bmabey/Bayesian Update.ipynb

## Bayesian Update.ipynb
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      "# run this first!\n",
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     "input": [
      "# you will probably need to `pip install seaborn` before running this\n",
      "%matplotlib inline\n",
      "import math \n",
      "\n",
      "import numpy as np\n",
      "import numpy.random as rand\n",
      "import matplotlib.pyplot as plt\n",
      "import pandas as pd\n",
      "import scipy as sp\n",
      "import seaborn\n",
      "\n",
      "\n",
      "from IPython.html.widgets import interact, interactive, fixed"
     ],
     "language": "python",
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     "source": [
      "# Bayesian Updates\n",
      "\n",
      "All of us are Bayesians, whether we would like to admit it or not. Each of us subconsciously computes probabilities of different outcomes every day. At any given point in time, we have a belief (a *prior*) of how probable any given outcome is, though we may not be able to quantize it. If it occurs or not (our data), we update our belief accordingly (once again, subconsciously). Essentially, this updated belief is called our *posterior*.\n",
      "\n",
      "As mathematicians and statisticians, we would like to quantify this notion, and it all follows from Bayes Rule:\n",
      "\\begin{equation*}\n",
      "\\mathbb{P}(H | D) = \\frac{\\mathbb{P}(D | H)\\mathbb{P}(H)}{\\mathbb{P}(D)}\n",
      "\\end{equation*}\n",
      "\n",
      "Where $H$ is short for *hypothesis* and $D$ for *data*. $\\mathbb{P}(H)$ is our *prior* belief in our hypothesis and $\\mathbb{P}(H | D)$ is our *posterior* updated belief that has incorporated the data. $\\mathbb{P}(D | H)$ is the *likelihood* of that data occuring given the hypothesis while $\\mathbb{P}(D)$ is the probability of the data occuring regardless of the hypothesis."
     ]
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     "metadata": {},
     "source": [
      "### Beta-Binomial Model\n",
      "\n",
      "Let us suppose we are going to run a medical trial on a given treatment, where we consider each sample to be a Bernoulli random variable, i.e. success or failure, with parameter $\\theta$. Recall that the parameter of a Bernoulli distribution, $\\theta$, is the probability of success for a sample.  Note that the sum of $n$ Bernoulli($\\theta$) random variables is itself a random variable from a Binomial distribution with parameters $n$ and $\\theta$, the same parameter for probability of success. The frequentist approach is to suppose there is no prior knowledge about the chance of success of the treatment. However, we often have some prior knowledge about similar treatments which we can carry over, giving us a *prior* on our parameter $\\theta$. This prior essentially relates our belief about the true, unknown value of $\\theta$, including its expected value, variance, etc.\n",
      "\n",
      "Let us suppose that our prior is a beta distribution over $\\theta$ with parameters $\\alpha$ and $\\beta$, i.e. \n",
      "\\begin{equation*}\n",
      "\\mathbb{P}(\\theta) = \\frac{\\theta^{\\alpha - 1}(1 - \\theta)^{\\beta - 1}}{B(\\alpha,\\beta)}\n",
      "\\end{equation*}\n",
      "where $B(\\alpha,\\beta)$ is the Beta function, such that \n",
      "\\begin{equation*}\n",
      "\\int_0^1 \\theta^{\\alpha - 1}(1 - \\theta)^{\\beta - 1} d\\theta = B(\\alpha,\\beta) = \\frac{\\Gamma(\\alpha)\\Gamma(\\beta)}{\\Gamma(\\alpha +\\beta)}\n",
      "\\end{equation*}\n",
      "\n",
      " Given a sequence of successes or failures $\\mathbf{y} = y_{1},\\cdots,y_{n}$, where there are $\\sum_{i=1}^n y_i = n_{1}$ successes and $(n - \\sum_{i=1}^n y_i) = n_{2}$ failures, we can compute our *posterior* distribution of $\\theta$, given the data $\\mathbf{y}$, as follows:\n",
      "\n",
      "\\begin{align*}\n",
      "\\mathbb{P}(\\theta | \\mathbf{y}) & = \\frac{\\mathbb{P}(\\mathbf{y} | \\theta)\\mathbb{P}(\\theta)}{\\mathbb{P}(\\mathbf{y})} \\\\\n",
      "& = \\frac{\\mathbb{P}(\\mathbf{y} | \\theta)\\mathbb{P}(\\theta)}{\\int_{0}^{1} \\mathbb{P}(\\mathbf{y} | \\theta)\\mathbb{P}(\\theta) d\\theta} \\\\\n",
      "& = \\frac{{n_{1} + n_{2} \\choose n_{1}} \\theta^{n_{1}}(1-\\theta)^{n_{2}} \\frac{\\theta^{\\alpha - 1}(1 - \\theta)^{\\beta - 1}}{B(\\alpha,\\beta)}}{\\int_{0}^{1} {n_{1} + n_{2} \\choose n_{1}} \\theta^{n_{1}}(1-\\theta)^{n_{2}} \\frac{\\theta^{\\alpha - 1}(1 - \\theta)^{\\beta - 1}}{B(\\alpha,\\beta)} d\\theta} \\\\\n",
      "& = \\frac{ \\theta^{n_{1} + \\alpha - 1}(1-\\theta)^{n_{2} + \\beta - 1}}{\\int_{0}^{1} \\theta^{n_{1} + \\alpha - 1}(1-\\theta)^{n_{2} + \\beta - 1} d\\theta} \\\\\n",
      "& = \\frac{\\frac{1}{B(n_{1} + \\alpha - 1, n_{2} + \\beta - 1)}}{\\frac{1}{B(n_{1} + \\alpha - 1, n_{2} + \\beta - 1)}} \\frac{ \\theta^{n_{1} + \\alpha - 1}(1-\\theta)^{n_{2} + \\beta - 1}}{\\int_{0}^{1} \\theta^{n_{1} + \\alpha - 1}(1-\\theta)^{n_{2} + \\beta - 1} d\\theta} \\\\\n",
      "& = \\frac{ \\theta^{n_{1} + \\alpha - 1}(1-\\theta)^{n_{2} + \\beta - 1}}{B(n_{1} + \\alpha - 1, n_{2} + \\beta - 1)}\n",
      "\\end{align*}\n",
      "\n",
      "This is simply the density of the Beta distribution over $\\theta$ with parameters $\\alpha + n_{1} - 1$ and $\\beta + n_{2} - 1$. Notice that both the prior and posterior are Beta distributions. When the prior and posterior distributions are from the same family, we call the prior \"conjugate\" to the distribution of the data. \n",
      "\n",
      "Python has the Beta density built in. Given parameters $\\alpha$ and $\\beta$, we can evaluate and plot the density for many values of $\\theta$:"
     ]
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     "input": [
      "#Evaluate this code cell to get an interactive plot of beta!\n",
      "def gen_plot(success=8, failure=5):\n",
      "    alpha = success\n",
      "    beta = failure\n",
      "    fig = plt.figure(figsize=(8, 6))\n",
      "    x = np.linspace(0, 1, 100)\n",
      "    ax = fig.add_subplot(111, xlabel='Chance of success (of the treatment)', \n",
      "                         ylabel='Probability of hypothesis', \n",
      "                         title=r'Posterior probability distribution of $\\theta$')\n",
      "    ax.plot(x, sp.stats.beta(alpha, beta).pdf(x), linewidth=3.)\n",
      "    ax.set_xticklabels(['0%', '20%', '40%', '60%', '80%', '100%']);\n",
      "\n",
      "\n",
      "interactive(gen_plot, success=(0,100), failure=(0,100))"
     ],
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VRRTnZ3oaj9yYtNQUFrvWP1C3fvIZM+Fba78CVFhr3xG862Zr7T9HNywR8doO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\n",
       "text": [
        "<matplotlib.figure.Figure at 0x115f52810>"
       ]
      }
     ],
     "prompt_number": 28
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "For now, let us suppose that we can assume a true value for one of the two parameters. Supposing first that we know the true variance $\\sigma^{2}$ for the distribution of scores, the posterior distribution of $\\mu$ given the data $\\mathbf{y}$ is\n",
      "$$\\mu | \\mathbf{y} \\sim N\\left(\\frac{\\frac{\\mu_{0}\\sigma^{2}}{n} + \\overline{\\mathbf{y}}\\sigma_{0}^{2}}{\\frac{\\sigma^{2}}{n} + \\sigma_{0}^{2}}, \\left(\\frac{n}{\\sigma^{2}} + \\frac{1}{\\sigma_{0}^{2}}\\right)^{-1}\\right),$$\n",
      "where $\\overline{\\mathbf{y}}$ is the average of the data points in $\\mathbf{y}$. Now, if we know the true mean $\\mu$ for the score distribution, the posterior distribution of $\\sigma^{2}$ given the data $\\mathbf{y}$ is $$\\sigma^{2} | \\mathbf{y} \\sim IG \\left(\\alpha + \\frac{n}{2}, \\beta + \\frac{\\sum_{i=1}^{n} (y_{i} - \\mu)^{2}}{2}\\right).$$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "<div class='problem'>\n",
      "<legend>Problem 1</legend>\n",
      "Lets translate some of this math into code! Complete the function `bernoulli_posterior`.  This function accepts a data set of i.i.d. Bernoulli random variables, `data`, and parameters $\\alpha$ and $\\beta$ (for a beta prior over $\\theta$) and returns the values of the distribtuion at the optionaly provided `x` range.\n",
      "\n",
      "We will be testing this function with the data in the file *trial.csv*:\n",
      "</div>"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "trial = np.fromfile('data/trial.csv', dtype=int, sep='\\n')\n",
      "trial"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 25,
       "text": [
        "array([0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1])"
       ]
      }
     ],
     "prompt_number": 25
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def bernoulli_posterior(data, alpha, beta, x=np.arange(0, 1.01, step=.01)):\n",
      "    # CODE HERE\n",
      "    # y = ....\n",
      "    return x, y"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 63
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Run the following code cell to test your code:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "## Test for Problem 1\n",
      "from numpy.testing import assert_array_almost_equal\n",
      "\n",
      "\n",
      "small_x = np.array([0., 0.4, 0.8])\n",
      "x_res, y_res = bernoulli_posterior(trial, 5, 10, x=small_x)\n",
      "assert_array_almost_equal(x_res, small_x)\n",
      "assert_array_almost_equal(y_res, np.array([0., 4.11392265e+00, 5.87174591e-04]))\n",
      "\n",
      "def sol(alpha, beta):\n",
      "    return np.fromfile('data/solutions/bernoulli_posterior_%d_%d.csv' % (alpha, beta),sep='\\n')\n",
      "\n",
      "assert_array_almost_equal(bernoulli_posterior(trial, 5, 10)[1], sol(5, 10))\n",
      "assert_array_almost_equal(bernoulli_posterior(trial, 20, 34)[1], sol(20, 34))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 83
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Using the completed `bernoulli_posterior` function we can plot the posterior and play around with the prior parameters, $\\alpha$ and $\\beta$.\n",
      "\n",
      "How does your belief about the treatment success rate ($\\theta$) change?"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "#Evaluate this code cell to get an interactive plot of beta!\n",
      "def plot_bernoulli_posterior(alpha=5, beta=10, data=trial):\n",
      "    fig = plt.figure(figsize=(8, 6))\n",
      "    x, y = bernoulli_posterior(data, alpha, beta, x=np.linspace(0, 1, 100))\n",
      "    ax = fig.add_subplot(111, xlabel='Chance of success (of the treatment)', \n",
      "                         ylabel='Probability of hypothesis', \n",
      "                         title=r'Posterior probability distribution of $\\theta$')\n",
      "    ax.plot(x, y, linewidth=3.)\n",
      "    ax.set_xticklabels(['0%', '20%', '40%', '60%', '80%', '100%']);\n",
      "\n",
      "actual_data = trial\n",
      "interactive(plot_bernoulli_posterior, alpha=(0,100), beta=(0,100), data=fixed(trial))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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58oxm9lGSqVQZo5FKUMho9huAG8ysHXgF8CMz6wMuA77s7hp9ISJVoVaa2dta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\nIiIiIiIiIiIiIhH/P9ci9pYSzXrIAAAAAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x115e4b090>"
       ]
      }
     ],
     "prompt_number": 84
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "In addition to the $\\alpha$ and $\\beta$ parameters now try adjusting the counts of the actual successes and failures.  What you'll notice is that the prior parameters and the actual counts have the same effect. This is why the $\\alpha$ and $\\beta$ are called *pseudocounts*."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def plot_bernoulli_posterior_with_actual_counts(alpha=5, beta=5, actual_successes=5, actual_failures=5):\n",
      "    data = np.concatenate([np.ones(actual_successes), np.zeros(actual_failures)])\n",
      "    plot_bernoulli_posterior(alpha,beta,data)\n",
      "    \n",
      "interactive(plot_bernoulli_posterior_with_actual_counts, \n",
      "            alpha=(0,100), beta=(0,100), \n",
      "            actual_successes=(0,100), actual_failures=(0,100))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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DsRb884HnAtsBn4w8PkuQ4EVEGl50DH6gnhK81sLLEhVM8O5+JoCZvcndf1S9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\n",
       "text": [
        "<matplotlib.figure.Figure at 0x115fdb090>"
       ]
      }
     ],
     "prompt_number": 85
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "### Extending the model to the Multinomial\n",
      "\n",
      "Let's now suppose that we run a fruit stand, selling apples, bananas, mangos, oranges, and watermelons. We assume that when a person comes and buys fruit, he selects one piece of fruit. This can be modeled as a draw from a categorical distribution with parameter $\\theta$. The Categorical distribution is a discrete probability distribution over the $k$ possible events, the multiple-event version of the Bernoulli distribution.  The combination/mutual occurence of $n$ Categorical variables  is called a Multinomial variable with probability mass function:\n",
      "\n",
      "\\begin{align}\n",
      "f(x_1,\\ldots,x_k;n,p_1,\\ldots,p_k) & {} = \\Pr(X_1 = x_1\\mbox{ and }\\dots\\mbox{ and }X_k = x_k) \\\\  \\\\\n",
      "& {} = \\begin{cases} { \\displaystyle {n! \\over x_1!\\cdots x_k!}p_1^{x_1}\\cdots p_k^{x_k}}, \\quad &\n",
      "\\mbox{when } \\sum_{i=1}^k x_i=n \\\\  \\\\\n",
      "0 & \\mbox{otherwise,} \\end{cases}\n",
      "\\end{align}\n",
      "\n",
      "for non-negative integers $x_1, ..., x_k$.\n",
      "\n",
      "The probability mass function can also be expressed using the [gamma function](https://en.wikipedia.org/wiki/Gamma_function) as:\n",
      "\n",
      "$f(x_1,\\dots, x_{k}; p_1,\\ldots, p_k) = \\frac{\\Gamma(\\sum_i x_i + 1)}{\\prod_i \\Gamma(x_i+1)} \\prod_{i=1}^k p_i^{x_i}.$\n",
      "\n",
      "Thus, in our five kinds of fruit example $\\theta$ is a vector of length five, with non-negative entries summing to $1$, each entry being the probability of selecting the corresponding fruit.\n",
      "\n",
      "If we know $\\theta$ we can use `numpy.random.choice` or `numpy.random.multinomial` to draw samples:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "fruits_pmf = {'Apple': 0.2, 'Banana': 0.3, 'Mango': 0.1, 'Orange': 0.3, 'Watermelon': 0.1}\n",
      "fruits = fruits_pmf.keys()\n",
      "theta = fruits_pmf.values()\n",
      "\n",
      "categorical_draw = rand.choice(fruits, 1, p=theta)[0]\n",
      "print('A single categorical draw: ' + categorical_draw)\n",
      "\n",
      "many_draws = dict(zip(fruits, rand.multinomial(20, theta)))\n",
      "print('20 draws: %s' % many_draws)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "A single categorical draw: Mango\n",
        "20 draws: {'Orange': 5, 'Mango': 2, 'Watermelon': 1, 'Apple': 4, 'Banana': 8}\n"
       ]
      }
     ],
     "prompt_number": 95
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "In the case we don't know what $\\theta$ is, but we have some prior belief of the populace's fruit preferences, which we represent as a [Dirichlet](https://en.wikipedia.org/wiki/Dirichlet_distribution) prior over $\\theta$ with parameter vector $\\alpha$, that is, $$\\mathbb{P}(\\theta) = \\frac{1}{B(\\alpha)} \\prod_{i=1}^{5} \\theta_{i}^{\\alpha_{i} - 1}$$ where $B(\\alpha)$ is the multinomial Beta function, $$B(\\alpha) = \\frac{\\prod_{i=1}^{5} \\Gamma(\\alpha_{i})}{\\Gamma(\\sum_{i=1}^{5} \\alpha_{i})}.$$ Given a sequence of fruit selections $\\mathbf{y} = y_{1}, \\cdots, y_{n}$ from $n$ people, where each $$y_{i} \\in \\{\\text{Apple, Banana, Mango, Orange, Watermelon}\\},$$ we let\n",
      "\\begin{align*}\n",
      "n_{1} & = |\\{1 \\leq i \\leq n | y_{i} = \\text{Apple}\\}| \\\\\n",
      "n_{2} & = |\\{1 \\leq i \\leq n | y_{i} = \\text{Banana}\\}| \\\\\n",
      "\\vdots & = \\; \\; \\; \\; \\; \\; \\; \\; \\; \\; \\; \\; \\; \\; \\; \\; \\; \\; \\vdots \\\\\n",
      "n_{5} & = |\\{1 \\leq i \\leq n | y_{i} = \\text{Watermelon}\\}|\n",
      "\\end{align*}\n",
      "\n",
      "Given this new data of people's purchases, we can compute our posterior distribution of $\\theta$ as follows:\n",
      "\\begin{align*}\n",
      "\\mathbb{P}(\\theta | \\mathbf{y}) & = \\frac{\\mathbb{P}(\\mathbf{y} | \\theta) \\mathbb{P}(\\theta)}{\\int \\mathbb{P}(\\mathbf{y} | \\theta) \\mathbb{P}(\\theta) d\\theta} \\\\\n",
      "& = \\frac{\\prod_{i=1}^{5} \\theta_{i}^{n_{i}} \\frac{1}{B(\\alpha)} \\prod_{i=1}^{5} \\theta_{i}^{\\alpha_{i}-1}}{\\int \\cdots \\int \\prod_{i=1}^{5} \\theta_{i}^{n_{i}} \\frac{1}{B(\\alpha)} \\prod_{i=1}^{5} \\theta_{i}^{\\alpha_{i}-1} d\\theta_{1}\\cdots d\\theta_{5}} \\\\\n",
      "& = \\frac{\\prod_{i=1}^{5} \\theta_{i}^{n_{i} + \\alpha_{i} - 1}}{\\int \\cdots \\int \\prod_{i=1}^{5} \\theta_{i}^{n_{i} + \\alpha_{i} - 1} d\\theta_{1} \\cdots d\\theta_{5}} \\\\\n",
      "& = \\frac{\\frac{1}{B(n + \\alpha)}}{\\frac{1}{B(n+\\alpha)}} \\frac{\\prod_{i=1}^{5} \\theta_{i}^{n_{i} + \\alpha_{i} - 1}}{\\int \\cdots \\int \\prod_{i=1}^{5} \\theta_{i}^{n_{i} + \\alpha_{i} - 1} d\\theta_{1} \\cdots d\\theta_{5}} \\\\\n",
      "& = \\frac{\\prod_{i=1}^{5} \\theta_{i}^{n_{i} + \\alpha_{i} - 1}}{B(n + \\alpha)}\n",
      "\\end{align*}\n",
      "where $$n = \\left( \\begin{array}{ccccc} n_{1} & n_{2} & n_{3} & n_{4} & n_{5} \\end{array} \\right).$$\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "<div class=\"problem\">\n",
      "<legend>Problem 2</legend>\n",
      "Write a multinomial Beta function that accepts a vector of nonnegative reals of arbitrary length. For reference the defintion of the multinomial beta is: $$B(\\alpha) = \\frac{\\prod_{i=1}^{n} \\Gamma(\\alpha_{i})}{\\Gamma(\\sum_{i=1}^{n} \\alpha_{i})}.$$\n",
      "\n",
      "\n",
      "<p/>\n",
      "*Hint: You can use `math.gamma`*\n",
      "</div>"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def multinomial_beta(alpha):\n",
      "    # CODE HERE\n",
      "    # return ...\n",
      "\n",
      "alpha = [0.5, 0.5]\n",
      "multinomial_beta(alpha) "
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 140,
       "text": [
        "3.1415926535897927"
       ]
      }
     ],
     "prompt_number": 140
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Verify your solution by running this test:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "## Test for Problem 2\n",
      "from numpy.testing import assert_almost_equal\n",
      "\n",
      "assert_almost_equal(multinomial_beta([0.5, 0.5]), math.pi)\n",
      "assert_almost_equal(multinomial_beta([0.3, 0.1, 0.1, 0.2, 0.3]), 3718.5352344)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 268
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "<div class=\"problem\">\n",
      "<legend>Problem 3</legend>\n",
      "Write a function that accepts a data set of i.i.d. draws from a Categorical($\\theta$) distribution, a parameter vector $\\alpha$ for the Dirichlet prior over $\\theta$, and a set of unique categories (in our example, fruits). Have this function return the Dirichlet parameter for the posterior distribution of $\\theta$. \n",
      "</div>"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "fruit = pd.read_csv('data/fruit.csv', header=None)[0].values\n",
      "fruit"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 176,
       "text": [
        "array(['Mango', 'Apple', 'Mango', 'Watermelon', 'Apple', 'Orange', 'Apple',\n",
        "       'Banana', 'Apple', 'Orange', 'Mango', 'Mango', 'Watermelon',\n",
        "       'Orange', 'Apple', 'Apple', 'Mango', 'Apple', 'Mango', 'Banana',\n",
        "       'Mango', 'Mango', 'Apple', 'Orange', 'Apple', 'Apple', 'Banana',\n",
        "       'Apple', 'Mango', 'Orange', 'Mango', 'Apple', 'Mango', 'Mango',\n",
        "       'Watermelon', 'Orange', 'Apple', 'Mango', 'Mango', 'Apple'], dtype=object)"
       ]
      }
     ],
     "prompt_number": 176
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def multinomial_posterior(data, prior):\n",
      "    # CODE HERE\n",
      "    # return ...\n",
      "\n",
      "def counts_dict_to_vector(counts):\n",
      "    return pd.Series(counts).sort_index().values\n",
      "\n",
      "prior_counts = {'Apple': 5, 'Banana': 3, 'Mango': 4, 'Orange': 2, 'Watermelon': 1}\n",
      "alpha =  counts_dict_to_vector(prior_counts)\n",
      "multinomial_posterior(fruit, alpha)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 225,
       "text": [
        "array([19,  6, 18,  8,  4])"
       ]
      }
     ],
     "prompt_number": 225
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "## Test for Problem 3\n",
      "from numpy.testing import assert_array_almost_equal\n",
      "\n",
      "assert_array_almost_equal(multinomial_posterior(['Foo','Foo','Bar'], \n",
      "                                                counts_dict_to_vector({'Bar': 5, 'Foo': 2})),\n",
      "                            [6, 4])\n",
      "        \n",
      "\n",
      "fruit = pd.read_csv('data/fruit.csv', header=None)[0].values\n",
      "prior_counts = {'Apple': 5, 'Banana': 3, 'Mango': 4, 'Orange': 2, 'Watermelon': 1}\n",
      "\n",
      "\n",
      "assert_array_almost_equal(multinomial_posterior(fruit, \n",
      "                                                counts_dict_to_vector(prior_counts)),\n",
      "                            [19,  6, 18,  8,  4])\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 228
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Now suppose that we have a set of exam scores $\\mathbf{y} = y_{1}, \\cdots, y_{n}$ from the university's calculus final. Historically, we know how people have scored on the exam, giving us information to make a prior on the distribution of scores. We also know that the scores should be Normally distributed, $X_i \\sim N(\\mu,\\sigma^2)$, $i=1,...,n$. Here we are encountering something new: a prior for a distribution with two parameters instead of one. What we do is place a prior on each parameter. We let $\\mu \\sim N(\\mu_{0}, \\sigma_{0}^{2})$ and $\\sigma^{2} \\sim IG(\\alpha, \\beta)$ be priors for our mean and variance, respectively, where $IG$ is the [Inverse Gamma](https://en.wikipedia.org/wiki/Inverse-gamma_distribution) distribution, with probability density function:\n",
      "\n",
      "$$\\mathbb{P}(\\sigma^{2}) = \\frac{\\beta^{\\alpha}}{\\Gamma(\\alpha)}(\\sigma^{2})^{-\\alpha - 1}e^{-\\frac{\\beta}{\\sigma^{2}}}\\text{; } \\alpha,\\beta,\\sigma^2 >0.$$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "<div class='problem'>\n",
      "<legend>Problem 4</legend>\n",
      "Write your own function to compute an Inverse Gamma probability density function given parameters $\\alpha$ and $\\beta$. It should be able to compute the density at an array of values. Call it *invgammapdf*.\n",
      "<p/>\n",
      "Your function should handle the case where $x$ is a scalar or an array. The function may be useful `np.isscalar`.\n",
      "</div>"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def invgammapdf(x, alpha, beta):\n",
      "    # CODE HERE\n",
      "    # return ..."
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 232
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "invgammapdf(5, 10, 20)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 233,
       "text": [
        "0.010584953352840239"
       ]
      }
     ],
     "prompt_number": 233
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Writing `invgammapdf` is to provide practice writing these type of functions. Normally, you would just use [scipy.stats.invgamma](http://docs.scipy.org/doc/scipy-0.14.0/reference/generated/scipy.stats.invgamma.html). This is what is used to test your version below:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "## Test for Problem 4\n",
      "from numpy.testing import assert_array_almost_equal, assert_almost_equal\n",
      "\n",
      "\n",
      "assert_almost_equal(invgammapdf(5, 10, 20), stats.invgamma.pdf(5, 10, scale=20))\n",
      "\n",
      "x = rand.randint(0,1000,50)\n",
      "assert_array_almost_equal(invgammapdf(x, 33, 5), stats.invgamma.pdf(x, 33, scale=5))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 246
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Let's look at how to plot each of these prior distributions:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from scipy.stats import norm\n",
      "def plot_priors(mu_0=74., sigma2_0 = 25., alpha=2., beta = 25.):\n",
      "    x = np.arange(0, 100.1, step=0.1)\n",
      "    y = norm.pdf(x, loc=mu_0, scale=5)\n",
      "    plt.plot(x, y)\n",
      "    plt.title('$\\mu \\sim N(%d, %d)$' % (mu_0, sigma2_0))\n",
      "    plt.show()\n",
      "\n",
      "    x = np.arange(0.1, 100.1, step=0.1)\n",
      "    y = invgammapdf(x, alpha=alpha, beta=beta)\n",
      "    plt.title('$\\sigma^2 \\sim IG(%d, %d)$' % (alpha,beta))\n",
      "    plt.plot(x, y)\n",
      "    plt.show()\n",
      "    \n",
      "\n",
      "plot_priors()\n",
      "#interactive(plot_priors, mu_0=(0,100), sigma2_0=(1,40), alpha=(0,30), beta=(10,40))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x11837df10>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0x11850cdd0>"
       ]
      }
     ],
     "prompt_number": 267
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "For now, let us suppose that we can assume a true value for one of the two parameters. Supposing first that we know the true variance $\\sigma^{2}$ for the distribution of scores, the posterior distribution of $\\mu$ given the data $\\mathbf{y}$ is\n",
      "$$\\mu | \\mathbf{y} \\sim N\\left(\\frac{\\frac{\\mu_{0}\\sigma^{2}}{n} + \\overline{\\mathbf{y}}\\sigma_{0}^{2}}{\\frac{\\sigma^{2}}{n} + \\sigma_{0}^{2}}, \\left(\\frac{n}{\\sigma^{2}} + \\frac{1}{\\sigma_{0}^{2}}\\right)^{-1}\\right),$$\n",
      "where $\\overline{\\mathbf{y}}$ is the average of the data points in $\\mathbf{y}$. "
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "<div class='problem'>\n",
      "<legend>Problem 5</legend>\n",
      "Write a function that accepts a data set of real numbers, a prior ($\\mu_{0}$ and $\\sigma_{0}^{2}$) for the mean $\\mu$ of a normal distribution, and the true variance $\\sigma^{2}$ of the distribution, and returns the updated posterior parameters for $\\mu$. Test it on the data in the file *examscores.csv* with prior $\\mu_{0} = 74, \\sigma_{0}^{2} = 25$ and true variance $\\sigma^{2} = 36$.\n",
      "</div>"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "df = pd.read_csv('data/examscores.csv', header=None, names=['score'])\n",
      "examscores = df['score'].values\n",
      "#print(examscores)\n",
      "#df.describe()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 285
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def mean_posterior(data, prior, variance):\n",
      "    mu_0, sigma2_0 = prior\n",
      "    # CODE HERE\n",
      "    # posterior_mean = ...\n",
      "    # posterior_variance = ...\n",
      "    return posterior_mean, posterior_variance\n",
      "\n",
      "mean_posterior(examscores, (74, 25), 36)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 307,
       "text": [
        "(67.447837150127228, 1.1450381679389312)"
       ]
      }
     ],
     "prompt_number": 307
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Test for Problem 5\n",
      "assert_array_almost_equal(mean_posterior(examscores, (74, 25), 36), (67.4478371, 1.1450381))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 292
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Lets use your `mean_posterior` function to plot the posterier and see how it changes as you adjust the priors:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "print('The mean and std of the data is: %s, %s' % (examscores.mean(), examscores.std()))\n"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The mean and std of the data is: 67.1333333333, 15.0615035401\n"
       ]
      }
     ],
     "prompt_number": 304
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def plot_mean_variance(data, mu_0=74, sigma2_0=25, true_variance=36):\n",
      "    posterior_mean, posterior_variance = mean_posterior(data, (mu_0, sigma2_0), true_variance)\n",
      "    x = np.arange(0, 100.1, step=0.1)\n",
      "    y = norm.pdf(x, loc=posterior_mean, scale=np.sqrt(posterior_variance))\n",
      "    plt.plot(x, y)\n",
      "    plt.show()\n",
      "    \n",
      "interactive(plot_mean_variance, data=fixed(examscores), mu_0=(0,99), sigma2_0=(1,50), true_variance=(5,50))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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60/846VVfsY1FftXSa/4msqt67gP+mvT18gl3j2PdahERERERERERERERERER\nERERERERERERERGRneb/Aw2Y+1ur4Mh5AAAAAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x11800eb10>"
       ]
      }
     ],
     "prompt_number": 306
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "\n",
      "Now, if we know the true mean $\\mu$ for the score distribution, the posterior distribution of $\\sigma^{2}$ given the data $\\mathbf{y}$ is $$\\sigma^{2} | \\mathbf{y} \\sim IG \\left(\\alpha + \\frac{n}{2}, \\beta + \\frac{\\sum_{i=1}^{n} (y_{i} - \\mu)^{2}}{2}\\right).$$\n"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "<div class='problem'>\n",
      "<legend>Problem 6</legend>\n",
      "Write a function that accepts a dataset of real numbers, a prior ($\\alpha$ and $\\beta$) for the variance $\\sigma^{2}$ of a normal distribution, and the true mean $\\mu$ of the distribution, and returns the updated posterior parameters for $\\sigma^{2}$. Test it on the data in the file *examscores.csv* with prior $\\alpha=2, \\beta = 25$ and true mean $\\mu = 62$.\n",
      "</div>"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def variance_posterior(data, prior, mean):\n",
      "    alpha, beta = prior\n",
      "    # CODE HERE\n",
      "    # posterior_alpha = ...\n",
      "    # posterior_beta = ...\n",
      "    return posterior_alpha, posterior_beta\n",
      "\n",
      "variance_posterior(examscores, (2., 25.), 62)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 308,
       "text": [
        "(17.0, 3823.0)"
       ]
      }
     ],
     "prompt_number": 308
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# Test for Problem 5\n",
      "assert_array_almost_equal(variance_posterior(examscores, (2, 25), 62), (17, 3823))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 310
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "For the final part of this lab write the code needed to plot the variance."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# code goes here to plot variance_posterior"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 311
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "\n",
      "What if we can't assume a true value for either of the parameters? This is where Gibbs sampling comes in as you will see in the next lab. :)"
     ]
    }
   ],
   "metadata": {}
  }
 ]
}