
 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Gaussian Probabilities"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "#format the book\n",
      "%matplotlib inline\n",
      "from __future__ import division, print_function\n",
      "import matplotlib.pyplot as plt\n",
      "import book_format\n",
      "book_format.load_style()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "html": [
        "<style>\n",
        "@import url('http://fonts.googleapis.com/css?family=Source+Code+Pro');\n",
        "\n",
        "    div.cell{\n",
        "        width: 850px;\n",
        "        margin-left: 0% !important;\n",
        "        margin-right: auto;\n",
        "    }\n",
        "    div.text_cell code {\n",
        "        background: #FFFFFF;\n",
        "        color: #000000;\n",
        "        font-weight: 600;\n",
        "        font-size: 13pt;\n",
        "        font-style: bold;\n",
        "        font-family:  'Source Code Pro', Consolas, monocco, monospace;\n",
        "   }\n",
        "    h1 {\n",
        "        font-family: 'Open sans',verdana,arial,sans-serif;\n",
        "\t}\n",
        "\t\n",
        "    div.input_area {\n",
        "        background: #F6F6F9;\n",
        "        border: 1px solid #586e75;\n",
        "    }\n",
        "\n",
        "    .text_cell_render h1 {\n",
        "        font-weight: 200;\n",
        "        font-size: 30pt;\n",
        "        line-height: 100%;\n",
        "        color:#c76c0c;\n",
        "        margin-bottom: 0.5em;\n",
        "        margin-top: 1em;\n",
        "        display: block;\n",
        "        white-space: wrap;\n",
        "    } \n",
        "    h2 {\n",
        "        font-family: 'Open sans',verdana,arial,sans-serif;\n",
        "    }\n",
        "    .text_cell_render h2 {\n",
        "        font-weight: 200;\n",
        "        font-size: 20pt;\n",
        "        font-style: italic;\n",
        "        line-height: 100%;\n",
        "        color:#c76c0c;\n",
        "        margin-bottom: 0.5em;\n",
        "        margin-top: 1.5em;\n",
        "        display: block;\n",
        "        white-space: nowrap;\n",
        "    } \n",
        "    h3 {\n",
        "        font-family: 'Open sans',verdana,arial,sans-serif;\n",
        "    }\n",
        "    .text_cell_render h3 {\n",
        "        font-weight: 300;\n",
        "        font-size: 18pt;\n",
        "        line-height: 100%;\n",
        "        color:#d77c0c;\n",
        "        margin-bottom: 0.5em;\n",
        "        margin-top: 2em;\n",
        "        display: block;\n",
        "        white-space: nowrap;\n",
        "    }\n",
        "    h4 {\n",
        "        font-family: 'Open sans',verdana,arial,sans-serif;\n",
        "    }\n",
        "    .text_cell_render h4 {\n",
        "        font-weight: 300;\n",
        "        font-size: 16pt;\n",
        "        color:#d77c0c;\n",
        "        margin-bottom: 0.5em;\n",
        "        margin-top: 0.5em;\n",
        "        display: block;\n",
        "        white-space: nowrap;\n",
        "    }\n",
        "    h5 {\n",
        "        font-family: 'Open sans',verdana,arial,sans-serif;\n",
        "    }\n",
        "    .text_cell_render h5 {\n",
        "        font-weight: 300;\n",
        "        font-style: normal;\n",
        "        color: #1d3b84;\n",
        "        font-size: 16pt;\n",
        "        margin-bottom: 0em;\n",
        "        margin-top: 1.5em;\n",
        "        display: block;\n",
        "        white-space: nowrap;\n",
        "    }\n",
        "    div.text_cell_render{\n",
        "        font-family: 'Open sans',verdana,arial,sans-serif;\n",
        "        line-height: 135%;\n",
        "        font-size: 110%;\n",
        "        width:750px;\n",
        "        margin-left:auto;\n",
        "        margin-right:auto;\n",
        "        text-align:justify;\n",
        "        text-justify:inter-word;\n",
        "    }\n",
        "    div.output_subarea.output_text.output_pyout {\n",
        "        overflow-x: auto;\n",
        "        overflow-y: scroll;\n",
        "        max-height: 300px;\n",
        "    }\n",
        "    div.output_subarea.output_stream.output_stdout.output_text {\n",
        "        overflow-x: auto;\n",
        "        overflow-y: scroll;\n",
        "        max-height: 300px;\n",
        "    }\n",
        "    code{\n",
        "      font-size: 70%;\n",
        "    }\n",
        "    .rendered_html code{\n",
        "    background-color: transparent;\n",
        "    }\n",
        "    ul{\n",
        "        margin: 2em;\n",
        "    }\n",
        "    ul li{\n",
        "        padding-left: 0.5em; \n",
        "        margin-bottom: 0.5em; \n",
        "        margin-top: 0.5em; \n",
        "    }\n",
        "    ul li li{\n",
        "        padding-left: 0.2em; \n",
        "        margin-bottom: 0.2em; \n",
        "        margin-top: 0.2em; \n",
        "    }\n",
        "    ol{\n",
        "        margin: 2em;\n",
        "    }\n",
        "    ol li{\n",
        "        padding-left: 0.5em; \n",
        "        margin-bottom: 0.5em; \n",
        "        margin-top: 0.5em; \n",
        "    }\n",
        "    ul li{\n",
        "        padding-left: 0.5em; \n",
        "        margin-bottom: 0.5em; \n",
        "        margin-top: 0.2em; \n",
        "    }\n",
        "    a:link{\n",
        "       font-weight: bold;\n",
        "       color:#447adb;\n",
        "    }\n",
        "    a:visited{\n",
        "       font-weight: bold;\n",
        "       color: #1d3b84;\n",
        "    }\n",
        "    a:hover{\n",
        "       font-weight: bold;\n",
        "       color: #1d3b84;\n",
        "    }\n",
        "    a:focus{\n",
        "       font-weight: bold;\n",
        "       color:#447adb;\n",
        "    }\n",
        "    a:active{\n",
        "       font-weight: bold;\n",
        "       color:#447adb;\n",
        "    }\n",
        "    .rendered_html :link {\n",
        "       text-decoration: underline; \n",
        "    }\n",
        "    .rendered_html :hover {\n",
        "       text-decoration: none; \n",
        "    }\n",
        "    .rendered_html :visited {\n",
        "      text-decoration: none;\n",
        "    }\n",
        "    .rendered_html :focus {\n",
        "      text-decoration: none;\n",
        "    }\n",
        "    .rendered_html :active {\n",
        "      text-decoration: none;\n",
        "    }\n",
        "    .warning{\n",
        "        color: rgb( 240, 20, 20 )\n",
        "    } \n",
        "    hr {\n",
        "      color: #f3f3f3;\n",
        "      background-color: #f3f3f3;\n",
        "      height: 1px;\n",
        "    }\n",
        "    blockquote{\n",
        "      display:block;\n",
        "      background: #fcfcfc;\n",
        "      border-left: 5px solid #c76c0c;\n",
        "      font-family: 'Open sans',verdana,arial,sans-serif;\n",
        "      width:680px;\n",
        "      padding: 10px 10px 10px 10px;\n",
        "      text-align:justify;\n",
        "      text-justify:inter-word;\n",
        "      }\n",
        "      blockquote p {\n",
        "        margin-bottom: 0;\n",
        "        line-height: 125%;\n",
        "        font-size: 100%;\n",
        "      }\n",
        "</style>\n",
        "<script>\n",
        "    MathJax.Hub.Config({\n",
        "                        TeX: {\n",
        "                           extensions: [\"AMSmath.js\"]\n",
        "                           },\n",
        "                tex2jax: {\n",
        "                    inlineMath: [ ['$','$'], [\"\\\\(\",\"\\\\)\"] ],\n",
        "                    displayMath: [ ['$$','$$'], [\"\\\\[\",\"\\\\]\"] ]\n",
        "                },\n",
        "                displayAlign: 'center', // Change this to 'center' to center equations.\n",
        "                \"HTML-CSS\": {\n",
        "                    styles: {'.MathJax_Display': {\"margin\": 4}}\n",
        "                }\n",
        "        });\n",
        "</script>\n"
       ],
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 3,
       "text": [
        "<IPython.core.display.HTML at 0x27e1e90>"
       ]
      }
     ],
     "prompt_number": 3
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Introduction"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The last chapter ended by discussing some of the drawbacks of the Discrete Bayesian filter. For many tracking and filtering problems our desire is to have a filter that is *unimodal* and *continuous*. That is, we want to model our system using floating point math (continuous) and to have only one belief represented (unimodal). For example, we want to say an aircraft is at (12.34381, -95.54321,2389.5) where that is latitude, longitude, and altidue. We do not want our filter to tell us \"it might be at (1,65,78) or at (34,656,98)\" That doesn't match our physical intuition of how the world works, and as we discussed, it is prohibitively expensive to compute.\n",
      "\n",
      ">So we desire a unimodal, continuous way to represent probabilities that models how the real world works, and that is very computationally efficient to calculate. As you might guess from the chapter name, Gaussian distributions provide all of these features.\n",
      "\n",
      "Before we go into the math, lets just look at a graph of the Gaussian distribution to get a sense of what we are talking about."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from stats import plot_gaussian\n",
      "\n",
      "plot_gaussian(mean=100, variance=15*15, xlabel='IQ', ylabel='percent')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAY8AAAEUCAYAAADTO7pnAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nCCGEEEIY4v8Dyc+UlrAZi9QAAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x300a290>"
       ]
      }
     ],
     "prompt_number": 4
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Probably this is immediately recognizable to you as a 'bell curve'. This curve is ubiquitious because under real world conditions most observations are distributed in such a manner. In fact, this is the bell curve for IQ (Intelligence Quotient). You've probably seen this before, and understand it. It tells us that the average IQ is 100, and that the number of people that have IQs higher or lower than that drops off as they get further away from 100. It's hard to see the exact number, but we can see that very few people have an IQ over 150 or under 50, but a lot have an IQ of 90 or 110. \n",
      "\n",
      "This curve is not unique to IQ distributions - a vast amount of natural phenomena exhibits this sort of distribution, including the sensors that we use in filtering problems. As we will see, it also has all the attributes that we are looking for - it represents a unimodal belief or value as a probabilitiy, it is continuous, and it is computationally efficient. We will soon discover that it also other desirable qualities that we do not yet recognize we need.\n",
      "<p>\n",
      "<p>"
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Nomenclature"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "A bit of nomenclature before we continue - this chart depicts the probability of of a *random variable* having any value between ($-\\infty..\\infty)$. For example, for this chart the probability of the variable being 100 is roughly 2.7%, whereas the probability of it being 80 is around 1%.\n",
      "> *Random variable* will be precisely defined later. For now just think of it as a variable that can 'freely' and 'randomly' vary. A dog's position in a hallway, air temperature, and a drone's height above the ground are all random variables. The position of the North Pole is not, nor is a sin wave (a sin wave is anything but 'free').\n",
      "\n",
      "You may object that human IQs cannot be less than zero, let alone $-\\infty$. This is true, but this is a common limitation of mathematical modelling. \"The map is not the territory\" is a common expression, and it is true for Bayesian filtering and statistics. The Gaussian distribution above very closely models the distribution of IQ test results, but being a model it is necessarily imperfect. The difference between model and reality will come up again and again in these filters. \n",
      "\n",
      "You will see these distributions called *Gaussian distributions*, *normal distributions*, and *bell curves*. Bell curve is ambiguous because there are other distributions which also look bell shaped but are not Gaussian distributions, so we will not use it further in this book. But *Gaussian* and *normal* both mean the same thing, and are used interchangeably. I will use both throughout this book as different sources will use either term, and so I want you to be used to seeing both. Finally, as in this paragraph, it is typical to shorten the name and just talk about a *Gaussian* or *normal* - these are both typical shortcut names for the *Gaussian distribution*. "
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Gaussian Distributions"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "So let us explore how Gaussians work. A Gaussian is a *continuous probability distribution* that is completely described with two parameters, the mean ($\\mu$) and the variance ($\\sigma^2$). It is defined as:\n",
      "$$ \n",
      "f(x, \\mu, \\sigma) = \\frac{1}{\\sigma\\sqrt{2\\pi}} e^{-\\frac{1}{2}{(x-\\mu)^2}/\\sigma^2 }\n",
      "$$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "<p> Don't be dissuaded by the equation if you haven't seen it before; you will not need to memorize or manipulate it. The computation of this function is stored in stats.py. \n",
      "\n",
      "> **Optional:** Let's remind ourselves how to look at a function stored in a file by using the *%load* magic. If you type *%load -s gaussian stats.py* into a code cell and then press CTRL-Enter, the notebook will create a new input cell and load the function into it.\n",
      "\n",
      "    %load -s gaussian stats.py\n",
      "    \n",
      "    def gaussian(x, mean, var):\n",
      "        \"\"\"returns normal distribution for x given a \n",
      "        gaussian with the specified mean and variance. \n",
      "        \"\"\"\n",
      "        return math.exp((-0.5*(x-mean)**2)/var) / \\\n",
      "                         math.sqrt(_two_pi*var)"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "<p><p><p><p>We will plot a Gaussian with a mean of 22 $(\\mu=22)$, with a variance of 4 $(\\sigma^2=4)$, and then discuss what this means. "
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from stats import gaussian\n",
      "plot_gaussian(22,4,mean_line=True,xlabel='$^{\\circ}C$',ylabel=\"Percent\")\n",
      "\n",
      "print('Probability of 22 is %.2f' % (gaussian(22,22,4)*100))\n",
      "print('Probability of 24 is %.2f' % (gaussian(24,22,4)*100))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Probability of 22 is 19.95\n",
        "Probability of 24 is 12.10\n"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAYoAAAEUCAYAAAA1EnEjAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nIYQQQgghhBBCWNL/Bypv6ZnCLdD0AAAAAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x302c590>"
       ]
      }
     ],
     "prompt_number": 5
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "So what does this curve *mean*? Assume for a moment that we have a themometer, which reads 22$\\,^{\\circ}C$. No thermometer is perfectly accurate, and so we normally expect that thermometer will read $\\pm$ that temperature by some amount each time we read it. Furthermore, a theorem called  **Central Limit Theorem** states that if we make many measurements that the measurements will be normally distributed. If that is true, then this chart can be interpreted as a continuous curve depicting our belief that the temperature is any given temperature. In this curve, we assign a probability of the temperature being exactly 22$^{\\circ}C$ is $19.95\\%$. Looking to the right, we assign the probability that the temperature is 24$^{\\circ}C$ is $12.10\\%$. Because of the curve's symmetry, the probability of 20$^{\\circ}$C is also $12.10\\%$.\n",
      "\n",
      "So the mean ($\\mu$) is what it sounds like - the average of all possible probabilities. Because of the symmetric shape of the curve it is also the tallest part of the curve. The thermometer reads $22^{\\circ}C$, so that is what we used for the mean.  \n",
      "\n",
      "> *Important*: I will repeat what I wrote at the top of this section: \"A Gaussian...is completely described with two parameters\"\n",
      "\n",
      "The standard notation for a normal distribution for a random variable $X$ is $X \\sim\\ \\mathcal{N}(\\mu,\\sigma^2)$. This means I can express the temperature reading of our thermometer as\n",
      "\n",
      "$$temp = \\mathcal{N}(22,4)$$\n",
      "\n",
      "This is an **extremely important** result. Gaussians allow me to capture an infinite number of possible values with only two numbers! With the values $\\mu=22$ and $\\sigma^2=4$ I can compute the probability of the temperature being $22\\,^{\\circ}C$, $20\\,^{\\circ}C$, $87.34\\,^{\\circ}C$, or any other arbitrary value.\n",
      "\n",
      "###### The Variance\n",
      "\n",
      "Since this is a probability distribution it is required that the area under the curve always equals one. This should be intuitively clear - the area under the curve represents all possible occurances, which must sum to one.\n",
      "\n",
      "This leads to an important insight. If the variance is small the curve will be narrow. To keep the area equal to 1, the curve must also be tall. On the other hand if the variance is large the curve will be wide, and thus it will also have to be short to make the area equal to 1.\n",
      "\n",
      "Let's look at that graphically:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import numpy as np\n",
      "import matplotlib.pyplot as plt\n",
      "\n",
      "xs = np.arange(15,30,0.05)\n",
      "p1, = plt.plot (xs,[gaussian(x, 23, .2) for x in xs],'b')\n",
      "p2, = plt.plot (xs,[gaussian(x, 23, 1) for x in xs],'g')\n",
      "p3, = plt.plot (xs,[gaussian(x, 23, 5) for x in xs],'r')\n",
      "plt.legend([p1,p2,p3], ['var = .2', 'var = 1', 'var = 5'])\n",
      "plt.show()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAXcAAAECCAYAAAAFL5eMAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x3203850>"
       ]
      }
     ],
     "prompt_number": 6
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "So what is this telling us? The blue gaussian is very narrow. It is saying that we believe $x=23$, and that we are very sure about that $(90%)$. In contrast, the red gaussian also believes that $x=23$, but we are much less sure about that $(18%)$. Our believe that $x=23$ is lower, and so our belief about the likely possible values for $x$ is spread out - we think it is quite likely that $x=20$ or $x=26$, for example. The blue gaussian has almost completely eliminated $22$ or $24$ as possible value - their probability is almost $0\\%$, whereas the red curve considers them nearly as likely as $23$.\n",
      "\n",
      "If we think back to the thermometer, we can consider these three curves as representing the readings from three different thermometers. The blue curve represents a very accurate thermometer, and the red one represents a fairly inaccurate one. Green of course represents one in between the two others. Note the very powerful property the Gaussian distribution affords us - we can entirely represent both the reading and the error of a thermometer with only two numbers - the mean and the variance.\n",
      "\n",
      "The standard notation for a normal distribution for a random variable $X$ is just $X \\sim\\ \\mathcal{N}(\\mu,\\sigma^2)$ where $\\mu$ is the mean and $\\sigma^2$ is the variance. It may seem odd to use $\\sigma$ squared - why not just $\\sigma$? We will not go into great detail about the math at this point, but in statistics $\\sigma$ is the *standard deviation* of a normal distribution. *Variance* is defined as the square of the standard deviation, hence $\\sigma^2$.\n",
      "\n",
      "It is worth spending a few words on standard deviation now. The standard deviation is a measure of how much variation from the mean exists. For Gaussian distributions, 68% of all the data falls within one standard deviation($1\\sigma$) of the mean, 95% falls within two standard deviations ($2\\sigma$), and 99.7% within three ($3\\sigma$). This is often called the 68-95-99.7 rule. So if you were told that the average test score in a class was 71 with a standard deviation of 9.4, you could conclude that 95% of the students received a score between 52.2 and 89.8 if the distribution is normal (that is calculated with $71 \\pm (2 * 9.4)$). "
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The following graph depicts the relationship between the standard deviation and the normal distribution. "
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from gaussian_internal import display_stddev_plot\n",
      "display_stddev_plot()\n",
      "plt.show()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAqwAAAFuCAYAAABECkoSAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\n0IfhEmLhUCQAAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0x35e8c50>"
       ]
      }
     ],
     "prompt_number": 7
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "<p>\n",
      "<p>\n",
      "<div style=\"border:1px dotted black;background-color:#DCDCDC#\">**Sidebar**: An equivalent formation for a Gaussian is $\\mathcal{N}(\\mu,1/\\tau)$ where $\\mu$ is the *mean* and $tau$ the *precision*. Here $1/\\tau = \\sigma^2$; it is the reciprocal of the variance. While we do not use this formulation in this book, it underscores that the variance is a measure of how precise our data is. A small variance yields large precision - our measurement is very precise. Conversely, a large variance yields low precision - our belief is spread out across a large area. You should become comfortable with thinking about Gaussians in these equivelant forms. Gaussians reflect our *belief* about a measurement, they express the *precision* of the measurement, and they express how much *variance* there is in the measurements. These are all different ways of stating the same fact.</div>"
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Interactive Gaussians"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "For those that are reading this in IPython Notebook, here is an interactive version of the Gaussian plots. Use the sliders to modify $\\mu$ and $\\sigma^2$. Adjusting $\\mu$ will move the graph to the left and right because you are adjusting the mean, and adjusting $\\sigma^2$ will make the bell curve thicker and thinner."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import math\n",
      "from IPython.html.widgets import interact, interactive, fixed\n",
      "import IPython.html.widgets as widgets\n",
      "\n",
      "def plt_g (mu,variance):\n",
      "    xs = np.arange(2,8,0.1)\n",
      "    ys = [gaussian (x, mu,variance) for x in xs]\n",
      "    plt.plot (xs, ys)\n",
      "    plt.ylim((0,1))\n",
      "    plt.show()\n",
      "\n",
      "interact (plt_g, mu=(0,10), variance=widgets.FloatSliderWidget(value=0.6,min=0.2,max=4.5))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAsMAAAFyCAYAAAAQ8tcPAAAABHNCSVQICAgIfAhkiAAAAAlwSFlz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\nAAAAAAAAhND/B39KSYW/Pub6AAAAAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x3037990>"
       ]
      },
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 8,
       "text": [
        "<function __main__.plt_g>"
       ]
      }
     ],
     "prompt_number": 8
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Computational Properties of the Gaussian"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Recall how our discrete Bayesian filter worked. We had a vector implemented as a numpy array representing our belief at a certain moment in time. When we performed another measurement using the `sense()` function we had to multiply probabilities together, and when we performed the motion step using the `update()` function we had to shift and add probabilities. I've promised you that the Kalman filter uses essentially the same process, and that  it uses Gaussians instead of histograms, so you might reasonable expect that we will be multipling, adding, and shifting Gaussians in the Kalman filter.\n",
      "\n",
      "A typical textbook would directly launch into a multipage proof of the behavior of Gaussians under these operations, but I don't see the value in that right now. I think the math will be much more intuitive and clear if we just start developing a Kalman filter using Gaussians. I will provide the equations for multiplying and shifting Gaussians at the appropriate time. You will then be able to develop a physical intuition for what these operations do, rather than be forced to digest a lot of fairly abstract math.\n",
      "\n",
      "The key point, which I will only assert for now, is that all the operations are very simple, and that they preserve the properties of the Gaussian. This is somewhat remarkable, in that the Gaussian is a nonlinear function, and typically if you multiply a nonlinear equation with itself you end up with a different equation. For example, the shape of `sin(x)sin(x)` is very different from `sin(x)`. But the result of multiplying two Gaussians is yet another Gaussian. This is a fundamental property, and the key reason why Kalman filters are possible."
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Computing Probabilities with scipy.stats"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "In this chapter I have used by custom written code for computing Gaussians, plotting, and so on. I chose to do that to give you a chance to look at the code and see how these functions are implemented.  However, Python comes with \"batteries included\" as the saying goes, and it comes with a wide range of statistics functions in the module scipy.stats. I find the performance of some of the functions rather slow (the scipy.stats documentation contains a warning to this effect), but this is offset by the fact that this is standard code available to everyone, and it is well tested. So let's walk through how to use scipy.stats to compute various things.\n",
      "\n",
      "The scipy.stats module contains a number of objects which you can use to compute attributes of various probability distributions. The full documentation for this module is here: http://http://docs.scipy.org/doc/scipy/reference/stats.html. However, we will focus on the  norm variable, which implements the normal distribution. Let's look at some code that uses `scipy.stats.norm` to compute a Gaussian, and compare its value to the value returned by the `gaussian()` function."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from scipy.stats import norm\n",
      "print (norm(2,3).pdf(1.5))\n",
      "print (gaussian(x=1.5, mean=2, var=3*3))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "0.131146572034\n",
        "0.131146572034\n"
       ]
      }
     ],
     "prompt_number": 9
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The call `norm(2,3)` creates what scipy calls a 'frozen' distribution - it creates and returns an object with a mean of 2 and a standard deviation of 3. You can then use this object multiple times to get the probability of various values, like so:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "n23 = norm(2,3)\n",
      "print ('probability of 1.5 is %.4f' % n23.pdf(1.5))\n",
      "print ('probability of 2.5 is also %.4f' % n23.pdf(2.5))\n",
      "print ('whereas probability of 2 is %.4f' % n23.pdf(2))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "probability of 1.5 is 0.1311\n",
        "probability of 2.5 is also 0.1311\n",
        "whereas probability of 2 is 0.1330\n"
       ]
      }
     ],
     "prompt_number": 10
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "If we look at the documentation for `scipy.stats.norm` <a href=\"http://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.norm.html#scipy.stats.normfor\">here</a>[1] we see that there are many other functions that norm provides.\n",
      "\n",
      "For example, we can generate $n$ samples from the distribution with the `rvs()` function."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "print (n23.rvs(size=15))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "[ 0.65050631  1.46864336  1.32158495 -2.73465823 -0.13679278 -1.33790321\n",
        "  5.72887237  3.33521646  2.35884602 -1.37062805 -0.06171522  1.94129846\n",
        "  0.5054121   4.00091453  2.946641  ]\n"
       ]
      }
     ],
     "prompt_number": 11
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We can get the *cumulative distribution function (CDF)*, which is the probability that a randomly drawn value from the distribution is less than or equal to $x$."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# probability that a random value is less than the mean 2\n",
      "print (n23.cdf (2))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "0.5\n"
       ]
      }
     ],
     "prompt_number": 12
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We can get various properties of the distribution:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "print ('variance is', n23.var())\n",
      "print ('standard deviation is', n23.std())\n",
      "print ('mean is', n23.mean())"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "variance is 9.0\n",
        "standard deviation is 3.0\n",
        "mean is 2.0\n"
       ]
      }
     ],
     "prompt_number": 13
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "There are many other functions available, and if you are interested I urge you to peruse the documentation. I find the documentation to be excessively terse, but with a bit of googling you can find out what a function does and some examples of how to use it. Most of this functionality is not of immediate interest to the book, so I will leave the topic in your hands to explore. The tutorial is quite approachable, and I suggest starting there. <a href=\"http://docs.scipy.org/doc/scipy/reference/tutorial/stats.html\">(tutorial)</a>[2]"
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Summary and Key Points"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The following points **must** be understood by you before we continue:\n",
      "\n",
      "* Normal distributions occur throughout nature\n",
      "* They express a continuous probability distribution\n",
      "* They are completely described by two parameters: the mean ($\\mu$) and variance ($\\sigma^2$)\n",
      "* $\\mu$ is the average of all possible values\n",
      "* $\\sigma^2$ represents how much our measurements vary from the mean"
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "References"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "[1] http://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.norm.html#scipy.stats.normfor\n",
      "\n",
      "[2] http://docs.scipy.org/doc/scipy/reference/tutorial/stats.html"
     ]
    }
   ],
   "metadata": {}
  }
 ]
}