ml-hw2

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  {
   "cells": [
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Assignment 2 - Machine Learning"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Miguel Vil\u00e1 Gonz\u00e1lez\n",
      "\n",
      "Juan Carlos Espinosa Moreno"
     ]
    },
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "1. Parametric classification"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "(a) First we load the dataset and shuffle it to avoid creating a biased training set (for example all elements from the $2$ class  are at the end of the file). After this we plot the dataset by class:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import math\n",
      "import pylab as pl\n",
      "import numpy as np\n",
      "from random import shuffle\n",
      "from itertools import product\n",
      "\n",
      "f = open('data.txt','r')\n",
      "lines = f.readlines()\n",
      "data_set = [ [ float(element.strip()) for element in line.split(',')] for line in lines ]\n",
      "#Shuffle the dataset to avoid creating a biased training set\n",
      "shuffle(data_set)\n",
      "f.close()\n",
      "\n",
      "X = [ arr[0:2] for arr in data_set ]\n",
      "Y = [ arr[2]   for arr in data_set ]\n",
      "\n",
      "n = int(0.8*len(X))\n",
      "\n",
      "X_train, Y_train = X[0:n] , Y[0:n]\n",
      "X_test, Y_test = X[n:], Y[n:]\n",
      "\n",
      "classes = xrange(len(set(Y))) #Counts the distinct number of classes\n",
      "\n",
      "def get_features_from_class(c,X,Y):\n",
      "    \"\"\"Extracts the features for elements with class 'c'\"\"\"\n",
      "    if len(X)!=len(Y):\n",
      "        raise Exception(\"'X' and 'Y' must have the same length\")\n",
      "    indices = [i for i in xrange(len(Y)) if Y[i]==c]\n",
      "    return [X[i] for i in indices]\n",
      "\n",
      "pl.clf()\n",
      "def plot_class(X,class_symbol,c):\n",
      "    pl.plot([x[0] for x in X],[x[1] for x in X], class_symbol, label=(\"Class \"+str(c) if c!=len(classes) else \"Rejected\"))\n",
      "    \n",
      "X_class0 = get_features_from_class(0,X,Y)\n",
      "plot_class(X_class0, 'ro',0)\n",
      "\n",
      "X_class1 = get_features_from_class(1,X,Y)\n",
      "plot_class(X_class1, 'go',1)\n",
      "\n",
      "X_class2 = get_features_from_class(2,X,Y)\n",
      "plot_class(X_class2, 'bo',2)\n",
      "pl.legend()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 64,
       "text": [
        "<matplotlib.legend.Legend at 0xa7fa18c>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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BQDx3W6AbObWArjRuWjZ5MoNLSxUBeBHwusq+i+LjCWluZuulS/7jTJrEYKdT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\nEwRBWID/B2I2ypMUN6v6AAAAAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0xa69b8ec>"
       ]
      }
     ],
     "prompt_number": 64
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Then we create an estimator for the whole training set and one for each class. We  create a function that encodes the maximum-likelihood estimator given samples in an array $x$:\n",
      "$$\n",
      "f(x) = \\frac{1}{\\sqrt{(2\\pi)^k|\\Sigma |}}\\cdot exp\\left(-\\frac{1}{2}(x-\\mu)^t\\Sigma^{-1}(x-\\mu)\\right)\n",
      "$$\n",
      "Here $\\mu$ is the mean vector, $\\Sigma$ is the covariance matrix and $|\\Sigma |$ the determinant of the covariance (taken from [Wikipedia](http://en.wikipedia.org/wiki/Multivariate_normal_distribution#Estimation_of_parameters))"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "class Multivariate_Normal_Estimator:\n",
      "    \n",
      "    def __init__(self, X):\n",
      "        self.k = X.shape[1]\n",
      "        mean = np.mean(X,0)\n",
      "        self.mean = mean.reshape((1,len(mean)))\n",
      "        self.cov = np.matrix(np.cov(X.T))\n",
      "        self.cov_inv = self.cov.I\n",
      "        self.det = np.linalg.det(self.cov)\n",
      "        self.normalizer = 1.0 / np.sqrt(np.power(2*np.pi,self.k)*self.det)\n",
      "        \n",
      "    def estimate(self,x):\n",
      "        if len(x)!=self.k:\n",
      "            raise Exception(\"Incompatible array: it must have length:\"+str(self.k))\n",
      "        diff = (x-self.mean)\n",
      "        return self.normalizer*math.pow(math.e, -0.5* (diff * self.cov_inv * diff.T))\n",
      "    \n",
      "    def __repr__(self):\n",
      "        return \"(1/sqrt(2*pi^\"+str(self.k)+\"*\"+str(self.det)+\"))*exp(-0.5*(x-\"+str(self.mean)+\")^t*\"+str(self.cov_inv)+\"*(x-\"+str(self.mean)+\"))\"\n",
      "\n",
      "class_estimators = []\n",
      "#The classes are sorted thus the array index matches the class\n",
      "\n",
      "X_of_class = [np.array(get_features_from_class(i,X_train,Y_train)) for i in classes]\n",
      "for i in classes:\n",
      "    X_i = X_of_class[i]\n",
      "    X_i_estimator = Multivariate_Normal_Estimator(X_i)\n",
      "    class_estimators.append(X_i_estimator)\n",
      "    print \"Discriminant function for class \"+str(i)+\": \"+str(X_i_estimator)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "Discriminant function for class 0: (1/sqrt(2*pi^2*0.688981859757))*exp(-0.5*(x-[[ 3.84079436  5.17519078]])^t*[[ 2.24660503 -0.66254263]\n",
        " [-0.66254263  0.8414384 ]]*(x-[[ 3.84079436  5.17519078]]))\n",
        "Discriminant function for class 1: (1/sqrt(2*pi^2*1.04509721656))*exp(-0.5*(x-[[ 9.16240144  5.9441285 ]])^t*[[ 1.02632396  0.80456602]\n",
        " [ 0.80456602  1.56303013]]*(x-[[ 9.16240144  5.9441285 ]]))\n",
        "Discriminant function for class 2: (1/sqrt(2*pi^2*1.68427807441))*exp(-0.5*(x-[[ 7.03967513  3.63726588]])^t*[[ 0.70491909 -0.02088859]\n",
        " [-0.02088859  0.84288045]]*(x-[[ 7.03967513  3.63726588]]))\n"
       ]
      }
     ],
     "prompt_number": 65
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Using the estimator we can compute $P(X|C)$ for each class and thus we can express $P(X)$ using the total probability law:\n",
      "\n",
      "$$\\sum_{C_i} P(X|C_i)P(C_i)$$\n",
      "\n",
      "Finally we compute the probability for each class by simply counting the number of examples of that class in the training set:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def P_C(c):\n",
      "    return float(len(X_of_class[c]))/len(X_train)\n",
      "\n",
      "def P_X_giv_C(x,c):\n",
      "    estimator = class_estimators[c]\n",
      "    return estimator.estimate(x)\n",
      "\n",
      "def P_X(x):\n",
      "    return sum(P_C(c)*P_X_giv_C(x,c) for c in classes)\n",
      "\n",
      "for c in classes:\n",
      "    print P_X_giv_C(np.array([3.79421804 , 5.09090221]),c)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "0.191200680251\n",
        "8.36332055196e-10\n",
        "0.00111369347248\n"
       ]
      }
     ],
     "prompt_number": 66
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Finally we can compute the probability of each class using Bayes:\n",
      "$$P(C|X)=\\frac{P(X|C)P(C)}{P(X)}$$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def P_C_giv_X(c,x):\n",
      "    return (P_X_giv_C(x,c)*P_C(c))/P_X(x)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 67
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "(d) Using this function we can classify new data points by looking for the class that maximizes the conditional probability. Also we can classify taking into acount the risk of rejection $\\lambda$ by"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def classify(x):\n",
      "    return max(classes, key=(lambda c: P_C_giv_X(c,x)))\n",
      "\n",
      "#print P_C_giv_X(\n",
      "\n",
      "def classify_with_rejection(x,lamb):\n",
      "    c_max = classify(x)\n",
      "    if P_C_giv_X(c_max,x)>1-lamb:\n",
      "        return c_max\n",
      "    else:\n",
      "        return len(classes) #Reject class"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 68
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "(e) Also we can plot the regions according to the classifier:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x_sp = np.linspace(0.,15, num=75)\n",
      "y_sp = np.linspace(0.,10, num=75)\n",
      "\n",
      "lamb = 0.05\n",
      "points = [np.array(p) for p in product(x_sp,y_sp)]\n",
      "\n",
      "classification = [classify_with_rejection(p,lamb) for p in points]\n",
      "\n",
      "plot_class(get_features_from_class(0,points,classification), \"ro\", 0)\n",
      "plot_class(get_features_from_class(1,points,classification), \"go\", 1)\n",
      "plot_class(get_features_from_class(2,points,classification), \"bo\", 2)\n",
      "rejected = get_features_from_class(3,points,classification)\n",
      "plot_class(rejected, \"wo\", 3)\n",
      "pl.legend()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 69,
       "text": [
        "<matplotlib.legend.Legend at 0xa9b840c>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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333yDFi1aYMyYMQG1eGuwJHXklCAgsEbJznkfjZGpUbJGHh7Nk+VolByMh8Vi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\nCrJN7REAAAAASUVORK5CYII=\n",
       "text": [
        "<matplotlib.figure.Figure at 0xaa6f58c>"
       ]
      }
     ],
     "prompt_number": 69
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "(f) Finally we use the test set to evaluate the classifier. First we compute a confussion matrix using the classifier that doesn't reject:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "Y_classified=[classify(example) for example in X_test]\n",
      "n = len(classes)\n",
      "confussion_matrix = np.zeros((n,n))\n",
      "for correct_class, classified_class in zip(Y_test,Y_classified):\n",
      "    confussion_matrix[correct_class][classified_class]+=1\n",
      "print 'confussion matrix:\\n',confussion_matrix\n",
      "overall_accuracy = round(100.0*sum(diag(confussion_matrix))/sum(confussion_matrix),2)\n",
      "print 'overall accuracy:',overall_accuracy,'% (missclassified:',100.0-overall_accuracy,'%)'\n",
      "for c in classes:\n",
      "    class_accuracy = round(100.0*confussion_matrix[c][c]/sum(confussion_matrix[c]),2)\n",
      "    print 'class ',c,' accuracy:',class_accuracy,'% (missclassified:',100.0-class_accuracy,'%)'"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "confussion matrix:\n",
        "[[ 42.   0.   0.]\n",
        " [  0.  35.   0.]\n",
        " [  1.   0.  42.]]\n",
        "overall accuracy: 99.17 % (missclassified: 0.83 %)\n",
        "class  0  accuracy: 100.0 % (missclassified: 0.0 %)\n",
        "class  1  accuracy: 100.0 % (missclassified: 0.0 %)\n",
        "class  2  accuracy: 97.67 % (missclassified: 2.33 %)\n"
       ]
      }
     ],
     "prompt_number": 70
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "First we can see that the missclassificaton is very low. The class with more missclasifications was the $2$ class, possibly because its the class that overlaps the most with the other two classes. \n",
      "\n",
      "But by observing the regions graph we can note that points in the upper right corner (values with $x>12$ and $y>8$) are classified as being in the class $2$. But following the visual tendency in the input dataset the points in said region should be classified in the class $1$. As the test set doesn't contain any point in this region we can't observe these shortcomings of the classifier. The same observation can be applied to points in the upper left region which should be classified in the $0$ class but the classiffier recognizes as being in the $1$ class."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Also we compute another confussion matrix taking into account the possibility of rejection:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "lamb = 0.05\n",
      "confussion_matrix_with_rejection = np.zeros((n,n+1))\n",
      "Y_classified_with_rejection=[classify_with_rejection(example, lamb) for example in X_test]\n",
      "for correct_class, classified_class in zip(Y_test,Y_classified_with_rejection):\n",
      "    confussion_matrix_with_rejection[correct_class][classified_class]+=1\n",
      "print 'confussion matrix with rejection:\\n',confussion_matrix_with_rejection\n",
      "rejected = confussion_matrix_with_rejection.T[len(classes)]\n",
      "overall_accuracy = round(100.0*sum(diag(confussion_matrix_with_rejection))/(sum(confussion_matrix_with_rejection)-sum(rejected)),2)\n",
      "print 'overall accuracy:',overall_accuracy,'% (missclassified:',100.0-overall_accuracy,'%)'\n",
      "for c in classes:\n",
      "    class_accuracy = round(100.0*confussion_matrix_with_rejection[c][c]/sum(confussion_matrix_with_rejection[c][0:c+1]),2)\n",
      "    print 'class ',c,' accuracy (not counting rejected):',class_accuracy,'% (missclassified:',100.0-class_accuracy,'%)'\n",
      "    rejected_percentage =round(100.0*confussion_matrix_with_rejection[c][len(classes)]/sum(confussion_matrix_with_rejection[c]),2)\n",
      "    print 'Percentage of true class ',c,' rejected elements:',rejected_percentage,'%'"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "confussion matrix with rejection:\n",
        "[[ 41.   0.   0.   1.]\n",
        " [  0.  24.   0.  11.]\n",
        " [  0.   0.  39.   4.]]\n",
        "overall accuracy: 100.0 % (missclassified: 0.0 %)\n",
        "class  0  accuracy (not counting rejected): 100.0 % (missclassified: 0.0 %)\n",
        "Percentage of true class  0  rejected elements: 2.38 %\n",
        "class  1  accuracy (not counting rejected): 100.0 % (missclassified: 0.0 %)\n",
        "Percentage of true class  1  rejected elements: 31.43 %\n",
        "class  2  accuracy (not counting rejected): 100.0 % (missclassified: 0.0 %)\n",
        "Percentage of true class  2  rejected elements: 9.3 %\n"
       ]
      }
     ],
     "prompt_number": 71
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Classifying with rejection (using $\\lambda=0.05$) the accuracy (without counting rejected items) is still very high. But there is a high percentage of elements that are rejected for classification. Adding the posibility to reject increases accuracy with respect to the number of classified elements and produces a more confident classifier, but at the cost of not being able to classify some elements."
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "2. [Alp10] Excercise 1 (chapter 16)"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "By the definition of conditional probability we have that $P(x_1|x_2)$ is equal to:\n",
      "\n",
      "$$\\frac{P(x_1,x_2)}{P(x_2)}$$\n",
      "\n",
      "By the total probability law:\n",
      "\n",
      "$$\\frac{\\sum_i P(x_1,x_2|C_i)P(C_i)}{\\sum_i P(x_2|C_i)P(C_i)}$$\n",
      "\n",
      "And by the conditional independence of $x_1$ and $x_2$ this is equal to:\n",
      "\n",
      "$$\\frac{\\sum_i P(x_1|C_i)P(x_2|C_i)P(C_i)}{\\sum_i P(x_2|C_i)P(C_i)}$$\n",
      "\n",
      "As $p(x_j | C_i) \\sim \\mathcal{N}( \\mu_{ij} , \\sigma^2_{ij})$ :\n",
      "\n",
      "$$\n",
      "\\left(\\frac{1}{2\\pi}\\sum_i \\frac{P(C_i)}{\\sigma_{i1}\\sigma_{i2}} exp\\left(-\\frac{(x_1-\\mu_{i1})^2}{2\\sigma^2_{i1}}-\\frac{(x_2-\\mu_{i2})^2}{2\\sigma^2_{i2}}\\right)\\right)\\div\\left(\\frac{1}{\\sqrt{2\\pi}} \\sum_i \\frac{P(C_i)}{\\sigma_{i2}} exp\\left(-\\frac{(x_2-\\mu_{i2})^2}{2\\sigma^2_{i2}}\\right)\\right)\n",
      "$$\n",
      "\n",
      "Finally we have:\n",
      "\n",
      "$$\n",
      "P(x_1|x_2) = \\left(\\frac{\\sqrt{2\\pi}}{2\\pi}\\sum_i \\frac{P(C_i)}{\\sigma_{i1}\\sigma_{i2}} exp\\left(-\\frac{(x_1-\\mu_{i1})^2}{2\\sigma^2_{i1}}-\\frac{(x_2-\\mu_{i2})^2}{2\\sigma^2_{i2}}\\right)\\right)\\div\\left(\\sum_i \\frac{P(C_i)}{\\sigma_{i2}} exp\\left(-\\frac{(x_2-\\mu_{i2})^2}{2\\sigma^2_{i2}}\\right)\\right)\n",
      "$$\n"
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "3. [DHS00] Computer exercise 9 (sect 2.11, page 81)"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The bayesian network is shown below:"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "<img src=\"http://i.imgur.com/NNEDsTi.png\">"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "(a) The first question is asking for the conditional probability $P(\\mbox{Salmon } | \\mbox{ Dark, Thin, North Atlantic, Summer})$. Using the software we computed a probability of 0.3182.\n",
      "\n",
      "(b) The second question is asking for the conditional probabilities  $P(season\\mbox{ }| \\mbox{ Medium, Thin, North Atlantic})$ for each $season$ .The result was:\n",
      "$$P(\\mbox{winter }| \\mbox{ Medium, Thin, North Atlantic})=0.2529$$\n",
      "$$P(\\mbox{spring }| \\mbox{ Medium, Thin, North Atlantic})=0.2558$$\n",
      "$$P(\\mbox{summer }| \\mbox{ Medium, Thin, North Atlantic})=0.2490$$\n",
      "$$P(\\mbox{autumn }| \\mbox{ Medium, Thin, North Atlantic})=0.2412$$\n",
      "\n",
      "(c) The last question asked for the conditional probability $P(\\mbox{North Atlantic } | \\mbox{ Light, Wide, Autumn})$ which was equal to $0.5893$"
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "4. [Bis06] Exercise 8.11"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Consider the example of the car fuel system shown in Figure 8.21, and suppose that instead of observing the state of the fuel gauge $G$ directly, the gauge is seen by the driver $D$ who reports to us the reading on the gauge. This report is either that the gauge shows full $D = 1$ or that it shows empty $D = 0$. Our driver is a bit unreliable,\n",
      "as expressed through the following probabilities:\n",
      "\n",
      "$p(D = 1|G = 1) = 0.9 $ \n",
      "\n",
      "$p(D = 0|G = 0) = 0.9 $\n",
      "\n",
      "Suppose that the driver tells us that the fuel gauge shows empty, in other words that we observe $D = 0$. Evaluate the probability that the tank is empty given only this observation. Similarly, evaluate the corresponding probability given also the observation that the battery is flat, and note that this second probability is lower.\n",
      "Discuss the intuition behind this result, and relate the result to Figure 8.54.\n",
      "\n",
      "Solution:\n",
      "\n",
      "The first step is to see what is the figure 8.54:\n",
      "\n",
      "<img src=\"http://i.imgur.com/Oq9TAHv.png\">\n",
      "\n",
      "From the example in the book we have:\n",
      "\n",
      "''The variables are called $B$, representing the state of a battery that is either charged $(B = 1)$ or flat $(B = 0)$, $F$ representing the state of the fuel tank that is either full $(F = 1)$ or empty $(F = 0)$, and $G$, which is the state of an electric fuel gauge and which indicates either full $(G = 1)$ or empty $(G=0)$.\""
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "$p(B=1) = 0.9$\n",
      "\n",
      "$p(F=1) = 0.9$\n",
      "\n",
      "$p(G=0) = 0.315$\n",
      "\n",
      "$p(D=1|G=1) = 0.9$\n",
      "\n",
      "$p(D=0|G=0) = 0.9$\n",
      "\n",
      "$p(G=1 | B=1, F=1) = 0.84$\n",
      "\n",
      "$p(G=1 | B=1, F=0) = 0.2$\n",
      "\n",
      "$p(G=1 | B=0, F=1) = 0.2$\n",
      "\n",
      "$p(G=1 | B=0, F=0) = 0.1$\n",
      "\n",
      "* Evaluate $ P(F=0 | D=0). $\n",
      "\n",
      "$p(F=0|D=0) = \\dfrac{p(F=0,D=0)}{p(D=0)}$\n",
      "\n",
      "By the total probability theorem we have:\n",
      "\n",
      "$$\n",
      "\\begin{align*}\n",
      "p(D=0) &= p(D=0|G=0)p(G=0) + p(D=0|G=1)p(G=1) \\\\\n",
      "&= 0.9(0.315) + 0.1(0.685) = 0.352\n",
      "\\end{align*}\n",
      "$$\n",
      "\n",
      "Marginalizing we have:\n",
      "\n",
      "$$\n",
      "\\begin{align*}\n",
      "p(F=0, D=0) &= \\displaystyle\\sum_{g=0}^1 \\displaystyle\\sum_{b=0}^1 p(F=0,D=0,G=g,B=b) \n",
      "\\end{align*}\n",
      "$$\n",
      "\n",
      "We are going to calculate one of these probabilities. The others where calculated in the same way.\n",
      "\n",
      "$$\n",
      "\\begin{align*}\n",
      "&p(F=0,D=0,G=0,B=0) \\\\\n",
      "&= p(F=0)p(B=0)p(D=0|G=0)p(G=0|B=0,F=0)\t\\\\\n",
      "&= 0.1(0.1)(0.9)(0.9) = 0.0081.\n",
      "\\end{align*}\n",
      "$$\n",
      "\n",
      "After this operations we get:\n",
      "\n",
      "$$\n",
      "\\begin{align*}\n",
      "p(F=0,D=0,G=0,B=1) &= 0.0648\t\\\\\n",
      "p(F=0,D=0,G=1,B=0) &= 0.0001\t\\\\\n",
      "p(F=0,D=0,G=1,B=1) &= 0.0018\t\\\\\n",
      "\\end{align*}\n",
      "$$\n",
      "\n",
      "Finally:\n",
      "\n",
      "$$\n",
      "\\begin{align*}\n",
      "p(F=0|D=0) = \\dfrac{0.0081+0.0648+0.0001+0.0018}{0.352} = \\dfrac{0.0748}{0.352} = 0.2125\n",
      "\\end{align*}\n",
      "$$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Now we are interested in evaluating $p(F=0|D=0,B=0)$\n",
      "\n",
      "We will calculate this probability with the same methods that the last exercise.\n",
      "\n",
      "$$\n",
      "p(F=0|D=0,B=0) = \\dfrac{p(F=0,D=0,B=0)}{p(D=0,B=0)}\n",
      "$$\n",
      "\n",
      "$p(F=0,D=0,B=0) = p(F=0,D=0,B=0,G=0) + p(F=0,D=0,B=0,G=1)$\n",
      "\n",
      "$$\n",
      "\\begin{align*}\n",
      "p(F=0,D=0,B=0,G=0) = & p(F=0)p(B=0)p(D=0|G=0)p(G=0|F=0,B=0)\t\\\\\n",
      "= & 0.1(0.1)(0.9)(0.9) = 0.0081.\n",
      "\\end{align*}\n",
      "$$\n",
      "\n",
      "Similarly we calculate\n",
      "\n",
      "$ p(F=0,D=0,B=0,G=1) = 0.0001 $"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Now we have to calculate $p(D=0,B=0)$\n",
      "\n",
      "Marginalizing:\n",
      "\n",
      "$$p(D=0,B=0) = \\displaystyle\\sum_{f=0}^1\\displaystyle\\sum_{g=0}^1 p(D=0,B=0,F=f,G=g) $$\n",
      "\n",
      "We realize that sum of probabilities was calculated in the last exercise. We found this probability was $0.0748$.\n",
      "\n",
      "Finally we have:\n",
      "\n",
      "$ p(F=0|D=0,B=0) = \\dfrac{0.0082}{0.0748} = 0.1097 $"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We now see that this second probability is lower than the first one. Intuitively we hoped this second probability was lower because we realize that the probability of having one of these variables taking value 0 is very low, so we think that if we use more variables taking value of 0 as conditionals of that one we will reduce the probability of this event.\n",
      "\n",
      "In the other hand, the relation between this exercise and the graphic is because we can replace the names to the nodes and we have a diagram of our problem. If we replace $a$ by $B$, $b$ by $F$, $c$ by $G$ and $d$ by $D$ we can see the network that shows our problematic."
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "5. Belief propagation"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We define the following functions:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def pi(i):\n",
      "    if i==1:\n",
      "        return 0.5\n",
      "    else:\n",
      "        return ((i/(i+1.))+(1./(i+1.)))*pi(i-1)\n",
      "\n",
      "#We cache the function to avoid repeated computations: \n",
      "cache = {}\n",
      "def lamb(i,n,x):\n",
      "    key = str(i)+\" \"+str(n)+\" \"+str(x)\n",
      "    if not cache.has_key(key):\n",
      "        value = 0\n",
      "        if n == i+1:\n",
      "            value = (i+1.)/(i+2.) if x else 1./(i+2.)\n",
      "        else:\n",
      "            value = ((i+1.)/(i+2.))*lamb(i+1,n,x)+(1./(i+2.))*lamb(i+1,n,not(x))\n",
      "        cache[key] = value\n",
      "    return cache[key]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 72
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We have that $\\mbox{pi}(x=0)\\mbox{ == pi}(x=1)$  because $p(x_{i+1}|x_i)=1-p(x_{i+1}|\\sim x_i)$ thus the function pi doesn't need to take an x parameter"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "$\\alpha=\\frac{p(x_0)}{p(x_0,x_n)}$.\n",
      "\n",
      "$p(x_0,x_n)$ by marginalization is equal to $p(x_0)\\cdot\\sum_{i=1}^{n}\\left(p(x_i|x_{i-1})+p(x_i|\\sim x_{i-1})\\right)$and as $p(x_{i+1}|x_i)=\\frac{i+1}{i+2}$ and $p(x_{i+1}|\\sim x_i )=\\frac{1}{i+2}$ each term of the sum is equal to $1$. As $p(x_0)=0.5$ we can conclude that $\\alpha=1$:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def alpha(i,n):\n",
      "    return 1."
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 77
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "(1) For $n=10$ we have $p(x_{n/2} |x_0 = 1, x_n = 0)$, $p(x_{n/2} |x_0 = 0, x_n = 1)$ and $p(x_{n/2} |x_0 = 1, x_n = 1)$ are respectively equal to:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "print alpha(5,10)*pi(5)*lamb(5,10,False)\n",
      "print alpha(5,10)*pi(5)*lamb(5,10,True)\n",
      "print alpha(5,10)*pi(5)*lamb(5,10,True)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "0.181818181818\n",
        "0.318181818182\n",
        "0.318181818182\n"
       ]
      }
     ],
     "prompt_number": 78
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "(2) For $n = 30$ we have that $p(x_{n/2} |x_0 = 1, x_n = 0)$, $p(x_{n/2} |x_0 = 0, x_n = 1)$ and $p(x_{n/2} |x_0 = 1, x_n = 1) $ are respectively equal to:\n"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "print alpha(15,30)*pi(15)*lamb(15,30,False)\n",
      "print alpha(15,30)*pi(15)*lamb(15,30,True)\n",
      "print alpha(15,30)*pi(15)*lamb(15,30,True)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "0.185483870968\n",
        "0.314516129032\n",
        "0.314516129032\n"
       ]
      }
     ],
     "prompt_number": 75
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "(3) For $n = 30$:"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plot_range = range(1,30)\n",
      "plot_value = [alpha(i,30)*pi(i)*lamb(i,30,True) for i in plot_range]\n",
      "pl.plot([0]+plot_range , [0.5]+plot_value, 'ro')"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 80,
       "text": [
        "[<matplotlib.lines.Line2D at 0xb4db04c>]"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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       "text": [
        "<matplotlib.figure.Figure at 0xae96a4c>"
       ]
      }
     ],
     "prompt_number": 80
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": []
    }
   ],
   "metadata": {}
  }
 ]
}