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     "source": [
      "Review of Probability Distributions: Bayesian Context"
     ]
    },
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     "metadata": {},
     "source": [
      "The material for this notebook is provided via this free online <a href='http://nbviewer.ipython.org/github/CamDavidsonPilon/Probabilistic-Programming-and-Bayesian-Methods-for-Hackers/blob/master/Chapter1_Introduction/Chapter1_Introduction.ipynb'>textbook</a> :  \n",
      "\n",
      "I am writing this notebook to document my learning, and hopefully to help others learn machine learning. You can think of it as my personal lecture notes. I would love suggestions / corrections / feedback for these notebooks.\n",
      "\n",
      "<b><a target=\"_parent\" href=\"http://rmdk.ca\">Visit my webpage for more</a></b>. \n",
      "\n",
      "Email me: <a href=\"mailto:email.ryan.kelly@gmail.com?Subject=Hey\" target=\"_top\">email.ryan.kelly@gmail.com</a>\n",
      "\n",
      "I'd love for you to share if you liked this post."
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Related Notebooks"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "* <a target=\"_parent\" href=\"http://nbviewer.ipython.org/github/rmdk/IPython-Notebooks/blob/master/Machine%20Learning/Bayesian%20Methods%20for%20Hackers.ipynb\">Intro to Bayesian Inference</a>\n",
      "\n",
      "* <a target=\"_parent\" href=\"http://nbviewer.ipython.org/github/RMDK/IPython-Notebooks/blob/master/Machine%20Learning/Probability%20Distribution%20Review.ipynb\">Probability Distribution Review</a>\n",
      "\n",
      "* <a target=\"_parent\" href=\"http://nbviewer.ipython.org/github/rmdk/IPython-Notebooks/blob/master/Machine%20Learning/Bayesian%202.ipynb\">Bayesian Inference 2</a>\n",
      "\n",
      "* <a target=\"_parent\" href=\"http://nbviewer.ipython.org/github/rmdk/IPython-Notebooks/blob/master/Machine%20Learning/Bayesian%203.ipynb\">Bayesian Inference 3</a>\n",
      "\n",
      "* <a target=\"_parent\" href=\"http://nbviewer.ipython.org/github/rmdk/IPython-Notebooks/blob/master/Machine%20Learning/Bayesian%20Inference%204.ipynb\">Bayesian Inference 4</a>\n",
      "\n",
      "* <a target=\"_parent\" href=\"http://rmdk.ca\">Bayesian Inference 6</a>"
     ]
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    {
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        "\n",
        "    <a style='float:left; margin-right:5px;' href=\"https://twitter.com/share\" class=\"twitter-share-button\" data-text=\"Check this out\" data-via=\"Ryanmdk\">Tweet</a>\n",
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        "    <a style='float:left; margin-right:5px;' href=\"https://twitter.com/Ryanmdk\" class=\"twitter-follow-button\" data-show-count=\"false\">Follow @Ryanmdk</a>\n",
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    {
     "cell_type": "heading",
     "level": 3,
     "metadata": {},
     "source": [
      "This notebook covers or includes:"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "* Probability Distribution Review"
     ]
    },
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Probability Distributions"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "This notebook is meant as a review to prepare a reader, and myself for more advanced Bayesian modeling techniques. \n",
      "\n",
      "Let's recall what a probability distribution is: Let $Z$ be some random variable. Then associated with $Z$ is a _probability distribution function_ that assigns probabilities to the different outcomes $Z$ can take. Graphically, a probability distribution is a curve where the probability of an outcome is proportional to the height of the curve. \n",
      "\n",
      "Random variables can be divided into three classifications:\n",
      "\n",
      "* $Z$ __is discrete__: Discrete random variables may only assume values on a specified list. Things like populations, movie ratings, and number of votes are discrete random variables.\n",
      "\n",
      "* $Z$ __is continuous__: Continuous random variable can take on arbitrarily exact values. For example, temperature, speed, time, and color are all modeled as continuous variables because you can progressively make the values more and more precise.\n",
      "\n",
      "* $Z$ __is mixed__: Mixed random variables assign probabilities to both discrete and continuous random variables as a combination of the two above categories."
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Discrete Case"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "If $Z$ is discrete, then is distribution is called a _probability mass function_, which measures the probability $Z$ on the value of $K$, denoted $P(Z=k)$. Note that the probability mass function completely describes the random variable $Z$, that is, if we know the mass function, we know how $Z$ should behave. There are popular probability mass functions that consistently appear: we will introduce them as needed. Let's introduce the first one: we say $Z$ is _Poisson_-distributed if\n",
      "\n",
      "$$P(Z=k)=\\frac{\\lambda^k e^{-k}}{k!},  \\ k=0,1,2,...$$\n",
      "\n",
      "$\\lambda$ is called a parameter of the distribution, and it controls the distribution's shape. For the Poisson distribution, $\\lambda$ can be any positive number. By increasing $\\lambda$, we add more probability to larger values, and conversely we can apply the opposite effect. Formally, $\\lambda$ describes the intensity of the Poisson distribution.\n",
      "\n",
      "Unlike $\\lambda$, which can be any positive number, the value $k$ in the formula must be a non-negative integer. This is important, since if you want to model a population you could not make sense of populations with 4.99 or 9.11 individuals.\n",
      "\n",
      "If a random variable $Z$ has a Poisson mass distribution, we denote it as: $Z$ ~ $Poi(\\lambda)$\n",
      "\n",
      "Once useful property of the Poisson distribution is that its expected value is equal to its parameter\n",
      "\n",
      "$$E[Z|\\lambda]=\\lambda$$\n",
      "\n",
      "This will come up often, so it is necessary to remember. Below is a plot of the probability mass distribution for different $\\lambda$ values. The first thing to notice is that by increasing $\\lambda$, we add more probability of lager values occurring. Secondly, though the graph ends at 15, the distributions do not, they assign positive probability to every non-negative integer."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import scipy.stats as stats\n",
      "import numpy as np\n",
      "\n",
      "\n",
      "%matplotlib inline\n",
      "from matplotlib import pyplot as plt\n",
      "\n",
      "import seaborn as sns\n",
      "# Some initial styling\n",
      "rc={'axes.labelsize': 16.0, 'font.size': 16, \n",
      "    'legend.fontsize': 16.0, 'axes.titlesize': 18,\n",
      "    'xtick.labelsize': 15.0, 'ytick.labelsize': 15.0}\n",
      "\n",
      "sns.set(context = 'poster', style = 'whitegrid', rc=rc)\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 31
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "a = np.arange(16)\n",
      "poi = stats.poisson\n",
      "lambda_ = [1.5, 4.25]\n",
      "colors = [\"#9b59b6\", \"#3498db\"]\n",
      "\n",
      "plt.bar(a, poi.pmf(a, lambda_[0]), color=colors[0],\n",
      "        label='$\\lambda = %.1f$' % lambda_[0], alpha=0.6)\n",
      "\n",
      "plt.bar(a, poi.pmf(a, lambda_[1]), color=colors[1],\n",
      "         label='$\\lambda = %.1f$' % lambda_[1], alpha=0.6)\n",
      "\n",
      "plt.legend()\n",
      "plt.xlabel('$k$')\n",
      "plt.title('Probability mass funtion of a Poisson random variable;\\n \\\n",
      "differing $\\lambda$ values')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
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lPOxnNRGxF/BoZv6qafE9lKdVPpsy7V3zg3DIzEURcR99dFPqyZ133tnnw3ck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\n",
       "text": [
        "<matplotlib.figure.Figure at 0x10e733e10>"
       ]
      }
     ],
     "prompt_number": 32
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Continuous Case"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Instead of a probability mass function, a continuous random variable has a _probability density function_. Don't be fooled, the density function and the mass function are very different creatures. An example of a continuous random variable is a random variable with _exponetial density_. The density function for an exponential random variable looks like this:\n",
      "\n",
      "$$f_Z(z|\\lambda) = \\lambda \\exp^{-\\lambda z},\\ \\ z >= 0$$\n",
      "\n",
      "Like a Poisson random variable, an exponential random variable can take on only non-negative values. But unlike a Poisson variable, the exponential can take on _any_ non-negative values, including non-negative integral values such as 4.103 or 5.12401. This property makes a poor choice for count data, which must be and integer, but a great choice for time data, temperature data (measured in Kelvins), or any other precise and _positive_ variable. The graph below shows two probability density functions with different $\\lambda$ values.\n",
      "\n",
      "When a random variable $Z$ has an exponential distribution with parameter $\\lambda$, we say $Z$ is _exponential_ and write $Z$ ~ Exp($\\lambda$).\n",
      "\n",
      "Given a specific $\\lambda$, the expect value of an exponential random variable is equal to the inverse of $\\lambda$, that is:\n",
      "\n",
      "$$E[Z|\\lambda] = \\frac{1}{\\lambda}$$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "a = np.linspace(0, 4, 100)\n",
      "expo = stats.expon\n",
      "lambda_ = [1, 0.5]\n",
      "\n",
      "for l, c in zip(lambda_, colors):\n",
      "    plt.plot(a, expo.pdf(a, scale=1. /l), color=c,\n",
      "             label='$\\lambda = %.1f$' % l,\n",
      "             lw=3)\n",
      "    plt.fill_between(a, expo.pdf(a, scale=1./l), color=c, \n",
      "                     alpha=0.33)\n",
      "plt.legend()\n",
      "plt.ylabel('PDF at $z$')\n",
      "plt.xlabel('$z$')\n",
      "plt.ylim(0, 1.2)\n",
      "plt.title('Probability density function of an Expoential random variable: \\n \\\n",
      "differing $\\lambda$')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 44,
       "text": [
        "<matplotlib.text.Text at 0x1115338d0>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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ly5Zx+umn06dP+Q/tbbmQJEmSGjFkyBDOOussZs6cydy5c0te/qxZszj//PO5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\n",
       "text": [
        "<matplotlib.figure.Figure at 0x1106d6b90>"
       ]
      }
     ],
     "prompt_number": 44
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "But what is $\\lambda$ ?"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "__This question is what motivates statistics__. In the real world, $\\lambda$ is hidden from us. We see only $Z$, and must go backwards to try and determine $\\lambda$. The problem is difficult because there is no one-toone mapping from $Z$ to $\\lambda$. Many different methods have been created to solve the problem of estimating $\\lambda$, but since it is never actually observed, no one can say for certain which method is best!\n",
      "\n",
      "Bayesian inference is concerned with _beliefs_ about what $\\lambda$ might be. Rather than try to guess exactly, we can only talk about what $\\lambda$ is likely to be by assigning a probability distribution to $\\lambda$.\n",
      "\n",
      "This might seem odd at first, after all, $\\lambda$ is fixed, but not necessarily random. How can we assign probabilities to values of a non-random variable. Recall that under Bayesian theory, we _can_ assign probabilities if we interpret them as beliefs, contrary to the frequentist theory."
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Example: Inferring behaviour from text messages"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Here we model a more interesting example, from data that records the rate at which a user sends and receives text messages.\n",
      "\n",
      "> The data are daily text message counts from a single user over time."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import pandas as pd\n",
      "count_data = pd.read_csv('https://raw.githubusercontent.com/CamDavidsonPilon/Probabilistic-Programming-and-Bayesian-Methods-for-Hackers/master/Chapter1_Introduction/data/txtdata.csv',\n",
      "    header=None, names=['msgs'])\n",
      "N = len(count_data)\n",
      "\n",
      "plt.bar(np.arange(N), count_data['msgs'], color=\"#9b59b6\", alpha=0.8)\n",
      "plt.xlabel('Time (days)')\n",
      "plt.ylabel('count of msgs received')\n",
      "plt.title('Did the texting habits change over time?')\n",
      "plt.xlim(0, n)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 87,
       "text": [
        "(0, 74)"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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k1C05OB34aES8FXhCNe2REfES4EjgzElu95mU3o7mtvz7cZU4nA+cDTwbOA84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\n3QkzOjm49N+vaTu90fJb0sahTs9eu7z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       "text": [
        "<matplotlib.figure.Figure at 0x113bc3d50>"
       ]
      }
     ],
     "prompt_number": 87
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    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Before we start modeling, what can you see just by looking at the chart above? Is there a change in behaviour going on here?\n",
      "\n",
      "How can we start to model this? As we have seen, a Poisson random variable is a very appropriate model for this type of _count_ data. Denoting day $i$'s test message count by $C_i$\n",
      "\n",
      "$$C_i \\sim Poisson(\\lambda)$$\n",
      "\n",
      "We are not sure what the value of $\\lambda$ parameter really is, however. Looking at the chart above it appears the rate may become higher later in the observation period (recall that higher $\\lambda$ assigns more probability to larger outcomes). \n",
      "\n",
      "How would we represent this mathematically? Let's assume that on some day during the observation period (call it $\\tau$), the parameter $\\lambda$ suddenly jumps to a higher value. We we really have two $\\lambda$ parameters: one for the period before $\\tau$, and one for the rest of the observation period. In the literature, a sudden transition would be called a _switchpoint_:\n",
      "\n",
      "\n",
      "$$\n",
      "\\lambda=\\begin{cases}\n",
      "\\ \\lambda _1  \\ \\ if  \\ t < \\tau \\\\\n",
      "\\ \\lambda _2  \\ \\ if \\ t < \\tau \\\\\n",
      "\\end{cases}\n",
      "$$\n",
      "\n",
      "if, in reality, no sudden change occurred and indeed $\\lambda _1 = \\lambda _2$, then the $\\lambda$s posterior distributions should look about equal.\n",
      "\n",
      "We are interesting in inferring the unknown $\\lambda$s. To use Bayesian inference, we need to assign prior probabilities to the different possible values of $\\lambda$. What we be good prior probability distributions for them? Recall that $\\lambda$ can be any positive number. As we saw earlier, the _expoential_ distribution provides a continuous density function for positive numbers, so it might be a good choice for modeling $\\lambda _i$. Yet, recall that the exponential distribution takes a parameter of its own, so we also need to include that variable in the model. The's call that parameter $\\alpha$.\n",
      "\n",
      "$$\n",
      "\\begin{align}\n",
      "\\lambda _1  \\ \\sim  \\ Exp(\\alpha) \\\\\n",
      "\\lambda _2 \\ \\sim \\ Exp(\\alpha) \\\\\n",
      "\\end{align}\n",
      "$$\n",
      "\n",
      "$\\alpha$ is called a _hyper-parameter_ or _parent variable_. It is a parameter that influences other parameters. Our initial guess at $\\alpha$ does not influence the model too strongly, so we have some flexibility in our choice. A good rule of thing is to set the exponential parameter equal to the inverse of the average count data. Since we are modeling $\\lambda$ using an exponential distribution, we can expect the value identity shown earlier to get:\n",
      "\n",
      "$$\\frac{1}{N}\\sum\\limits_{i=0}^n C_i \u2248 E[ \\ \u03bb \\ | \\ \u03b1]=\\frac{1}{\u03b1}$$\n",
      "\n",
      "An alternative, would be to have two prior: one for each $\\lambda$. Creating two exponential distributions with different $\\alpha$ values reflect our prior belief that the rate changed at some point during the observations.\n",
      "\n",
      "What about $\\tau$? Because of the noisiness in the data, it is hard to pick out a prior when $\\tau$ might have occurred. Instead, we can assign a _uniform prior belief_ to every possible day, equivalent to saying:\n",
      "\n",
      "$$\n",
      "\\begin{align}\n",
      "\\tau \\ \\sim \\ DiscreteUniform(1, 70) \\\\\n",
      "-> P(\\tau = k) = \\frac{1}{70} \\\\\n",
      "\\end{align}\n",
      "$$\n",
      "\n",
      "So after all this, what does our overall prior distribution for the unknown variables look like? Frankly _it doesn't matter_. What we should understand is that it's an ugly complicated mess, and it only gets uglier. Regardless, all we really care about is the posterior distribution.\n",
      "\n",
      "Next we will use PyMC, a python library for Bayesian analysis. Click the link below to follow along."
     ]
    },
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "<a target=\"_parent\" href=\"http://nbviewer.ipython.org/github/rmdk/IPython-Notebooks/blob/master/Machine%20Learning/Bayesian%202.ipynb\">Up Next: Bayesian Methods 2</a>\n",
      "\n",
      "<hr>"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from IPython.core.display import HTML\n",
      "\n",
      "\n",
      "def css_styling():\n",
      "    styles = open(\"/users/ryankelly/desktop/custom_notebook2.css\", \"r\").read()\n",
      "\n",
      "    return HTML(styles)\n",
      "css_styling()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
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      "def social():\n",
      "    code = \"\"\"\n",
      "    <a style='float:left; margin-right:5px;' href=\"https://twitter.com/share\" class=\"twitter-share-button\" data-text=\"Check this out\" data-via=\"Ryanmdk\">Tweet</a>\n",
      "<script>!function(d,s,id){var js,fjs=d.getElementsByTagName(s)[0],p=/^http:/.test(d.location)?'http':'https';if(!d.getElementById(id)){js=d.createElement(s);js.id=id;js.src=p+'://platform.twitter.com/widgets.js';fjs.parentNode.insertBefore(js,fjs);}}(document, 'script', 'twitter-wjs');</script>\n",
      "    <a style='float:left; margin-right:5px;' href=\"https://twitter.com/Ryanmdk\" class=\"twitter-follow-button\" data-show-count=\"false\">Follow @Ryanmdk</a>\n",
      "<script>!function(d,s,id){var js,fjs=d.getElementsByTagName(s)[0],p=/^http:/.test(d.location)?'http':'https';if(!d.getElementById(id)){js=d.createElement(s);js.id=id;js.src=p+'://platform.twitter.com/widgets.js';fjs.parentNode.insertBefore(js,fjs);}}(document, 'script', 'twitter-wjs');</script>\n",
      "    <a style='float:left; margin-right:5px;'target='_parent' href=\"http://www.reddit.com/submit\" onclick=\"window.location = 'http://www.reddit.com/submit?url=' + encodeURIComponent(window.location); return false\"> <img src=\"http://www.reddit.com/static/spreddit7.gif\" alt=\"submit to reddit\" border=\"0\" /> </a>\n",
      "<script src=\"//platform.linkedin.com/in.js\" type=\"text/javascript\">\n",
      "  lang: en_US\n",
      "</script>\n",
      "<script type=\"IN/Share\"></script>\n",
      "\"\"\"\n",
      "    return HTML(code)"
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  }
 ]
}