{
 "metadata": {
  "name": "2.1.1-beta-distribution"
 },
 "nbformat": 3,
 "nbformat_minor": 0,
 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%pylab inline\n",
      "import sympy as sym\n",
      "%load_ext sympy.interactive.ipythonprinting\n",
      "from IPython.display import display, Math, Latex\n",
      "maths = lambda s: display(Math(s))\n",
      "latex = lambda s: display(Latex(s))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "\n",
        "Welcome to pylab, a matplotlib-based Python environment [backend: module://IPython.zmq.pylab.backend_inline].\n",
        "For more information, type 'help(pylab)'.\n"
       ]
      }
     ],
     "prompt_number": 151
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Define the Beta distribution (symbolically using sympy) which will act as our prior distribution"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "a,b,mu = sym.symbols('a b mu')\n",
      "G = sym.gamma\n",
      "\n",
      "# The normalisation factor\n",
      "Beta_norm = G(a + b)/(G(a)*G(b))\n",
      "# The functional form, note the similarity\n",
      "# to the Binomial likelihood\n",
      "Beta_f = mu**(a-1) * (1-mu)**(b-1)\n",
      "\n",
      "# Turn Beta into a function\n",
      "Beta = sym.Lambda((mu,a,b), Beta_norm * Beta_f)\n",
      "\n",
      "maths(r\"\\operatorname{Beta}(\\mu|a,b) = \")\n",
      "display(Beta.expr)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\operatorname{Beta}(\\mu|a,b) = $$"
       ],
       "output_type": "display_data",
       "text": [
        "<IPython.core.display.Math object at 0x8361e50>"
       ]
      },
      {
       "latex": [
        "$$\\frac{\\mu^{a -1} \\left(- \\mu + 1\\right)^{b -1} \\Gamma\\left(a + b\\right)}{\\Gamma\\left(a\\right) \\Gamma\\left(b\\right)}$$"
       ],
       "output_type": "display_data",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAJEAAAAoCAYAAAD+HRieAAAABHNCSVQICAgIfAhkiAAABRVJREFU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\nTkSuQmCC\n",
       "text": [
        " a - 1         b - 1         \n",
        "\u03bc     \u22c5(-\u03bc + 1)     \u22c5\u0393(a + b)\n",
        "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "          \u0393(a)\u22c5\u0393(b)          "
       ]
      }
     ],
     "prompt_number": 204
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "mus = arange(0,1,.01)\n",
      "\n",
      "# Plot for various values of a and b\n",
      "for ab in [(.1,.1),(8,4),(2,3), (1,1)]:\n",
      "    plot(mus, vectorize(Beta)(mus,*ab), label=\"a=%s b=%s\" % ab)\n",
      "\n",
      "legend(loc=0)\n",
      "xlabel(r\"$\\mu$\", size=22)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "pyout",
       "prompt_number": 205,
       "text": [
        "Text(0.5,0,'$\\\\mu$')"
       ]
      },
      {
       "output_type": "display_data",
       "png": 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r1w8BAQGwsLCAlpYWdu/eLfbgzLAzFI0/Q//EIKtBaKbbrOrGDLERCAQYZkPd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\n"
      }
     ],
     "prompt_number": 205
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Now define the Binomial, which gives our likelihood (as a function of $\\mu$)"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "N, m = sym.symbols('N m')\n",
      "\n",
      "Bin_norm = sym.binomial(N,m)\n",
      "Bin_f = mu**m * (1-mu) ** (N-m)\n",
      "\n",
      "Bin = sym.Lambda((m, N, mu), Bin_norm * Bin_f)\n",
      "\n",
      "maths(r\"\\operatorname{Bin}(m|N,\\mu) = \")\n",
      "display(Bin.expr)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\operatorname{Bin}(m|N,\\mu) = $$"
       ],
       "output_type": "display_data",
       "text": [
        "<IPython.core.display.Math object at 0x7f2e910>"
       ]
      },
      {
       "latex": [
        "$$\\mu^{m} \\left(- \\mu + 1\\right)^{N - m} {\\binom{N}{m}}$$"
       ],
       "output_type": "display_data",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAKgAAAAbCAYAAADoFcRvAAAABHNCSVQICAgIfAhkiAAABSJJREFU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\nwzNbgvvGdgAAAABJRU5ErkJggg==\n",
       "text": [
        " m         N - m \u239bN\u239e\n",
        "\u03bc \u22c5(-\u03bc + 1)     \u22c5\u239c \u239f\n",
        "                 \u239dm\u23a0"
       ]
      }
     ],
     "prompt_number": 209
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Ignoring normalisation for now, we can find the posterior distribution\n",
      "by multiplying our Beta prior by the Binomial likelihood"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "p = Beta_f * Bin_f\n",
      "p = p.powsimp()\n",
      "\n",
      "maths(r\"\\operatorname{Beta}(\\mu|a,b) \\times \\operatorname{Bin}(m|N,\\mu) \\propto %s\" % sym.latex(p))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\operatorname{Beta}(\\mu|a,b) \\times \\operatorname{Bin}(m|N,\\mu) \\propto \\mu^{a + m -1} \\left(- \\mu + 1\\right)^{N + b - m -1}$$"
       ],
       "output_type": "display_data",
       "text": [
        "<IPython.core.display.Math object at 0x7f538d0>"
       ]
      }
     ],
     "prompt_number": 208
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Let $l = N-m$ (the number of tails, if we are talking about coin tosses) then we can write it as"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "l = sym.Symbol('l')\n",
      "p = p.subs(N-m, l)\n",
      "p"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\mu^{a + m -1} \\left(- \\mu + 1\\right)^{b + l -1}$$"
       ],
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAALYAAAAbCAYAAADRcnU0AAAABHNCSVQICAgIfAhkiAAABKVJREFU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\nSUVORK5CYII=\n",
       "prompt_number": 210,
       "text": [
        " a + m - 1         b + l - 1\n",
        "\u03bc         \u22c5(-\u03bc + 1)         "
       ]
      }
     ],
     "prompt_number": 210
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "So the posterior distribution has the same functional dependence on $\\mu$ as the prior, it is just another Beta distribution"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "maths(\"p(\\mu|m,l,a,b) \\propto %s\" % sym.latex(p))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$p(\\mu|m,l,a,b) \\propto \\mu^{a + m -1} \\left(- \\mu + 1\\right)^{b + l -1}$$"
       ],
       "output_type": "display_data",
       "text": [
        "<IPython.core.display.Math object at 0x83910d0>"
       ]
      }
     ],
     "prompt_number": 211
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "maths(\"p(\\mu|m,l,a,b) = \\operatorname{Beta}(\\mu|a+m,b+l) =b\")\n",
      "Beta(mu,a+m,b+l)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$p(\\mu|m,l,a,b) = \\operatorname{Beta}(\\mu|a+m,b+l) =$$"
       ],
       "output_type": "display_data",
       "text": [
        "<IPython.core.display.Math object at 0x835f6d0>"
       ]
      },
      {
       "latex": [
        "$$\\frac{\\mu^{a + m -1} \\left(- \\mu + 1\\right)^{b + l -1} \\Gamma\\left(a + b + l + m\\right)}{\\Gamma\\left(a + m\\right) \\Gamma\\left(b + l\\right)}$$"
       ],
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAN0AAAAoCAYAAAB6m/+wAAAABHNCSVQICAgIfAhkiAAAB0RJREFU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\n",
       "prompt_number": 216,
       "text": [
        " a + m - 1         b + l - 1                 \n",
        "\u03bc         \u22c5(-\u03bc + 1)         \u22c5\u0393(a + b + l + m)\n",
        "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "              \u0393(a + m)\u22c5\u0393(b + l)              "
       ]
      }
     ],
     "prompt_number": 216
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Let's see how this works graphically, with prior a=2, b=2 and likelihood N=1, m=1"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "prior = Beta(mu,2,2)\n",
      "\n",
      "plot(mus, (sym.lambdify(mu,prior))(mus), 'r')\n",
      "xlabel(\"$\\mu$\", size=22)\n",
      "\n",
      "maths(\"\\mathbf{Prior}: \\operatorname{Beta}(\\mu|a=2,b=2) = %s\" % sym.latex(prior))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\mathbf{Prior}: \\operatorname{Beta}(\\mu|a=2,b=2) = 6 \\mu \\left(- \\mu + 1\\right)$$"
       ],
       "output_type": "display_data",
       "text": [
        "<IPython.core.display.Math object at 0xaa1c250>"
       ]
      },
      {
       "output_type": "display_data",
       "png": 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u3LnUr1+f6667jszMTGJiYs55fHJyMl26dOGuu+46+0KeKIVctw46doStW81e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\nAUjJXUQkACm5i4gEICV3EZEA9P9oBF+ZswXA7wAAAABJRU5ErkJggg==\n"
      }
     ],
     "prompt_number": 259
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "likelihood = Bin(1,1,mu)\n",
      "\n",
      "plot(mus, (sym.lambdify(mu,likelihood))(mus), 'b')\n",
      "xlabel(\"$\\mu$\", size=22)\n",
      "\n",
      "maths(\"\\mathbf{Likelihood}: \\operatorname{Bin}(m=1|N=1,\\mu) = %s\" % sym.latex(likelihood))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\mathbf{Likelihood}: \\operatorname{Bin}(m=1|N=1,\\mu) = \\mu$$"
       ],
       "output_type": "display_data",
       "text": [
        "<IPython.core.display.Math object at 0x9ada7d0>"
       ]
      },
      {
       "output_type": "display_data",
       "png": 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      }
     ],
     "prompt_number": 260
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "posterior = prior * likelihood\n",
      "\n",
      "plot(mus, (sym.lambdify(mu,posterior))(mus), 'r')\n",
      "xlabel(\"$\\mu$\", size=22)\n",
      "\n",
      "maths(r\"\\mathbf{Posterior (unnormalised)}: \\operatorname{Beta}(\\mu|2,2) \\times \\operatorname{Bin}(1|1,\\mu) = %s\" % sym.latex(posterior))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\mathbf{Posterior (unnormalised)}: \\operatorname{Beta}(\\mu|2,2) \\times \\operatorname{Bin}(1|1,\\mu) = 6 \\mu^{2} \\left(- \\mu + 1\\right)$$"
       ],
       "output_type": "display_data",
       "text": [
        "<IPython.core.display.Math object at 0xa5c1f90>"
       ]
      },
      {
       "output_type": "display_data",
       "png": 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Rhw8D11xj92nYc1erU6fEsMfPPmOwE3mS5s3FOjMOmG1qD4a7O1MUIC4OGDEC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\nAElFTkSuQmCC\n"
      }
     ],
     "prompt_number": 247
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "posterior = Beta(mu, 2+1, 2+0)\n",
      "\n",
      "plot(mus, (sym.lambdify(mu,posterior))(mus), 'r')\n",
      "xlabel(\"$\\mu$\", size=22)\n",
      "\n",
      "maths(r\"\\mathbf{Posterior}: p(\\mu|m,l,a,b) = p(\\mu|1,0,2,2) = \\operatorname{Beta}(\\mu,2+1,2+0) = %s\" % sym.latex(posterior))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\mathbf{Posterior}: p(\\mu|m,l,a,b) = p(\\mu|1,0,2,2) = \\operatorname{Beta}(\\mu,2+1,2+0) = 12 \\mu^{2} \\left(- \\mu + 1\\right)$$"
       ],
       "output_type": "display_data",
       "text": [
        "<IPython.core.display.Math object at 0xa132150>"
       ]
      },
      {
       "output_type": "display_data",
       "png": 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dBQsWcNVVV1GvXj2WLVvGhRdeaGOlIjWjcJeAkpmZyf79+ykoKGDP\nnj2AWdpu3Lhx1KtXr3Ro5L///W/i4uLsLFWkRvxzKXGRszjRb7/vvvu45557iIyMpKioiPHjxzNu\n3DhGjhzJvffeS58+fahTp47N1YpUnxbrkIByyy23kJGRwb59++wuRcSt1JaRgFFcXMzy5cvp27ev\n3aWIuJ3CXQLG2rVrOXz4MDfeeKPdpYi4ncJdAsbevXu5/PLLueWWW+wuRcTt1HMXEfFDunIXEfFD\nCncRET+kcBcR8UMKdxERP6RwFxHxQwp3ERE/pHAXEfFDCncRET/0/wGu7a0uvoOIEAAAAABJRU5E\nrkJggg==\n"
      }
     ],
     "prompt_number": 268
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left\\{\\mu\\right\\}$$"
       ],
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAABoAAAATCAYAAACORR0GAAAABHNCSVQICAgIfAhkiAAAAWBJREFU\nOI3t1L9L1lEUBvBP+kJK+EqbYIEgKa9/gUjqK4gJaUMgYkSL4N8hgqN7QxANCeLgHyAoGtKQg4Ob\nSCBYiJSbg5o13C/x9Xp9f7i4+Gz3Oec8zznccy93iFeYRfGW9ZOYQ1OlpE5s3NIgj2mh2f9oiBKe\n4vsNxS9viH3F24jbR0clo0qYwu+IK6EXP6sV12NUxnrE9eNcmKouo8asMMYztCeMBvANpxF/kWkl\njRoxgZWEURl/8CXi+7GZyN/BE+HOr2FBWO0UFrPiPDrwF6MooCeKt2MVrbHYA3zEeMLoB5YjbgaX\naEGfq5v3SJg+ORGM4EPEdWedb+W4krAAx9l5Ho9z8ef4nBcpRKJnCa4sXO4+PuEXjjCMJbzHLk4i\n3StLFYumMIRtvEvExmqoR23vaBBrtQrWanSIrty5hDbX3081dOKgWtJr4XKLeIE9NNdh8kb4vR/W\n2dw90vgH7kc6XmsY2sQAAAAASUVORK5CYII=\n",
       "prompt_number": 256,
       "text": [
        "set([\u03bc])"
       ]
      }
     ],
     "prompt_number": 256
    },
    {
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 ]
}