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diff --git a/factor.jpg b/factor.jpg
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diff --git a/full_joint.jpg b/full_joint.jpg
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diff --git a/hw5.ipynb b/hw5.ipynb
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+{
+ "cells": [
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "metadata": {
+    "collapsed": true
+   },
+   "outputs": [],
+   "source": [
+    "import pandas as pd\n",
+    "import numpy as np\n",
+    "import math\n",
+    "import scipy.special as sps\n",
+    "from matplotlib import colors, ticker, cm\n",
+    "from matplotlib.mlab import bivariate_normal\n",
+    "from IPython.display import Image\n",
+    "from IPython.core.display import HTML "
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 2,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/html": [
+       "<img src=\"graphical.jpg\"/>"
+      ],
+      "text/plain": [
+       "<IPython.core.display.Image object>"
+      ]
+     },
+     "execution_count": 2,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "#Graphical Model\n",
+    "Image(url= \"graphical.jpg\")"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 3,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/html": [
+       "<img src=\"factor.jpg\"/>"
+      ],
+      "text/plain": [
+       "<IPython.core.display.Image object>"
+      ]
+     },
+     "execution_count": 3,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "#Factor Graph\n",
+    "Image(url= \"factor.jpg\")"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 4,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/html": [
+       "<img src=\"full_joint.jpg\"/>"
+      ],
+      "text/plain": [
+       "<IPython.core.display.Image object>"
+      ]
+     },
+     "execution_count": 4,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "#derivation of full joint distribution and individual factors\n",
+    "Image(url= \"full_joint.jpg\")"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 5,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/html": [
+       "<img src=\"conditional_1.jpg\"/>"
+      ],
+      "text/plain": [
+       "<IPython.core.display.Image object>"
+      ]
+     },
+     "execution_count": 5,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "#q(A)\n",
+    "Image(url= \"conditional_1.jpg\")"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 6,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/html": [
+       "<img src=\"conditional_2.jpg\"/>"
+      ],
+      "text/plain": [
+       "<IPython.core.display.Image object>"
+      ]
+     },
+     "execution_count": 6,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "#q(R)\n",
+    "Image(url= \"conditional_2.jpg\")"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 7,
+   "metadata": {
+    "collapsed": true
+   },
+   "outputs": [],
+   "source": [
+    "nu = 0.4\n",
+    "beta = 100\n",
+    "P = 1.2\n",
+    "T = 1000\n",
+    "x_0 = 1.0\n",
+    "x_1 = -6.0\n",
+    "E_A = -100.0\n",
+    "E_A2 = E_A*E_A\n",
+    "E_invR = 100.0"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 8,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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sqq7IJGTtSHJdd99bVZcP19+V5NYkr8xkM5qvJ3n9cn2Hoa9KcktVXZrks0leO/S5t6pu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\nhkVJkiRJUsdfAdI/YT8AleabAAAAAElFTkSuQmCC\n",
+      "text/plain": [
+       "<matplotlib.figure.Figure at 0x10eb2ba90>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "#Contour Plot\n",
+    "import numpy as np\n",
+    "import matplotlib.pyplot as plt\n",
+    "x = np.arange(-8.,8.,0.1)\n",
+    "y = np.arange(-4.,4.,0.1)\n",
+    "y = np.power(10.,y)\n",
+    "A, R = np.meshgrid(x,y)\n",
+    "Z = np.exp(-0.5*x_1*x_1/R + x_0*x_1*A/R - 0.5*x_0*x_0*A*A/R - 0.5*np.log(2*np.pi*R))*np.exp(-0.5*A*A/P - 0.5*np.log(np.abs(2*np.pi*P)))*np.exp(-(nu+1)*np.log(R) - nu/(beta*R) - sps.gamma(nu) + nu*np.log(nu/beta))\n",
+    "plt.figure(figsize=(16,10))\n",
+    "cp = plt.contourf(A, R, Z, cmap=plt.cm.jet)\n",
+    "plt.colorbar(cp)\n",
+    "plt.title('Filled Contours Plot')\n",
+    "plt.xlabel('A')\n",
+    "plt.yscale('log')\n",
+    "plt.ylabel('R')\n",
+    "plt.show()"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 9,
+   "metadata": {
+    "collapsed": true
+   },
+   "outputs": [],
+   "source": [
+    "#Implementation of EM algorithm to find MAP estimates of A and R\n",
+    "for t in range(2,T):\n",
+    "    sig = 1/(1/P + x_0*x_0*E_invR)\n",
+    "    mu = sig*x_0*x_1*E_invR\n",
+    "    E_A = mu\n",
+    "    E_A2 = mu*mu + sig\n",
+    "    a = nu + 0.5\n",
+    "    b = 0.5*(x_1*x_1 - 2*x_1*x_0*E_A + x_0*x_0*E_A2) + nu/beta\n",
+    "    E_invR = a/b"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 10,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "-0.3704347574553939"
+      ]
+     },
+     "execution_count": 10,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "E_A"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 11,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "0.05483471954102366"
+      ]
+     },
+     "execution_count": 11,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "E_invR"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 12,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [],
+   "source": [
+    "nu = 0.4\n",
+    "beta = 100\n",
+    "P = 1.2\n",
+    "x_0 = 1.0\n",
+    "x_1 = -6.0\n",
+    "T = 1000\n",
+    "R = np.zeros((1,1000))\n",
+    "A = np.zeros((1,1000))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 13,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [],
+   "source": [
+    "A[0][0] = -6\n",
+    "R[0][0] = 0.00001"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 14,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [],
+   "source": [
+    "#implementation of ICM\n",
+    "for t in range(1,T):\n",
+    "    sig = 1/(1/P + np.power(x_0,2)*(1.0/R[0][t-1]))\n",
+    "    mu = sig*x_0*x_1*(1/R[0][t-1])\n",
+    "    A[0][t] = math.sqrt(sig)*np.random.normal() + mu;\n",
+    "    b = 0.5*(np.power(x_1,2) - 2*x_1*x_0*A[0][t] + np.power(x_0,2)*np.power(A[0][t],2)) + nu/beta\n",
+    "    R[0][t] = 1/(np.random.gamma(nu+0.5, 1/b));"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 15,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "image/png": 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VdhjcP1y1lHItxRXAW3YfasksoPlzq5eg6KT33/Vk1cCSdAMwfHyEqzY/5SeBDMpkALDu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\n",
+      "text/plain": [
+       "<matplotlib.figure.Figure at 0x10f30e850>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "from matplotlib import pyplot\n",
+    "%matplotlib inline\n",
+    "import math\n",
+    "pyplot.scatter(A,R)\n",
+    "\n",
+    "pyplot.yscale('log')\n",
+    "pyplot.xlim([-8,6])\n",
+    "pyplot.ylim([0.0001,100000])\n",
+    "pyplot.show()"
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python [default]",
+   "language": "python",
+   "name": "python2"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 2
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython2",
+   "version": "2.7.13"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 2
+}