{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "$\\newcommand{\\ind}[1]{\\left[#1\\right]}$\n", "\n", "# Model\n", "\n", "- Data set\n", "$$ {\\cal D} = \\{ x_1, \\dots x_N \\} $$\n", "- Model with parameter $\\theta$\n", "$$ p(\\cal D | \\theta) $$\n", "\n", "\n", "\n", "\n", "## Maximum Likelihood\n", "\n", "- Maximum Likelihood (ML)\n", "$$ \\theta^{\\text{ML}} = \\arg\\max_{\\theta} \\log p({\\cal D} | \\theta) $$\n", "- Predictive distribution\n", "$$ p(x_{N+1} | {\\cal D} ) \\approx p(x_{N+1} | \\theta^{\\text{ML}}) $$\n", "\n", "## Maximum Aposteriori\n", "\n", "- Prior\n", "$$ p(\\theta) $$\n", "\n", "- Maximum a-posteriori (MAP) : Regularised Maximum Likelihood\n", "$$\n", "\\theta^{\\text{MAP}} = \\arg\\max_{\\theta} \\log p({\\cal D} | \\theta) p(\\theta)\n", "$$\n", "\n", "- Predictive distribution\n", "$$ p(x_{N+1} | {\\cal D} ) \\approx p(x_{N+1} | \\theta^{\\text{MAP}}) $$\n", "\n", "## Bayesian Learning\n", "\n", "- We treat parameters on the same footing as all other variables\n", "- We integrate over unknown parameters rather than using point estimates (remember the many-dice example)\n", " - Self-regularisation, avoids overfitting\n", " - Natural setup for online adaptation\n", " - Model selection\n", "\n", "\n", "- Predictive distribution\n", "\\begin{eqnarray}\n", "p(x_{N+1} , {\\cal D} ) &=& \\int d\\theta \\;\\; p(x_{N+1} | \\theta) p( {\\cal D}| \\theta) p(\\theta) \\\\\n", " &=& \\int d\\theta \\;\\; p(x_{N+1}| \\theta) p( {\\cal D}, \\theta) \\\\\n", " &=& \\int d\\theta \\;\\; p(x_{N+1}| \\theta) p( \\theta| {\\cal D}) p({\\cal D}) \\\\\n", " &=& p({\\cal D}) \\int d\\theta \\;\\; p(x_{N+1}| \\theta) p( \\theta| {\\cal D}) \\\\\n", "p(x_{N+1} | {\\cal D} ) &=& \\int d\\theta \\;\\; p(x_{N+1} | \\theta) p(\\theta | {\\cal D}) \n", "\\end{eqnarray}\n", "\n", "The interpretation is that past data provides an 'update' to the recent prior to be used for the current prediction.\n", "\n", "- Bayesian learning is just inference ...\n", "\n", "\n", "\n", "\n", "# Learning the parameter of a (possibly fake) coin\n", "\n", "Suppose we have a coin, flipped several times independently. A vague question one can ask is if one can predict the outcome of the next flip.\n", "\n", "It depends. If we already know that the coin is fair, there is nothing that we can learn from past data and indeed the future flips are independent of the previous flips. However, if we don't know the probability of the coin, we could estimate the parameter from past data to create a better prediction. Mathematically, the model is identical to \n", "\n", "\n", "\n", "Here, $\\theta$ is the parameter of the coin.\n", "\n", "## Maximum Likelihood Estimation\n", "\n", "We observe the outcome of $N$ coin flips $\\{x^{(n)}\\}_{n=1\\dots N}$ where $x^{(n)} \\in \\left\\{0,1\\right\\}$. The model is a Bernoulli distribution with parameter $\\pi = (\\pi_0, \\pi_1)$. We have $\\pi_0 = 1 - \\pi_1$ where $0 \\leq \\pi_1 \\leq 1$. \n", "\n", "\\begin{eqnarray}\n", "x^{(n)} & \\sim & p(x|\\pi) = (1-\\pi_1)^{1-x^{(n)} } \\pi_1^{x^{(n)} }\n", "\\end{eqnarray}\n", "\n", "The loglikelihood is \n", "\n", "\\begin{eqnarray}\n", "{\\cal L}(\\pi_1) & = & \\sum_{n=1}^N (1- x^{(n)}) \\log (1 - \\pi_1) + \\sum_{n=1}^N x^{(n)} \\log (\\pi_1) \\\\\n", "& = & \\log (1 - \\pi_1) \\sum_{n=1}^N (1- x^{(n)}) + \\log (\\pi_1) \\sum_{n=1}^N x^{(n)} \n", "\\end{eqnarray}\n", "\n", "We define the number of $0$'s \n", "\\begin{eqnarray}\n", "c_0 = \\sum_{n=1}^N (1- x^{(n)})\n", "\\end{eqnarray}\n", "and $1$'s as\n", "\\begin{eqnarray}\n", "c_1 = \\sum_{n=1}^N x^{(n)}\n", "\\end{eqnarray}\n", "\n", "\\begin{eqnarray}\n", "{\\cal L}(\\pi_1) & = & \\log (1 - \\pi_1) c_0 + \\log (\\pi_1) c_1 \n", "\\end{eqnarray}\n", "\n", "We compute the gradient \n", "\\begin{eqnarray}\n", "\\frac{\\partial}{\\partial \\pi_1} {\\cal L}(\\pi_1) & = & - \\frac{c_0}{1 - \\pi_1} + \\frac{c_1}{\\pi_1} = 0 \n", "\\end{eqnarray}\n", "\n", "The solution is quite predictable\n", "\\begin{eqnarray}\n", "\\pi_1 & = &\\frac{c_1}{c_0 + c_1} = \\frac{c_1}{N} \n", "\\end{eqnarray}\n", "\n", "## Maximum A-posteriori estimation\n", "\n", "We need a prior over the probability parameter. One choice is the beta distribution\n", "\n", "\\begin{eqnarray}\n", "p(\\pi_1) & = & \\mathcal{B}(\\pi_1; \\alpha, \\beta) = \\frac{\\Gamma(\\alpha + \\beta)}{\\Gamma(\\alpha) \\Gamma(\\beta) } \\pi_1^{\\alpha-1} (1-\\pi_1)^{\\beta-1}\n", "\\end{eqnarray}\n", "\n", "The log joint ditribution of data is\n", "\\begin{eqnarray}\n", "\\log p(X, \\pi_1) & = & \\log p(\\pi_1) + \\sum_{n=1}^N \\log p(x^{(n)}|\\pi_1) \\\\\n", "& = & \\log \\Gamma(\\alpha + \\beta) -\\log \\Gamma(\\alpha) - \\log \\Gamma(\\beta) \\\\\n", "& & + (\\alpha-1) \\log \\pi_1 + (\\beta-1) \\log(1-\\pi_1) \\\\\n", "& & + c_1 \\log (\\pi_1) + c_0 \\log (1 - \\pi_1) \\\\\n", "& = & \\log \\Gamma(\\alpha + \\beta) -\\log \\Gamma(\\alpha) - \\log \\Gamma(\\beta) \\\\\n", "& & + (\\alpha + c_1 -1) \\log \\pi_1 + (\\beta + c_0 -1) \\log(1-\\pi_1) \n", "\\end{eqnarray}\n", "\n", "The gradient is \n", "\n", "\\begin{eqnarray}\n", "\\frac{\\partial}{\\partial \\pi_1} \\log p(X, \\pi_1) & = & - \\frac{\\beta + c_0 -1}{1 - \\pi_1} + \\frac{\\alpha + c_1 -1}{\\pi_1} = 0 \n", "\\end{eqnarray}\n", "\n", "We can solve for the parameter.\n", "\\begin{eqnarray}\n", "\\pi_1 (\\beta + c_0 -1) & = & (1 - \\pi_1) (\\alpha + c_1 -1) \\\\ \n", "\\pi_1 \\beta + \\pi_1 c_0 - \\pi_1 & = & \\alpha + c_1 - 1 - \\pi_1 \\alpha - \\pi_1 c_1 + \\pi_1 \\\\ \n", "\\pi_1 & = & \\frac{\\alpha - 1 + c_1}{\\alpha + \\beta - 2 + c_0 + c_1} \\\\ \n", "\\end{eqnarray}\n", "\n", "When the prior is flat, i.e., when $\\alpha = \\beta = 1$, MAP and ML solutions coincide.\n", "\n", "## Full Bayesian inference\n", "\n", "We infer the posterior\n", "\n", "\\begin{eqnarray}\n", "p(\\pi_1| X) & = & \\frac{p(\\pi_1, X)}{p(X)} \n", "\\end{eqnarray}\n", "\n", "\n", "The log joint density is \n", "\\begin{eqnarray}\n", "\\log p(X, \\pi_1) & = & \\log \\Gamma(\\alpha + \\beta) -\\log \\Gamma(\\alpha) - \\log \\Gamma(\\beta) \\\\\n", "& & + (\\alpha + c_1 -1) \\log \\pi_1 + (\\beta + c_0 -1) \\log(1-\\pi_1) \n", "\\end{eqnarray}\n", "\n", "At this stage, we may try to evaluate the integral \n", "$$\n", "p(X) = \\int d\\pi_1 p(X, \\pi_1) \n", "$$\n", "\n", "Rather than trying to evaluate this integral directly, a simple approach is known as 'completing the square': we add an substract terms to obtain an expression that corresponds to a known, normalized density. This typically involves adding and substracting an expression that will make us identify a normalized density. \n", "\n", "\\begin{eqnarray}\n", "\\log p(X, \\pi_1) & = & \\log \\Gamma(\\alpha + \\beta) -\\log \\Gamma(\\alpha) - \\log \\Gamma(\\beta) \\\\\n", "& & - \\log \\Gamma(\\alpha + \\beta + c_0 + c_1) + \\log \\Gamma(\\alpha + c_1) + \\log \\Gamma(\\beta + c_0) \\\\\n", "& & + \\log \\Gamma(\\alpha + \\beta + c_0 + c_1) - \\log \\Gamma(\\alpha + c_1) - \\log \\Gamma(\\beta + c_0) \\\\\n", "& & + (\\alpha + c_1 -1) \\log \\pi_1 + (\\beta + c_0 -1) \\log(1-\\pi_1) \\\\\n", "& = & \\log \\Gamma(\\alpha + \\beta) -\\log \\Gamma(\\alpha) - \\log \\Gamma(\\beta) \\\\\n", "& & - \\log \\Gamma(\\alpha + \\beta + c_0 + c_1) + \\log \\Gamma(\\alpha + c_1) + \\log \\Gamma(\\beta + c_0) \\\\\n", "& & + \\log \\mathcal{B}(\\alpha + c_1, \\beta + c_0) \\\\\n", "& = & \\log p(X) + \\log p(\\pi_1| X)\n", "\\end{eqnarray}\n", "\n", "From the resulting expression, taking the exponent on both sides we see that \n", "\\begin{eqnarray}\n", "p(\\pi_1| X) & = & \\mathcal{B}(\\alpha + c_1, \\beta + c_0) \\\\\n", "p(X) & = & \\frac{\\Gamma(\\alpha + \\beta)}{\\Gamma(\\alpha)\\Gamma(\\beta)} \\frac{\\Gamma(\\alpha + c_1)\\Gamma(\\beta + c_0)}{\\Gamma(\\alpha + \\beta + c_0 + c_1)}\n", "\\end{eqnarray}\n", "\n", "Predictive distribution: Let $a=\\alpha + c_1$ and $b=\\beta+c_0$\n", "\n", "\\begin{eqnarray}\n", "\\int d\\pi_1 p(x|\\pi_1) p(\\pi_1| X) & = & \\int d\\pi_1 \\mathcal{BE}(x; \\pi_1) \\mathcal{B}(\\pi_1; a, b) \\\\\n", " & = & \\int d\\pi_1 \\pi_1^{\\ind{x=1}}(1-\\pi_1)^{\\ind{x=0}}\\mathcal{B}(a, b) \\\\\n", "\\end{eqnarray}\n", "\n", "\\begin{eqnarray}\n", "p(x) & = & \\frac{\\Gamma(a + b)}{\\Gamma(a)\\Gamma(b)} \\frac{\\Gamma(a + \\ind{x=1})\\Gamma(b + \\ind{x=0})}{\\Gamma(b + a + 1)}\n", "\\end{eqnarray}\n", "\n", "\\begin{eqnarray}\n", "p(x=1) & = & \\frac{\\Gamma(a + b)}{\\Gamma(a)\\Gamma(b)} \\frac{\\Gamma(a + 1)\\Gamma(b)}{\\Gamma(b + a + 1)} \\\\\n", "& = & \\frac{\\Gamma(a + b)}{\\Gamma(a)\\Gamma(b)} \\frac{a\\Gamma(a)\\Gamma(b)}{(b+a)\\Gamma(b + a)} \\\\\n", "& = & \\frac{a}{a+b}\n", "\\end{eqnarray}\n", "\n", "\n", "## Alternative Derivation\n", "Alternatively, we may directly write\n", "\\begin{eqnarray}\n", "p(X, \\pi_1) & = & \\frac{\\Gamma(\\alpha + \\beta)}{\\Gamma(\\alpha)\\Gamma(\\beta)} \\pi_1^{(\\alpha + c_1 -1)} (1-\\pi_1)^{(\\beta + c_0 -1)} \n", "\\end{eqnarray}\n", "\n", "\\begin{eqnarray}\n", "p(X) &=& \\int d\\pi_1 p(X, \\pi_1) = \\frac{\\Gamma(\\alpha + \\beta)}{\\Gamma(\\alpha)\\Gamma(\\beta)} \\int d\\pi_1 \\pi_1^{(\\alpha + c_1 -1)} (1-\\pi_1)^{(\\beta + c_0 -1)} \n", "\\end{eqnarray}\n", "\n", "\n", "From the definition of the beta distribution, we can arrive at the 'formula' for the integral \n", "\\begin{eqnarray}\n", "1 &=& \\int d\\pi \\mathcal{B}(\\pi; a, b) \\\\\n", "& = & \\int d\\pi \\frac{\\Gamma(a + b)}{\\Gamma(a)\\Gamma(b)} \\pi^{(a -1)} (1-\\pi)^{(b -1)} \\\\\n", "\\frac{\\Gamma(a)\\Gamma(b)}{\\Gamma(a + b)} & = & \\int d\\pi \\pi^{(a -1)} (1-\\pi)^{(b -1)}\n", "\\end{eqnarray}\n", "Just substitute $a = \\alpha + c_1$ and $b = \\beta + c_0$\n", "\n", "## An Approximation\n", "For large $x$, we have the following approximation\n", "\\begin{eqnarray}\n", "\\log \\Gamma(x + a) - \\log \\Gamma(x) & \\approx & a \\log(x) \\\\\n", "\\Gamma(x + a) & \\approx & \\Gamma(x) x^a \\\\\n", "\\end{eqnarray}\n", "\n", "When $c_0$ and $c_1$ are large, we obtain:\n", "\n", "\\begin{eqnarray}\n", "p(X) & \\approx & \\frac{\\Gamma(\\alpha + \\beta)}{\\Gamma(\\alpha)\\Gamma(\\beta)} \\frac{\\Gamma(c_1)\\Gamma(c_0)c_0^{\\beta}c_1^{\\alpha}}{\\Gamma(c_0 + c_1)(c_0+c_1)^{\\alpha + \\beta}}\n", "\\end{eqnarray}\n", "\n", "Let $\\hat{\\pi}_1 = c_1/(c_0+c_1)$ and $N = c_0 + c_1$, we have\n", "\\begin{eqnarray}\n", "p(X) & \\approx & \\frac{\\Gamma(c_1)\\Gamma(c_0)}{\\Gamma(c_0 + c_1)} (1-\\hat{\\pi}_1) \\hat{\\pi}_1 \\frac{\\Gamma(\\alpha + \\beta)}{\\Gamma(\\alpha)\\Gamma(\\beta)} (1-\\hat{\\pi}_1)^{\\beta-1}\\hat{\\pi}_1^{\\alpha-1}\n", "\\end{eqnarray}" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Illustration: Bayesian update of a Beta Distribution " ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "data": { "image/png": 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gsce0kl0IyJUrl+sI4gtZssDIkVCypJ3reN8+mD1bV3NEAoCGj0RCzd69UL8+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\ndAjOOy87Y7xpU1VvxLljXM5++lPYsgX+4R8sxZI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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "%matplotlib inline\n", "import matplotlib.pyplot as plt\n", "import numpy as np\n", "from scipy.special import gammaln\n", "\n", "def log_beta_pdf(x, a, b):\n", " return - gammaln(a) - gammaln(b) + gammaln(a+b) + np.log(x)*(a-1) + np.log(1-x)*(b-1) \n", "\n", "x = np.arange(0.01,1,0.01)\n", "\n", "a = 1\n", "b = 1\n", "c_0 = 1\n", "c_1 = 1\n", "N = c_0 + c_1\n", "\n", "pi_ML = c_1/N\n", "\n", "plt.figure(figsize=(12,8))\n", "plt.plot(x, np.exp(log_beta_pdf(x, a, b)), 'b')\n", "plt.plot(x, np.exp(log_beta_pdf(x, a+c_1, b+c_0)), 'r')\n", "yl = plt.gca().get_ylim()\n", "plt.plot([pi_ML, pi_ML], yl , 'k:')\n", "plt.legend([\"Prior $\\cal B$ $(a={}, b={})$\".format(a,b), \"Posterior $\\cal B$ $(a={}, b={})$\".format(a+c_1, b+c_0)], loc=\"best\")\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Illustration: Learning from a sequence of coin flips" ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "scrolled": false }, "outputs": [ { "name": "stderr", "output_type": "stream", "text": [ "/Users/cemgil/anaconda/envs/py35/lib/python3.5/site-packages/ipykernel/__main__.py:11: RuntimeWarning: divide by zero encountered in log\n", "/Users/cemgil/anaconda/envs/py35/lib/python3.5/site-packages/ipykernel/__main__.py:11: RuntimeWarning: invalid value encountered in multiply\n" ] }, { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "%matplotlib inline\n", "import matplotlib.pyplot as plt\n", "from scipy.special import gammaln\n", "import numpy as np\n", "import scipy.special as sps\n", "\n", "#savefigs = \n", "\n", "def log_beta_pdf(x, a, b):\n", " \n", " y = - gammaln(a) - gammaln(b) + gammaln(a+b) + np.log(x)*(a-1) + np.log(1-x)*(b-1) \n", " idx = np.where(x == 0)\n", " if a==1:\n", " y[idx] = - gammaln(a) - gammaln(b) + gammaln(a+b)\n", " elif a<1:\n", " y[idx] = np.Inf\n", " else:\n", " y[idx] = - np.Inf\n", " \n", " idx = np.where(x == 1)\n", " if b==1:\n", " y[idx] = - gammaln(a) - gammaln(b) + gammaln(a+b)\n", " elif b<1:\n", " y[idx] = np.Inf\n", " else:\n", " y[idx] = - np.Inf\n", " return y\n", "\n", "a = 1\n", "b = 1\n", "\n", "xx = [1,1,1,1,1,0,1,1, 1,0,1,0,1,1,1,1,0, 1,1,1,1,1,1,1]\n", "\n", "p = np.arange(0.0,1.01,0.01)\n", "\n", "c = [0,0]\n", "N = 0\n", "plt.figure(figsize=(5,3))\n", "plt.plot(p, np.exp(log_beta_pdf(p, a+c[1], b+c[0])), 'r')\n", "plt.xticks([0,1])\n", "plt.yticks([])\n", "plt.xlim([-0.1,1.1])\n", "plt.ylim([0,6])\n", "pi_ML = (a+c[1])/(a+b+N)\n", "yl = plt.gca().get_ylim()\n", "plt.plot([pi_ML, pi_ML], yl , 'k:')\n", "\n", "plt.xlabel('$\\lambda$')\n", "#plt.savefig('/Users/cemgil/Dropbox/tex/cam/talks/cmpe547/beta{n}.eps'.format(n=N), bbox_inches='tight')\n", "plt.show()\n", "\n", "for x in xx:\n", " c[x] += 1\n", " N += 1\n", " plt.figure(figsize=(5,3))\n", " plt.plot(p, np.exp(log_beta_pdf(p, a+c[1], b+c[0])), 'r')\n", "\n", " \n", " pi_ML = (a+c[1])/(a+b+N)\n", " pi_mode = (a+c[1]-1)/float(a+b+N-2)\n", " tmp = str(int(a+c[1]-1))+'/'+str(a+b+N-2)\n", " plt.xticks([0,pi_mode],('0',tmp))\n", " #plt.xticks([0,pi_mode])\n", " plt.yticks([])\n", " plt.xlim([-0.1,1.1])\n", " plt.ylim([0,6])\n", " \n", " yl = plt.gca().get_ylim()\n", " plt.plot([pi_ML, pi_ML], yl , 'k:')\n", " plt.plot([pi_mode, pi_mode], yl , 'b:')\n", " \n", " plt.xlabel('$\\lambda$')\n", " #plt.savefig('/Users/cemgil/Dropbox/tex/cam/talks/cmpe547/beta{n}.eps'.format(n=N), bbox_inches='tight')\n", " plt.show()\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Finding if a Coin is Fair or Fake \n", "\n", "We consider the folowing problem: Given a sequence of coin tosses $X = \\{x^{(n)}\\}_{n=1\\dots N}$, determine if the coin is fair or fake.\n", "\n", "This can be cast as a model selection problem:\n", "\n", "\\begin{eqnarray}\n", "\\pi_1|m & \\sim & \\left\\{ \\begin{array}{cc} \\delta(\\pi_1 - 0.5) & m = 0\\\\ \\mathcal{B}(\\pi; a, b) & m = 1 \\end{array} \\right.\n", "\\end{eqnarray}\n", "For $n = 1\\dots N$\n", "\\begin{eqnarray}\n", "x^{(n)}| \\pi_1 & \\sim & \\mathcal{BE}(x; \\pi_1)\n", "\\end{eqnarray}\n", "\n", "This model defines the following:\n", "- The indicator $m$, that denotes if the coin is fake,\n", "- What a fake coin is: a fake coin is one that has an arbitrary probability $\\pi_1$ between $0$ and $1$. \n", "- What a fair coin is: a fair coin has $\\pi_1 = 0.5$\n", "\n", "We need to calculate the marginal likelihoods for $m=0$ and $m=1$\n", "\\begin{eqnarray}\n", "p(X| m) & = & \\int d\\pi_1 p(X | \\pi_1) p(\\pi_1|m)\n", "\\end{eqnarray}\n", "\n", "###### Not Fake\n", "\\begin{eqnarray}\n", "p(X| m) & = & \\int d\\pi_1 p(X| \\pi_1) \\delta(\\pi_1 - 0.5) \\\\\n", "& = & \\prod_{n=1}^N \\left(\\frac{1}{2}\\right)^{x^{(n)}} \\left(\\frac{1}{2}\\right)^{1-x^{(n)}} = \\frac{1}{2^N}\n", "\\end{eqnarray}\n", "\n", "###### Fake\n", "\n", "\\begin{eqnarray}\n", "p(X| m) & = & \\int d\\pi_1 p(\\pi_1; a, b) \\prod_{n=1}^{N} p(x^{(n)}| \\pi_1) \\\\\n", "& = & \\int d\\pi_1 \\left(\\prod_{n=1}^N \\left(1-\\pi_1\\right)^{1-x^{(n)}} \\pi_1^{x^{(n)}} \\right) \\mathcal{B}(\\pi; a, b) \\\\\n", "& = & \\frac{\\Gamma(a + b)}{\\Gamma(a)\\Gamma(b)} \\int d\\pi_1 \\left(1-\\pi_1\\right)^{c_0+a-1} \\pi_1^{c_1+b-1} \\\\\n", "& = & \\frac{\\Gamma(a + b)}{\\Gamma(a)\\Gamma(b)} \\frac{\\Gamma(c_0+a)\\Gamma(c_1+b)}{\\Gamma(c_0 + c_1 +a + b)}\n", "\\end{eqnarray}" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The log-odds is the ratio of marginal likelihoods\n", "\n", "$$\n", "l(X) = \\log\\left( \\frac{p(X|m = \\text{Fair})}{p(X|m = \\text{Fake})} \\right)\n", "$$\n", "\n", "If $l(X)>0$, we may conclude that the coin is fair and biased when $l<0$." ] }, { "cell_type": "code", "execution_count": 42, "metadata": { "scrolled": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "log_odds = -0.826678573184\n" ] } ], "source": [ "import numpy as np\n", "import scipy.special as sps\n", "\n", "def log_odds(c_0, c_1, a, b):\n", " # Total number of tosses\n", " N = c_0 + c_1\n", " \n", " M_fair = N*np.log(0.5)\n", " M_fake = sps.gammaln(a+b) - sps.gammaln(a) - sps.gammaln(b) + sps.gammaln(c_0+a) + sps.gammaln(c_1+b) - sps.gammaln(N+a + b) \n", " return M_fair - M_fake\n", "\n", "# Number of Zeros observed\n", "c_0 = 6\n", "# Number of Ones\n", "c_1 = 1\n", "\n", "# Prior\n", "a = 1\n", "b = 1\n", "\n", "\n", "print('log_odds = ', log_odds(c_0, c_1, a, b) )\n" ] }, { "cell_type": "code", "execution_count": 43, "metadata": { "scrolled": true }, "outputs": [ { "data": { "image/png": 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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "a = 1\n", "b = 1\n", "N = 10\n", "\n", "l = np.zeros(N+1)\n", "\n", "for c in range(0,N+1):\n", " l[c] = log_odds(N-c, c, a, b)\n", "\n", "plt.plot(range(0,N+1), l, 'o')\n", "plt.plot(range(0,N+1), np.zeros(N+1), 'k:')\n", "ax = plt.gca()\n", "ax.set_xlabel('Number of ones $c_1$')\n", "ax.set_ylabel('log-odds $l(X)$')\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We can visualize the region where we would decide that the coin is fake by plotting the points where the log-odds is negative." ] }, { "cell_type": "code", "execution_count": 50, "metadata": { "scrolled": true }, "outputs": [ { "data": { "image/png": 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TzVix91VFkmWemUkNE/e2Riw/5UoIYQaYA35lWJkjR468+He73abdbk+8XZVT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\n", "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "a = 1\n", "b = 1\n", "\n", "for N in range(1, 25):\n", "\n", " l = np.zeros(N+1)\n", "\n", " for c in range(0,N+1):\n", " l[c] = log_odds(N-c, c, a, b)\n", " \n", " \n", " idx = np.where( np.array(l)<0 )\n", " p = np.arange(0,N+1)/N\n", " plt.plot(N*np.ones_like(p), p, '.k',markersize=4) \n", " plt.plot(N*np.ones_like(p[idx]), p[idx], '.r',markersize=20)\n", " \n", "\n", "ax = plt.gca()\n", "ax.set_ylim((0,1))\n", "ax.set_xlabel('$N$')\n", "ax.set_ylabel('$c_1/N$')\n", "plt.show()" ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.5.3" }, "toc": { "nav_menu": {}, "number_sections": true, "sideBar": true, "skip_h1_title": 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