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   "cells": [
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "This is a little mathematical background to justify why we might require [priors](http://en.wikipedia.org/wiki/Prior_probability) to be \"invariant\" under certain transformations of the parameter space.  This is not a new idea, but the derivation is my own (and hence so are errors...)\n",
      "\n",
      "I'll use `sympy` to do some examples."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%matplotlib inline\n",
      "import sympy\n",
      "sympy.init_printing(use_latex = True)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 3
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "As motivation, consider a model where $(X_i)_{i=1}^n$ are IID samples from $N(\\mu,\\sigma^2)$ where we wish to use a sample $(x_i)$ to make inferences about $\\mu, \\sigma$.  For some variety, I'll use the [precision](http://en.wikipedia.org/wiki/Precision_%28statistics%29) $\\tau = 1 / \\sigma^2$.\n",
      "\n",
      "So our random variable $x=(x_i)$ takes values in $\\Omega = \\mathbb R^n$.  Consider the transformation $\\phi : \\Omega\\rightarrow\\Omega; (x_i) \\mapsto (bx_i+b)$.  That is, for each sample, we scale by $b$ and add $a$.  Then $\\phi(x)$ is a sample of IID $N(b\\mu+a, b^2\\sigma^2)$.  If $\\Theta$ is the parameter space (in our case, say $\\{ (\\mu,\\tau) : \\tau > 0 \\}$) then consider the transformation $\\psi : \\Theta \\rightarrow \\Theta; (\\mu,\\tau) \\mapsto (b\\mu+a, b^{-2} \\tau )$.  It is hence reasonable to say the following:\n",
      "   - Suppose we make some inference about $\\theta = (\\mu,\\tau)$ given $x = (x_i)$.\n",
      "   - Then we should make the \"same\" inference about $\\psi(\\theta)$ given $\\phi(x)$.\n",
      "   \n",
      "As an example, if $\\phi(x) = ( 2x_i - 1 )$ and our inference about $(x_i)$ gives a value to $\\mathbb P( 0\\leq \\mu \\leq 2, 1 \\leq \\tau \\leq 3 )$ then as $\\psi(\\mu,\\tau) = (2\\mu-1, \\tau/2)$, the inference about $(2x_i-1)$ should give the sample value to $\\mathbb P( -1 \\leq \\mu \\leq 3, 1/4 \\leq \\tau \\leq 3/4 )$."
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "The assumption"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Let $f(x|\\theta)$ be the PDF of $x$ given the parameter value $\\theta$.  So $\\mathbb P(x\\in A | \\theta) = \\int_A f(x|\\theta) dx$ for measurable $A\\subseteq\\Omega$.  Our assumption is that apply $\\phi$ to $\\Omega$ is \"the same as\" applying $\\psi$ to $\\Theta$.  That is,\n",
      "$$ \\mathbb P(\\phi(x)\\in A | \\theta) = \\mathbb P(x\\in A | \\psi(\\theta) ). $$\n",
      "By the [change of variables formula](http://en.wikipedia.org/wiki/Integration_by_substitution#Application_in_probability) the distribution $y = \\phi(x)$ has density $f(\\phi^{-1}(y)|\\theta) |\\det J(\\phi^{-1})|$ here $J$ is the Jacobian.  Using that $|\\det J(\\phi^{-1})| = 1 / |\\det J(\\phi)|$ we find that\n",
      "$$ f(\\phi^{-1}(y)|\\theta) = f(y|\\psi(\\theta)) |\\det J(\\phi)|, $$\n",
      "or equivalently, with $x = \\phi^{-1}(y)$,\n",
      "$$ f(x|\\theta) = f(\\phi(x)|\\psi(\\theta)) |\\det J(\\phi)|. $$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Let's check this for our normal distribution example.  \n",
      "\n",
      "   - In `sympy` I don't know how to specify $x_1, x_2, \\cdots, x_n$ for an unknown $n$.\n",
      "   - So let's get around this with a hack.\n",
      "\n",
      "Write $\\sum_i (x_i - \\mu)^2 = \\sum_i x_i^2 - 2\\mu \\sum_i x_i + n\\mu^2 = S_{xx} - 2\\mu S_x + n\\mu^2$."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def density(Sxx, Sx, n, mu, tau):\n",
      "    \"\"\"Return the density (2\\pi\\sigma^2)^{-n/2} \\exp(-\\sum (x_i-\\mu)^2 / 2\\sigma^2)\n",
      "    = (2\\pi)^{-n/2} \\tau^{n/2} \\exp(-\\tau (S_{xx} - 2\\mu S_x + n\\mu^2)/2)\"\"\"\n",
      "    return (2 * sympy.pi)**(-n/2) * tau**(n/2) * sympy.exp( - tau * (Sxx - 2*mu*Sx + n*mu**2) / 2 )\n",
      "\n",
      "n = sympy.symbols(\"n\", integer = True, positive = True)\n",
      "Sx, mu = sympy.symbols(\"Sx mu\", real = True)\n",
      "Sxx, tau = sympy.symbols(\"Sxx tau\", real = True, positive = True)\n",
      "\n",
      "density(Sxx, Sx, n, mu, tau)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{\\tau^{\\frac{n}{2}}}{\\left(2 \\pi\\right)^{\\frac{n}{2}}} e^{- \\frac{\\tau}{2} \\left(- 2 Sx \\mu + Sxx + \\mu^{2} n\\right)}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAPIAAAA4BAMAAAA1NbhFAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nYCNe8Cac8Rph6v8CBa86+Ck7qlwAAAAASUVORK5CYII=\n",
       "prompt_number": 4,
       "text": [
        "                \u239b                 2  \u239e \n",
        " n      -n   -\u03c4\u22c5\u239d-2\u22c5Sx\u22c5\u03bc + Sxx + \u03bc \u22c5n\u23a0 \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\n",
        " 2       2               2             \n",
        "\u03c4 \u22c5(2\u22c5\u03c0)   \u22c5\u212f                          "
       ]
      }
     ],
     "prompt_number": 4
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Our transformation $\\phi$ is $x_i \\mapsto bx_i+a$ and so $S_{xx} \\mapsto \\sum_i (bx_i+a)^2 = b^2S_{xx} + 2abS_x + na^2$ and $S_x \\mapsto \\sum_i bx_i+a = bS_x+na$."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def phi(a, b):\n",
      "    def func(Sxx, Sx, n):\n",
      "        return b*b*Sxx + 2*a*b*Sx + n*a*a, b*Sx + n*a, n\n",
      "    return func\n",
      "    \n",
      "a, b = sympy.symbols(\"a b\", real = True)\n",
      "density(*phi(a,b)(Sxx,Sx,n),mu=mu,tau=tau)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{\\tau^{\\frac{n}{2}}}{\\left(2 \\pi\\right)^{\\frac{n}{2}}} e^{- \\frac{\\tau}{2} \\left(2 Sx a b + Sxx b^{2} + a^{2} n + \\mu^{2} n - 2 \\mu \\left(Sx b + a n\\right)\\right)}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
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       "prompt_number": 5,
       "text": [
        "                \u239b                2    2      2                     \u239e \n",
        " n      -n   -\u03c4\u22c5\u239d2\u22c5Sx\u22c5a\u22c5b + Sxx\u22c5b  + a \u22c5n + \u03bc \u22c5n - 2\u22c5\u03bc\u22c5(Sx\u22c5b + a\u22c5n)\u23a0 \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\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        " 2       2                              2                            \n",
        "\u03c4 \u22c5(2\u22c5\u03c0)   \u22c5\u212f                                                        "
       ]
      }
     ],
     "prompt_number": 5
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Our transformation $\\psi$ is $\\mu\\mapsto b\\mu+a, \\tau \\mapsto b^{-2} \\tau$."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def psi(a, b):\n",
      "    def func(mu, tau):\n",
      "        return b*mu+a, b**(-2) * tau\n",
      "    return func\n",
      "\n",
      "density(Sxx,Sx,n,*psi(a,b)(mu,tau))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{\\left(\\frac{\\tau}{b^{2}}\\right)^{\\frac{n}{2}}}{\\left(2 \\pi\\right)^{\\frac{n}{2}}} e^{- \\frac{\\tau}{2 b^{2}} \\left(- Sx \\left(2 a + 2 b \\mu\\right) + Sxx + n \\left(a + b \\mu\\right)^{2}\\right)}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAV0AAAA+BAMAAAB6sdXEAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nPR0gXtZHbXuGogdNn/43pWHVijNe2UMAAAAASUVORK5CYII=\n",
       "prompt_number": 6,
       "text": [
        "                   \u239b                                     2\u239e \n",
        "             n  -\u03c4\u22c5\u239d-Sx\u22c5(2\u22c5a + 2\u22c5b\u22c5\u03bc) + Sxx + n\u22c5(a + b\u22c5\u03bc) \u23a0 \n",
        "     -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",
        "     \u2500\u2500\u2500     2                         2                    \n",
        "      2  \u239b\u03c4 \u239e                       2\u22c5b                     \n",
        "(2\u22c5\u03c0)   \u22c5\u239c\u2500\u2500\u239f \u22c5\u212f                                            \n",
        "         \u239c 2\u239f                                               \n",
        "         \u239db \u23a0                                               "
       ]
      }
     ],
     "prompt_number": 6
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Unfortunately, our \"hack\" means we can't automate the calculation of the determinant.  We have $\\phi:\\mathbb R^n \\rightarrow \\mathbb R^n; (x_i) \\mapsto (bx_i+a)$ so $J(\\phi) = bI_n$ and hence $\\det J(\\phi) = b^n$."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "rhs = density(*(phi(a,b)(Sxx,Sx,n) + psi(a,b)(mu,tau))) * sympy.Abs(b)**n\n",
      "sympy.simplify(rhs)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{\\tau^{\\frac{n}{2}}}{2^{\\frac{n}{2}} \\pi^{\\frac{n}{2}}} e^{\\frac{\\tau}{2} \\left(2 Sx \\mu - Sxx - \\mu^{2} n\\right)}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAANcAAAAvBAMAAACCkkLyAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nI2LqmAwsg/5Y989MerOCYnRYf7N/pfAf2VAk42EwpdYAAAAASUVORK5CYII=\n",
       "prompt_number": 7,
       "text": [
        "                \u239b                2  \u239e\n",
        " -n   -n   n  \u03c4\u22c5\u239d2\u22c5Sx\u22c5\u03bc - Sxx - \u03bc \u22c5n\u23a0\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\n",
        "  2    2   2             2           \n",
        "2   \u22c5\u03c0   \u22c5\u03c4 \u22c5\u212f                       "
       ]
      }
     ],
     "prompt_number": 7
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "lhs = density(Sxx,Sx,n,mu,tau)\n",
      "lhs"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{\\tau^{\\frac{n}{2}}}{\\left(2 \\pi\\right)^{\\frac{n}{2}}} e^{- \\frac{\\tau}{2} \\left(- 2 Sx \\mu + Sxx + \\mu^{2} n\\right)}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAPIAAAA4BAMAAAA1NbhFAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nYCNe8Cac8Rph6v8CBa86+Ck7qlwAAAAASUVORK5CYII=\n",
       "prompt_number": 8,
       "text": [
        "                \u239b                 2  \u239e \n",
        " n      -n   -\u03c4\u22c5\u239d-2\u22c5Sx\u22c5\u03bc + Sxx + \u03bc \u22c5n\u23a0 \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\n",
        " 2       2               2             \n",
        "\u03c4 \u22c5(2\u22c5\u03c0)   \u22c5\u212f                          "
       ]
      }
     ],
     "prompt_number": 8
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Unfortunately, the following is a little distressing..."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "rhs == lhs"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 9,
       "text": [
        "False"
       ]
      }
     ],
     "prompt_number": 9
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "rhs - lhs == 0"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 10,
       "text": [
        "False"
       ]
      }
     ],
     "prompt_number": 10
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "sympy.simplify(rhs - lhs)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$0$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAOBAMAAADkjZCYAAAALVBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADAOrOgAAAADnRSTlMAEJl2Mt1EzasiVIlm7xEe\nOPwAAABJSURBVAgdY2AQMlZhYGAIYxB1YGAOYGAuYGBtYGB7ycCnwMDyimHeBgaWx2DyNRL5GCz7\nBqSS6yVIF3cAA0MUg6wDA4PQ5hYGAON0EpMbNABcAAAAAElFTkSuQmCC\n",
       "prompt_number": 11,
       "text": [
        "0"
       ]
      }
     ],
     "prompt_number": 11
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "sympy.simplify(rhs - lhs) == 0"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 12,
       "text": [
        "True"
       ]
      }
     ],
     "prompt_number": 12
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "What we wish to be true"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Okay, so what about our inference, about the _posterior distribution_, do we wish to be true?\n",
      "\n",
      "$$ \\mathbb P(\\theta \\in \\Psi(A) | \\phi(x) ) = \\mathbb P(\\theta\\in A | x). $$\n",
      "\n",
      "Again, if we write $p(\\theta|x)$ for the posterior density, then this is equivalent to:\n",
      "\n",
      "$$ p(\\theta|x) = p(\\psi(\\theta)| \\phi(x)) |\\det J(\\psi)|. $$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "By Bayes' Theorem, we want the following to be true:\n",
      "\n",
      "$$ \\frac{f(x|\\theta) p(\\theta)}{p(x)} = \\frac{f(\\phi(x)|\\psi(\\theta)) p(\\psi(\\theta))}{p(\\phi(x))} |\\det J(\\psi)|. $$\n",
      "\n",
      "By our assumption $f(x|\\theta) = f(\\phi(x)|\\psi(\\theta)) |\\det J(\\phi)|$, this is equivalent to:\n",
      "\n",
      "$$ \\frac{f(\\phi(x)|\\psi(\\theta)) |\\det J(\\phi)| p(\\theta)}{p(x)} = \\frac{f(\\phi(x)|\\psi(\\theta)) p(\\psi(\\theta))}{p(\\phi(x))} |\\det J(\\psi)|. $$"
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "A little arm waving."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We need to deal with the terms $p(x)$ and $p(\\phi(x))$.  Now, these are likely to be \"improper distributions\", much as we expect $p(\\theta)$ to be improper.  Thus, $\\int p(x) dx$ may fail to be finite, and so we more correctly think of $p(x)$ as measuring \"proportional\" or \"relative\" probability: $\\int_A p(x) dx$ cannot be interpretted as a probability, but we can make sense of the ratio $\\int_A p(x) dx / \\int_B p(x) dx$ as the \"ratio\" of more likely $x$ is to be in $A$ relative to $B$.\n",
      "\n",
      "With that out the way, it still makes sense to transform $x$.  Our philosophy is that $\\phi$ isn't \"changing\" the data, and so $\\phi(x)$ should be no more or less likely than $x$.  Formally then, we want\n",
      "$$ \\mathbb P(x \\in A) = \\mathbb P(\\phi(x)\\in A). $$\n",
      "By change of variables, this means that\n",
      "$$ p(x) = p(\\phi(x)) |\\det J(\\phi)|. $$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Combining this with the above, what we hope to be true is that:\n",
      "\n",
      "$$ \\frac{f(\\phi(x)|\\psi(\\theta)) |\\det J(\\phi)| p(\\theta)}{p(\\phi(x))|\\det J(\\phi)|} = \\frac{f(\\phi(x)|\\psi(\\theta)) p(\\psi(\\theta))}{p(\\phi(x))} |\\det J(\\psi)|. $$\n",
      "\n",
      "This cancels down to:\n",
      "\n",
      "$$ p(\\theta) = p(\\psi(\\theta)) |\\det J(\\psi)|. $$\n",
      "\n",
      "Thus we have derived a relationship which the _prior_ should have!"
     ]
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "For our example."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "We have $\\psi(\\mu,\\tau) = (b\\mu+a, b^{-2} \\tau)$ and so $|\\det J(\\psi)| = |b|^{-1}$.  Thus we need\n",
      "\n",
      "$$ p(\\mu,\\tau) = |b|^{-1} p(b\\mu+a, b^{-2} \\tau) $$\n",
      "\n",
      "for all $\\mu,\\tau,a,b$.  Set $b=1$ and $a=-\\mu$ to get\n",
      "\n",
      "$$ p(\\mu,\\tau) = p(0, \\tau) $$\n",
      "\n",
      "So $p(\\mu,\\tau) = h(\\tau)$ for some function $h$.  Then set $b = \\tau^{1/2}$ to find that\n",
      "\n",
      "$$ h(\\tau) = |b|^{-1} h(|b|^{-2}\\tau) \\implies h(\\tau) = \\tau^{-1/2} h(1) \\implies h(\\tau) \\propto \\tau^{-1/2} $$\n",
      "\n",
      "as $\\tau > 0$.  Thus $p(\\mu,\\tau) \\propto \\tau^{-1/2}$."
     ]
    },
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Some comments"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Firstly, let's check that at least the posterior is well-defined."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "posterior = density(Sxx, Sx, n, mu, tau) * tau**(-1/2)\n",
      "posterior"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{\\tau^{\\frac{n}{2}}}{\\tau^{0.5} \\left(2 \\pi\\right)^{\\frac{n}{2}}} e^{- \\frac{\\tau}{2} \\left(- 2 Sx \\mu + Sxx + \\mu^{2} n\\right)}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAARcAAAA4BAMAAAAlcKGwAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\ncHmvQ/JcuNvuMaDbpU7y/wGMaWFltd+AdAAAAABJRU5ErkJggg==\n",
       "prompt_number": 69,
       "text": [
        "                      \u239b                 2  \u239e \n",
        "       n      -n   -\u03c4\u22c5\u239d-2\u22c5Sx\u22c5\u03bc + Sxx + \u03bc \u22c5n\u23a0 \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\n",
        " -0.5  2       2               2             \n",
        "\u03c4    \u22c5\u03c4 \u22c5(2\u22c5\u03c0)   \u22c5\u212f                          "
       ]
      }
     ],
     "prompt_number": 69
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "int1 = sympy.integrate(posterior, (mu, -sympy.oo, sympy.oo))\n",
      "int1 = sympy.simplify(int1)\n",
      "int1"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{1}{\\sqrt{n}} 2^{- \\frac{n}{2} + \\frac{1}{2}} \\pi^{- \\frac{n}{2} + \\frac{1}{2}} \\tau^{\\frac{n}{2} - 1.0} e^{\\frac{\\tau}{2} \\left(\\frac{Sx^{2}}{n} - Sxx\\right)}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAU8AAAAxBAMAAACovDbEAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n9apexr8WMXIKCVfunQAAAABJRU5ErkJggg==\n",
       "prompt_number": 70,
       "text": [
        "                              \u239b  2      \u239e\n",
        "                              \u239cSx       \u239f\n",
        "                            \u03c4\u22c5\u239c\u2500\u2500\u2500 - Sxx\u239f\n",
        "   n   1    n   1  n          \u239d n       \u23a0\n",
        " - \u2500 + \u2500  - \u2500 + \u2500  \u2500 - 1.0  \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "   2   2    2   2  2              2      \n",
        "2       \u22c5\u03c0       \u22c5\u03c4       \u22c5\u212f             \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\n",
        "                    ___                  \n",
        "                  \u2572\u2571 n                   "
       ]
      }
     ],
     "prompt_number": 70
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "sympy.integrate(int1, (tau, 1, sympy.oo))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\begin{cases} \\frac{1}{\\sqrt{n}} \\left(2 \\pi\\right)^{- \\frac{n}{2} + \\frac{1}{2}} \\left(\\frac{2^{\\frac{n}{2}} e^{- \\frac{i \\pi}{2} n} \\Gamma{\\left(\\frac{n}{2} \\right)}}{\\operatorname{polar\\_lift}^{\\frac{n}{2}}{\\left (\\frac{Sx^{2}}{n} - Sxx \\right )}} + \\frac{2^{\\frac{n}{2} - 1} n e^{- \\frac{i \\pi}{2} n} \\Gamma{\\left(- \\frac{n}{2} \\right)} \\gamma\\left(\\frac{n}{2}, \\frac{e^{i \\pi}}{2} \\operatorname{polar\\_lift}{\\left (\\frac{Sx^{2}}{n} - Sxx \\right )}\\right)}{\\Gamma{\\left(- \\frac{n}{2} + 1 \\right)} \\operatorname{polar\\_lift}^{\\frac{n}{2}}{\\left (\\frac{Sx^{2}}{n} - Sxx \\right )}}\\right) & \\text{for}\\: \\left(\\left\\lvert{\\operatorname{periodic_{argument}}{\\left (e^{i \\pi} \\operatorname{polar\\_lift}{\\left (\\frac{Sx^{2}}{n} - Sxx \\right )},\\infty \\right )}}\\right\\rvert = \\frac{\\pi}{2} \\wedge - \\frac{n}{2} - \\frac{1}{2} > - \\frac{3}{2}\\right) \\vee \\left\\lvert{\\operatorname{periodic_{argument}}{\\left (e^{i \\pi} \\operatorname{polar\\_lift}{\\left (\\frac{Sx^{2}}{n} - Sxx \\right )},\\infty \\right )}}\\right\\rvert < \\frac{\\pi}{2} \\\\\\int_{1}^{\\infty} \\frac{1}{\\sqrt{n}} 2^{- \\frac{n}{2} + \\frac{1}{2}} \\pi^{- \\frac{n}{2} + \\frac{1}{2}} \\tau^{\\frac{n}{2} - 1.0} e^{\\frac{\\tau}{2} \\left(\\frac{Sx^{2}}{n} - Sxx\\right)}\\, d\\tau & \\text{otherwise} \\end{cases}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": 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2Bi6qE2exZaJxrUWHJ9g4OXK99tWJxRuVQi4BYOpEw6W6ZiMi/OgYro3zR0Ob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\nAElFTkSuQmCC\n",
       "prompt_number": 71,
       "text": [
        "\u23a7             \u239b                                                               \n",
        "\u23aa             \u239c                                             n        -\u2148\u22c5\u03c0\u22c5n   \n",
        "\u23aa             \u239c                                             \u2500 - 1    \u2500\u2500\u2500\u2500\u2500\u2500\u2500  \n",
        "\u23aa       n   1 \u239c n  -\u2148\u22c5\u03c0\u22c5n                                   2           2     \n",
        "\u23aa     - \u2500 + \u2500 \u239c \u2500  \u2500\u2500\u2500\u2500\u2500\u2500\u2500                   \u239b  2      \u239e   2     \u22c5n\u22c5\u212f       \u22c5\u0393\n",
        "\u23aa       2   2 \u239c 2     2     \u239bn\u239e           -n \u239cSx       \u239f                      \n",
        "\u23aa(2\u22c5\u03c0)       \u22c5\u239c2 \u22c5\u212f       \u22c5\u0393\u239c\u2500\u239f\u22c5polar_lift\u2500\u2500\u2500\u239c\u2500\u2500\u2500 - Sxx\u239f + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "\u23aa             \u239c             \u239d2\u23a0            2 \u239d n       \u23a0                      \n",
        "\u23aa             \u239c                                                               \n",
        "\u23aa             \u239d                                                               \n",
        "\u23aa\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\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",
        "\u23aa                                                                      ___    \n",
        "\u23aa                                                                    \u2572\u2571 n     \n",
        "\u23aa                                                                             \n",
        "\u23a8                                               \u221e                             \n",
        "\u23aa                                               \u2320                             \n",
        "\u23aa                                               \u23ae                             \n",
        "\u23aa                                               \u23ae                             \n",
        "\u23aa                                               \u23ae                             \n",
        "\u23aa                                               \u23ae    n   1    n   1  n        \n",
        "\u23aa                                               \u23ae  - \u2500 + \u2500  - \u2500 + \u2500  \u2500 - 1.0  \n",
        "\u23aa                                               \u23ae    2   2    2   2  2        \n",
        "\u23aa                                               \u23ae 2       \u22c5\u03c0       \u22c5\u03c4       \u22c5\u212f\n",
        "\u23aa                                               \u23ae \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",
        "\u23aa                                               \u23ae                     ___     \n",
        "\u23aa                                               \u23ae                   \u2572\u2571 n      \n",
        "\u23aa                                               \u2321                             \n",
        "\u23aa                                               1                             \n",
        "\u23a9                                                                             \n",
        "\n",
        "       \u239b                  \u239b  2      \u239e\u239e                         \u239e              \n",
        "       \u239c    \u2148\u22c5\u03c0           \u239cSx       \u239f\u239f                         \u239f              \n",
        "       \u239c   \u212f   \u22c5polar_lift\u239c\u2500\u2500\u2500 - Sxx\u239f\u239f              \u239b  2      \u239e\u239f              \n",
        "\u239b-n \u239e  \u239cn                 \u239d n       \u23a0\u239f           -n \u239cSx       \u239f\u239f              \n",
        "\u239c\u2500\u2500\u2500\u239f\u22c5\u03b3\u239c\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\u239f\u22c5polar_lift\u2500\u2500\u2500\u239c\u2500\u2500\u2500 - Sxx\u239f\u239f              \n",
        "\u239d 2 \u23a0  \u239d2              2             \u23a0            2 \u239d n       \u23a0\u239f              \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\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f              \n",
        "                  \u239b  n    \u239e                                    \u239f              \n",
        "                 \u0393\u239c- \u2500 + 1\u239f                                    \u239f      \u239b\u2502      \n",
        "                  \u239d  2    \u23a0                                    \u23a0      \u239c\u2502      \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\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500  for \u239c\u2502period\n",
        "                                                                      \u239d\u2502      \n",
        "                                                                              \n",
        "                                                                              \n",
        "                                                                              \n",
        "                                                                              \n",
        "  \u239b  2      \u239e                                                                 \n",
        "  \u239cSx       \u239f                                                                 \n",
        "\u03c4\u22c5\u239c\u2500\u2500\u2500 - Sxx\u239f                                                                 \n",
        "  \u239d n       \u23a0                                                                 \n",
        "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500                                                                 \n",
        "      2                                                                       \n",
        "                                                                              \n",
        "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 d\u03c4                                                              \n",
        "                                                                              \n",
        "                                                                              \n",
        "                                                                              \n",
        "                                                                              \n",
        "                                                                              \n",
        "\n",
        "                                                                              \n",
        "                                                                              \n",
        "                                                                              \n",
        "                                                                              \n",
        "                                                                              \n",
        "                                                                              \n",
        "                                                                              \n",
        "                                                                              \n",
        "           \u239b               \u239b  2      \u239e   \u239e\u2502                     \u239e   \u2502         \n",
        "           \u239c \u2148\u22c5\u03c0           \u239cSx       \u239f   \u239f\u2502   \u03c0     n   1       \u239f   \u2502         \n",
        "ic_argument\u239c\u212f   \u22c5polar_lift\u239c\u2500\u2500\u2500 - Sxx\u239f, \u221e\u239f\u2502 = \u2500 \u2227 - \u2500 - \u2500 > -3/2\u239f \u2228 \u2502periodic_\n",
        "           \u239d               \u239d n       \u23a0   \u23a0\u2502   2     2   2       \u23a0   \u2502         \n",
        "                                                                              \n",
        "                                                                              \n",
        "                                                                              \n",
        "                                                                              \n",
        "                                                                              \n",
        "                                                                              \n",
        "                                                                              \n",
        "                                                                              \n",
        "                                                                              \n",
        "                                                                              \n",
        "                                                                              \n",
        "                                                  otherwise                   \n",
        "                                                                              \n",
        "                                                                              \n",
        "                                                                              \n",
        "                                                                              \n",
        "                                                                              \n",
        "\n",
        "                                            \n",
        "                                            \n",
        "                                            \n",
        "                                            \n",
        "                                            \n",
        "                                            \n",
        "                                            \n",
        "                                            \n",
        "        \u239b               \u239b  2      \u239e   \u239e\u2502    \n",
        "        \u239c \u2148\u22c5\u03c0           \u239cSx       \u239f   \u239f\u2502   \u03c0\n",
        "argument\u239c\u212f   \u22c5polar_lift\u239c\u2500\u2500\u2500 - Sxx\u239f, \u221e\u239f\u2502 < \u2500\n",
        "        \u239d               \u239d n       \u23a0   \u23a0\u2502   2\n",
        "                                            \n",
        "                                            \n",
        "                                            \n",
        "                                            \n",
        "                                            \n",
        "                                            \n",
        "                                            \n",
        "                                            \n",
        "                                            \n",
        "                                            \n",
        "                                            \n",
        "                                            \n",
        "                                            \n",
        "                                            \n",
        "                                            \n",
        "                                            \n",
        "                                            "
       ]
      }
     ],
     "prompt_number": 71
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Again, `sympy` defeats me: I'm not sure how to tell it $|S_{x}| = |\\sum x_i| \\leq n^{1/2} S_{xx}^{1/2}$ by Cauchy-Schwarz and so $S_x^2 / n \\leq S_{xx}$."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "alpha = sympy.symbols(\"alpha\", real = True, positive = True)\n",
      "int2 = (1/sympy.sqrt(n)) * (2*sympy.pi)**((1-n)/2) * tau**((n-2)/2) * sympy.exp(-tau * alpha / 2)\n",
      "int2"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{\\tau^{\\frac{n}{2} - 1}}{\\sqrt{n} e^{\\frac{\\alpha \\tau}{2}}} \\left(2 \\pi\\right)^{- \\frac{n}{2} + \\frac{1}{2}}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAJ8AAAA0BAMAAABm7vf+AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nCJqB9/zmAAAAAElFTkSuQmCC\n",
       "prompt_number": 72,
       "text": [
        " n            n   1  -\u03b1\u22c5\u03c4 \n",
        " \u2500 - 1      - \u2500 + \u2500  \u2500\u2500\u2500\u2500\u2500\n",
        " 2            2   2    2  \n",
        "\u03c4     \u22c5(2\u22c5\u03c0)       \u22c5\u212f     \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\n",
        "            ___           \n",
        "          \u2572\u2571 n            "
       ]
      }
     ],
     "prompt_number": 72
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "sympy.simplify( sympy.integrate(int2, (tau,0,sympy.oo)) )"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{2 \\sqrt{2}}{\\alpha^{\\frac{n}{2}} n^{\\frac{3}{2}}} \\pi^{- \\frac{n}{2} + \\frac{1}{2}} \\Gamma{\\left(\\frac{n}{2} + 1 \\right)}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAANIAAAAyBAMAAADb85jzAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 73,
       "text": [
        "           n   1  -n          \n",
        "         - \u2500 + \u2500  \u2500\u2500\u2500         \n",
        "    ___    2   2   2   \u239bn    \u239e\n",
        "2\u22c5\u2572\u2571 2 \u22c5\u03c0       \u22c5\u03b1   \u22c5\u0393\u239c\u2500 + 1\u239f\n",
        "                       \u239d2    \u23a0\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\n",
        "              3/2             \n",
        "             n                "
       ]
      }
     ],
     "prompt_number": 73
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def normalise_my_prior(alpha, n):\n",
      "    return 2 * sympy.sqrt(2) * sympy.pi**((1-n)/2) * sympy.gamma((n+2)/2) / ( alpha**(n/2) * n * sympy.sqrt(n) )\n",
      "\n",
      "normalise_my_prior(alpha, n)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{2 \\sqrt{2}}{\\alpha^{\\frac{n}{2}} n^{\\frac{3}{2}}} \\pi^{- \\frac{n}{2} + \\frac{1}{2}} \\Gamma{\\left(\\frac{n}{2} + 1 \\right)}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAANIAAAAyBAMAAADb85jzAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 74,
       "text": [
        "           n   1  -n          \n",
        "         - \u2500 + \u2500  \u2500\u2500\u2500         \n",
        "    ___    2   2   2   \u239bn    \u239e\n",
        "2\u22c5\u2572\u2571 2 \u22c5\u03c0       \u22c5\u03b1   \u22c5\u0393\u239c\u2500 + 1\u239f\n",
        "                       \u239d2    \u23a0\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\n",
        "              3/2             \n",
        "             n                "
       ]
      }
     ],
     "prompt_number": 74
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "sympy.simplify( int1 / normalise_my_prior(alpha, n) )"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{\\alpha^{\\frac{n}{2}} \\tau^{\\frac{n}{2} - 1.0}}{2^{\\frac{n}{2}} \\Gamma{\\left(\\frac{n}{2} \\right)}} e^{\\frac{\\tau}{2} \\left(\\frac{Sx^{2}}{n} - Sxx\\right)}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAM0AAAA4BAMAAACszWHwAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nAAAAAElFTkSuQmCC\n",
       "prompt_number": 75,
       "text": [
        "                    \u239b  2      \u239e\n",
        "                    \u239cSx       \u239f\n",
        "                  \u03c4\u22c5\u239c\u2500\u2500\u2500 - Sxx\u239f\n",
        " -n   n  n          \u239d n       \u23a0\n",
        " \u2500\u2500\u2500  \u2500  \u2500 - 1.0  \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "  2   2  2              2      \n",
        "2   \u22c5\u03b1 \u22c5\u03c4       \u22c5\u212f             \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\n",
        "               \u239bn\u239e             \n",
        "              \u0393\u239c\u2500\u239f             \n",
        "               \u239d2\u23a0             "
       ]
      }
     ],
     "prompt_number": 75
    },
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Comparison with the Jeffreys prior"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The [Jeffreys Prior]() is one way to construct a \"noninformative\" prior which is \"invariant\": what this means is that you get given a prodecure to follow to construct a prior, and if you then transform your coordinates, and follow the procedure in the new coorindate system, then you get a prior which is the transformed version of your previous prior.  (There is some category theory here: this is a \"covariant\" prior in the catgeory sense, not the stats sense, of the word).\n",
      "\n",
      "From [Duke lecture notes](https://stat.duke.edu/courses/Fall11/sta114/jeffreys.pdf) we find that $p(s) = s^{-3/2}$ if $s=\\sigma^2$.  From [Berkeley lecture notes](http://www.cs.berkeley.edu/~jordan/courses/260-spring10/lectures/lecture7.pdf) we find $p(\\sigma) = \\sigma^{-2}$, though we're warned that this gives \"poor convergence properties\", though I don't fully understand what that means.\n",
      "\n",
      "Let's just check that there are consistent:\n",
      "\n",
      "$$ p_\\sigma(\\sigma) = p_s(s) \\Big|\\frac{ds}{d\\sigma}\\Big| = 2 s^{-3/2} \\sigma \\propto \\sigma^{-2}. $$\n",
      "\n",
      "My prior is in terms of $\\tau = \\sigma^{-2}$ and is $p(\\tau) = \\tau^{-1/2}$.  Thus\n",
      "\n",
      "$$ p_\\sigma(\\sigma) = p_\\tau(\\tau) \\Big|\\frac{d\\tau}{d\\sigma}\\Big| = 2\\tau^{-1/2} \\sigma^{-3} \\propto \\sigma^{-2}. $$\n",
      "\n",
      "So, this agrees with the Jeffreys prior.\n",
      "\n",
      "For comparison, if $p_\\sigma(\\sigma) = \\sigma^{-1}$ then $p_\\tau(\\tau) = \\sigma^{-1} \\Big|\\frac{d\\sigma}{d\\tau}\\Big|\n",
      "= \\sigma^{-1} \\frac12 \\sigma^3 \\propto \\sigma^2 = 1/\\tau$."
     ]
    },
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Convergence"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "The posterior distribution is then\n",
      "\n",
      "$$ p(\\mu,\\tau | x) = p(x | \\mu,\\tau) p(\\mu,\\tau) = \\Big(\\frac{\\tau}{2\\pi}\\Big)^{n/2} \\exp\\Big(-\\frac{\\tau}{2} \\sum (x_i-\\mu)^2 \\Big) \\tau^{-1/2}. $$\n",
      "\n",
      "With $\\alpha = S_{xx} - S_x^2 / n$, from the above calculation, the normalised posterior is\n",
      "\n",
      "$$ p(\\mu,\\tau | x) = \\frac{\\alpha^{n/2} n^{3/2}}{2\\sqrt 2} \\pi^{-1/2} \\Gamma\\Big(\\frac{n}{2}+1\\Big)^{-1} \\Big(\\frac{\\tau}{2}\\Big)^{n/2} \\exp\\Big(-\\frac{\\tau}{2} \\sum (x_i-\\mu)^2 \\Big) \\tau^{-1/2} $$\n",
      "\n",
      "What I do below is generate some samples from $N(0,1)$ to compare this distribution with the one from using the prior $p(\\sigma) = 1/\\sigma$.\n",
      "\n",
      "Firstly we find the normaisation for this new prior, repeating the work above."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "posterior = density(Sxx, Sx, n, mu, tau) * tau**(-1)\n",
      "sympy.simplify( sympy.integrate(posterior, (mu,-sympy.oo, sympy.oo)))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{1}{\\sqrt{n}} 2^{- \\frac{n}{2} + \\frac{1}{2}} \\pi^{- \\frac{n}{2} + \\frac{1}{2}} \\tau^{\\frac{n}{2} - \\frac{3}{2}} e^{\\frac{\\tau}{2} \\left(\\frac{Sx^{2}}{n} - Sxx\\right)}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
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       "prompt_number": 80,
       "text": [
        "                            \u239b  2      \u239e\n",
        "                            \u239cSx       \u239f\n",
        "                          \u03c4\u22c5\u239c\u2500\u2500\u2500 - Sxx\u239f\n",
        "   n   1    n   1  n   3    \u239d n       \u23a0\n",
        " - \u2500 + \u2500  - \u2500 + \u2500  \u2500 - \u2500  \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "   2   2    2   2  2   2        2      \n",
        "2       \u22c5\u03c0       \u22c5\u03c4     \u22c5\u212f             \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\n",
        "                   ___                 \n",
        "                 \u2572\u2571 n                  "
       ]
      }
     ],
     "prompt_number": 80
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "int1 = n**(-1/2) * (2*sympy.pi)**((1-n)/2) * tau**((n-3)/2) * sympy.exp(-tau * alpha / 2)\n",
      "sympy.integrate(int1, (tau,0,sympy.oo))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\begin{cases} \\frac{2}{\\alpha n^{0.5}} 2^{- \\frac{n}{2} + \\frac{1}{2}} \\pi^{- \\frac{n}{2} + \\frac{1}{2}} \\left(\\frac{\\alpha}{2}\\right)^{- \\frac{n}{2} + \\frac{3}{2}} \\Gamma{\\left(\\frac{n}{2} - \\frac{1}{2} \\right)} & \\text{for}\\: - \\frac{n}{2} + \\frac{3}{2} < 1 \\\\\\int_{0}^{\\infty} \\frac{\\tau^{\\frac{n}{2} - \\frac{3}{2}}}{n^{0.5} e^{\\frac{\\alpha \\tau}{2}}} \\left(2 \\pi\\right)^{- \\frac{n}{2} + \\frac{1}{2}}\\, d\\tau & \\text{otherwise} \\end{cases}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": 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       "prompt_number": 81,
       "text": [
        "\u23a7                               n   3                          \n",
        "\u23aa     n   1    n   1          - \u2500 + \u2500                          \n",
        "\u23aa   - \u2500 + \u2500  - \u2500 + \u2500            2   2                          \n",
        "\u23aa     2   2    2   2  -0.5 \u239b\u03b1\u239e         \u239bn   1\u239e                 \n",
        "\u23aa2\u22c52       \u22c5\u03c0       \u22c5n    \u22c5\u239c\u2500\u239f       \u22c5\u0393\u239c\u2500 - \u2500\u239f                 \n",
        "\u23aa                          \u239d2\u23a0         \u239d2   2\u23a0        n   3    \n",
        "\u23aa\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  for - \u2500 + \u2500 < 1\n",
        "\u23aa                      \u03b1                              2   2    \n",
        "\u23a8                                                              \n",
        "\u23aa    \u221e                                                         \n",
        "\u23aa    \u2320                                                         \n",
        "\u23aa    \u23ae        n   3        n   1  -\u03b1\u22c5\u03c4                         \n",
        "\u23aa    \u23ae        \u2500 - \u2500      - \u2500 + \u2500  \u2500\u2500\u2500\u2500\u2500                        \n",
        "\u23aa    \u23ae  -0.5  2   2        2   2    2                          \n",
        "\u23aa    \u23ae n    \u22c5\u03c4     \u22c5(2\u22c5\u03c0)       \u22c5\u212f      d\u03c4         otherwise   \n",
        "\u23aa    \u2321                                                         \n",
        "\u23a9    0                                                         "
       ]
      }
     ],
     "prompt_number": 81
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def normalise_other_prior(alpha, n):\n",
      "    return ( 2 / (alpha * n**(1/2)) ) * (2*sympy.pi)**((1-n)/2) * (alpha/2)**((3-n)/2) * sympy.gamma((n-1)/2)\n",
      "\n",
      "normalise_other_prior(alpha, n)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{2}{\\alpha n^{0.5}} \\left(\\frac{\\alpha}{2}\\right)^{- \\frac{n}{2} + \\frac{3}{2}} \\left(2 \\pi\\right)^{- \\frac{n}{2} + \\frac{1}{2}} \\Gamma{\\left(\\frac{n}{2} - \\frac{1}{2} \\right)}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAVsAAAA0BAMAAADWNIblAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIpnNu0SrdlQQ3e8y\niWbzIQYJAAAHWklEQVRoBc1YfYhUVRQ/M7Pz/amCUEI7qf2hYZoryVbmI/pAjJgMoQ/IBckwCicT\nF8vaoQL/KPD9If4jMUtERRS70ZeRxdAf/iMxi9CnLI1FBYXsWmhtlNs559775t47b95TG9cu7L3n\nd86555x5++655x2Anoz48MM9sTNHRmKVB+bIU0/cRGDyP9v5ItDCQTdQHCr8ft9PrJPi+fNVoRtC\nFKITgQrxQUscWbymYrGC4D4ny+I+oXRvkO75yL7srrSBRD9Y8oWQ/NNiBcFEYwGLOdw8FEtByuGy\n1GhXnQMzJEqUTYXtAGMmJxBl4Nn7SYHDPQT9gcrhwkWNbjoHt3G4cNhU+NiFHRfwNhyAUy0ywOEW\nhh8zjV0wWtd9R1KE+6JrqKx3YWVnuMmWUkrj4/cZ8t31kQSxEqZvSImQfLfIcGNlWzpmGUH5LvxL\nT+6m3yFOU+o2HBugOItjGrYO3DQwUAUCtrEgnDtRMsXZCcTxXfOGHzf5hGS48dOWKPePxQDgE3Cs\nkj6SbkHS0cXxZ+5iyE9XAV0hkM5Y4U65qJ5PrIV6o2OfDBdWW5KYYzEADjXw4X4IcEcS7d2ii2Nw\njCGHq4CuYNEqT9eYb4f7GnE3ZxyoV1iuTypc/kmaYKlGS/JWXBPTAMvzSPSXNIWMW8RfII+aAprc\nJlWedlhghRs/Q1wXY93PRllHTSrcYlVxeE0OGZBAnE5A0QE4sQeJ6BBO7SHKBH66AKE1g8rTDhuw\nwo2OMxdj3dq2rygVbmJUcXh9DiIlgwHAp7Eff9V3NZREjJc9fT0rJ3iWgGn/SeVph8VWuNkyc5cA\nnEszpU8q3JThPleGgh1ukxjFFj5d3m+87EbRYQDdl0erPO0QJ/f0156AiGaN4SdQmKG3zhwq3PQf\nOv/KycnlOiaaGXj9FXbkXIRTmjy97OU2MkCb7Uc5fswRsg6RM5B3XuqQq3DhVV00Njv7l46JxpyA\nY8Xwnr6viKjXaBYjo2t7IF5Vclp/xL8XKE+7Mk8nBgbeHVgLHXl6I++Kj0PkzhqT2hS99vfrBNyh\ncf3I3N8mN1M2cSc6CjB/ye2KH20oCk+tzNMOsSQ4OvbBtkcp2b9HzNAxQq9mwEhOm0LrbJpCRoMQ\nGYKRmpKsUgSeWpmnHWIp0BwC2I2ZvfN+Ii177K/ZHBP3jZrYDt+UIkpWAauBjPcrT7U1VGp2iKUA\nhYupJX6OmKGjWQ1WwVvGGHHr5TCEBLIuZD+CBCd9wodoUkOkZkdAAShcTAWF8yu062Wxt9tctOS5\nMLNYBcVmIOqF21drm5apeYg5ElC4+PWUDHsMwkrREWu3uW49fTPz+exawbzMDKzEEu0sPrlyW8lI\nzRJQuDiixgXALL8pqx0FP3l/y+KGHQlRBDUnonv3XXWyhE+u7cBIzQrIcBMB1a4WQWZUAz5ks2Ux\nf7OwDd9nxhq4D55IYrSAmVQNLzUTQwEV7rTSClwz45ywqbi2BxsQd6MUkaWwcPmljTqo6XAtkQuJ\nwwxXOKItkRvfpFFBkr8DRO0fG0ccMES4msJ5hYuJFLNvhrZdWLiaI18yNNyWtS0s3MOonyrDPIi2\nirRVexksSwKaT9dXRWPiy2COyZ2EN13xs2BT6WiMsKN2N2p/BnAS6m6xgXRhlbG9A8hwoxd31JIT\n8SE0uWWWZhzFCbGqORJ2+xzH1+DtyeOj8JTYmy+rrf6rCvfiElmmAfQ7H7pHGpdVs+cq9JrAuyqD\nB2YUP+v6yrgtVvH2+hHz15+9mvgXeU3UXXgDd29WpuW3ptc9M+t5paWt2ZIGkDQuYVOko4K6BrVq\nzpN73sG+hJsuvI5qm4aXCWX5CeV1z/LTnhF/omC9PVwl+6vqXFXiGNWcUvC8g13ibBHhzoPnha7s\nrWxX3bO+VcpGt3XQEKRD9YW6KiCNak5Z8ryDXUDKp4ufDQ1WTouSxuueWZ/OyqC2LtRorA1LBuwO\nZHluVHNK2/MOdnleL3Gf72YoVoTyal7Wq+5ZM9R9pKq80LpXB0H0RiE0qjml73kH+vgRDTDRnso2\ngFLKEa8B8qvaM+YS9a2CvV5H2DxZxWqucwjv/IkkGmCiPYV514kMwjf0QcJDtYBk9+zTTku94bRv\n++ZEp0XpncpX2QCT7anhnW7uNOSPqz15R2wWGS013mmqN5x2gl/jY1Dm0wL+52UDrEt7Ki3L/KVs\nJNPysdUTlsyYWKg7PvaEd6APW/qgoAbYft/2FDzJu5NDvPzC86WYvI9A9RbqTqR3vtZVA2yJb3sK\nYi3aKLpnkXd0I72lXxHmUlTN2UP17qbwPBZbogHm356C9Fu4OSe6Z7Gabah3eEqY4mrOsiq9A9yA\nAtkA69KeAlhQApDds0csO72E2SpZE9WcZVf17kSzXzTAsI72aU+RhQrewNw9SyN1yYZIOqKas5yo\n3h2XeJbsssF1oZ4fdENV5k5hUSnMF5W2/5uhdST8Y4qW/fmXibs4xK+saUO05kzcVw10FTcr6UDd\nORFeE+jlAB+0fwEmXPcQsvHMgAAAAABJRU5ErkJggg==\n",
       "prompt_number": 82,
       "text": [
        "             n   3                      \n",
        "           - \u2500 + \u2500        n   1         \n",
        "             2   2      - \u2500 + \u2500         \n",
        "   -0.5 \u239b\u03b1\u239e               2   2  \u239bn   1\u239e\n",
        "2\u22c5n    \u22c5\u239c\u2500\u239f       \u22c5(2\u22c5\u03c0)       \u22c5\u0393\u239c\u2500 - \u2500\u239f\n",
        "        \u239d2\u23a0                      \u239d2   2\u23a0\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\n",
        "                   \u03b1                    "
       ]
      }
     ],
     "prompt_number": 82
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "sympy.simplify( int1 / normalise_other_prior(alpha, n) )"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{2^{- \\frac{n}{2} + \\frac{1}{2}} \\alpha^{\\frac{n}{2} - \\frac{1}{2}} \\tau^{\\frac{n}{2} - \\frac{3}{2}}}{e^{\\frac{\\alpha \\tau}{2}} \\Gamma{\\left(\\frac{n}{2} - \\frac{1}{2} \\right)}}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAKkAAAA6BAMAAAA5CWTxAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nn4mH+JgCAAAAAElFTkSuQmCC\n",
       "prompt_number": 84,
       "text": [
        "   n   1  n   1  n   3  -\u03b1\u22c5\u03c4 \n",
        " - \u2500 + \u2500  \u2500 - \u2500  \u2500 - \u2500  \u2500\u2500\u2500\u2500\u2500\n",
        "   2   2  2   2  2   2    2  \n",
        "2       \u22c5\u03b1     \u22c5\u03c4     \u22c5\u212f     \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",
        "            \u239bn   1\u239e          \n",
        "           \u0393\u239c\u2500 - \u2500\u239f          \n",
        "            \u239d2   2\u23a0          "
       ]
      }
     ],
     "prompt_number": 84
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "import numpy as np\n",
      "%matplotlib inline\n",
      "import matplotlib.pyplot as plt"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 85
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def make_example(n):\n",
      "    x = np.random.normal(size = n)\n",
      "    Sxx = np.sum(x*x)\n",
      "    Sx = np.sum(x)\n",
      "\n",
      "    def posterior(mu, tau):\n",
      "        return ( (tau / (2*sympy.pi))**(n/2) * sympy.exp( - tau * (Sxx - 2*mu*Sx + n*mu*mu) / 2 ) * tau**(-1/2)\n",
      "                / normalise_my_prior(Sxx-Sx*Sx/n, n) )\n",
      "\n",
      "    def posterior1(mu, tau):\n",
      "        return ( (tau / (2*sympy.pi))**(n/2) * sympy.exp( - tau * (Sxx - 2*mu*Sx + n*mu*mu) / 2 ) * tau**(-1)\n",
      "                / normalise_other_prior(Sxx-Sx*Sx/n, n) )\n",
      "\n",
      "    return posterior, posterior1"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 86
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "posterior, posterior1 = make_example(10)\n",
      "mubins = np.linspace(-1, 1, 50)\n",
      "taubins = np.linspace(0.1, 2, 50)\n",
      "\n",
      "# Slow as our functions use sympy primitives not numpy primitives.\n",
      "Z = np.zeros([len(mubins),len(taubins)])\n",
      "Z1 = np.zeros([len(mubins),len(taubins)])\n",
      "for x, mux in enumerate(mubins):\n",
      "    for y, tauy in enumerate(taubins):\n",
      "        Z[x,y] = posterior(mux, tauy)\n",
      "        Z1[x,y] = posterior1(mux, tauy)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 87
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plt.contour(mubins, taubins, Z.T)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 88,
       "text": [
        "<matplotlib.contour.QuadContourSet at 0x17e40c50>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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dBbCjscZFMqtglAmPvdGMxy5RpbAnviEhNFc5dzmipooKhoSZuNDu3qZCMY0m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\nYgx2SSrmA2Xl6ItKEpdWAAAAAElFTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x151eb9b0>"
       ]
      }
     ],
     "prompt_number": 88
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plt.contour(mubins, taubins, Z1.T)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 89,
       "text": [
        "<matplotlib.contour.QuadContourSet at 0x17e9e198>"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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VpmAdjBT2bsLOnZnMnBmDg+gx0VtobIHDedovtghFFOGJJ14IHO0vPArBY8Qu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\npTebHW/zkAAAAABJRU5ErkJggg==\n",
       "text": [
        "<matplotlib.figure.Figure at 0x15047ba8>"
       ]
      }
     ],
     "prompt_number": 89
    },
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Marginalise out $\\mu$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Comparing two dimensional distributions is hard: what do \"confidence intervals\" mean, for example?  So, let's marginalise out $\\mu$ and get the posterior distribution on $\\tau$.  We copy some formula from above."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "posterior = (alpha * tau / 2)**(n/2) / (tau * sympy.gamma(n/2)) * sympy.exp(-alpha * tau / 2)\n",
      "posterior"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{\\left(\\frac{\\alpha \\tau}{2}\\right)^{\\frac{n}{2}}}{\\tau e^{\\frac{\\alpha \\tau}{2}} \\Gamma{\\left(\\frac{n}{2} \\right)}}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAF0AAAA+BAMAAAC7PwoEAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\ngg==\n",
       "prompt_number": 94,
       "text": [
        "     n       \n",
        "     \u2500  -\u03b1\u22c5\u03c4 \n",
        "     2  \u2500\u2500\u2500\u2500\u2500\n",
        "\u239b\u03b1\u22c5\u03c4\u239e     2  \n",
        "\u239c\u2500\u2500\u2500\u239f \u22c5\u212f     \n",
        "\u239d 2 \u23a0        \n",
        "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "       \u239bn\u239e   \n",
        "    \u03c4\u22c5\u0393\u239c\u2500\u239f   \n",
        "       \u239d2\u23a0   "
       ]
      }
     ],
     "prompt_number": 94
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "posterior1 = (alpha * tau / 2)**((n-1)/2) / (tau * sympy.gamma((n-1)/2)) * sympy.exp(-alpha * tau / 2)\n",
      "posterior1"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\frac{\\left(\\frac{\\alpha \\tau}{2}\\right)^{\\frac{n}{2} - \\frac{1}{2}}}{\\tau e^{\\frac{\\alpha \\tau}{2}} \\Gamma{\\left(\\frac{n}{2} - \\frac{1}{2} \\right)}}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAIMAAABBBAMAAAAAkwrqAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n8BO9K0v7mHafQgAAAABJRU5ErkJggg==\n",
       "prompt_number": 95,
       "text": [
        "     n   1       \n",
        "     \u2500 - \u2500  -\u03b1\u22c5\u03c4 \n",
        "     2   2  \u2500\u2500\u2500\u2500\u2500\n",
        "\u239b\u03b1\u22c5\u03c4\u239e         2  \n",
        "\u239c\u2500\u2500\u2500\u239f     \u22c5\u212f     \n",
        "\u239d 2 \u23a0            \n",
        "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n",
        "       \u239bn   1\u239e   \n",
        "    \u03c4\u22c5\u0393\u239c\u2500 - \u2500\u239f   \n",
        "       \u239d2   2\u23a0   "
       ]
      }
     ],
     "prompt_number": 95
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "So actually the only difference is that the \"other\" prior multiplies by $\\tau^{-1/2}$."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from scipy.special import gamma\n",
      "\n",
      "def make_example(n):\n",
      "    x = np.random.normal(size=n)\n",
      "    alpha = np.sum(x*x) - np.sum(x)**2/n\n",
      "\n",
      "    def p1(tau):\n",
      "        return (alpha * tau / 2)**(n/2) / (tau * gamma(n/2)) * np.exp(-alpha * tau / 2)\n",
      "    def p2(tau):\n",
      "        return (alpha * tau / 2)**((n-1)/2) / (tau * gamma((n-1)/2)) * np.exp(-alpha * tau / 2)\n",
      "    \n",
      "    return p1, p2\n",
      "\n",
      "p1, p2 = make_example(10)\n",
      "x = np.linspace(0.1, 2, 50)\n",
      "y1 = p1(x)\n",
      "y2 = p2(x)\n",
      "plt.plot(x, y1)\n",
      "plt.plot(x, y2)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 110,
       "text": [
        "[<matplotlib.lines.Line2D at 0x19b0f9e8>]"
       ]
      },
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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SaodSqoZrQ/QMc/fNpV3Zdihl/Res1t41tz0lmQMy83rD1/lg1Qcptnv+eYiKkrozQqSV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\nc6efaVm6JbW/qc3qyNVO9S9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TLPE1cSeHn6XW+rLWOjbx5wVAoFIqHSpOewX5XrqQfDed\np5QKBGYCU7XWs5No4tR31NXJ/daiJ6VUJsyipzl3tZkDPAW3VsAmuehJOP4slVIFVeIuBEqpupip\nrefcH6pXkO+lC8l30zmJn9VE4C+t9bhkmjn1HXXpsIyWRU8uY+WzBB4DnldKxWNq6XdJt4AzOKXU\nz0AzIFgpdRR4BzMLSb6XqeDo80S+m85qBDwJ7FRK/bsfxhCgBKTuOyqLmIQQwgvJFtNCCOGFJLkL\nIYQXkuQuhBBeSJK7EEJ4IUnuQgjhhSS5CyGEF5LkLoQQXkiSuxBCeKH/A+VAcC30j5WNAAAAAElF\nTkSuQmCC\n",
       "text": [
        "<matplotlib.figure.Figure at 0x19b0f828>"
       ]
      }
     ],
     "prompt_number": 110
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": []
    }
   ],
   "metadata": {}
  }
 ]
}