bernoulli_trials_bayesian.ipynb

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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Bernoulli trials are one of the simplest experimential setups: there are a number of iterations of some activity, where each iteration (or trial) may turn out to be a \"success\" or a \"failure\". From the data on T trials, we want to estimate the probability of \"success\".\n",
    "\n",
    "Since it is such a simple case, it is a nice setup to use to describe some of Python's capabilities for estimating statistical models. Here I show estimation from the Bayesian perspective, via Metropolis-Hastings MCMC methods.\n",
    "\n",
    "In [another post](./bernoulli_trials_classical.html) I show estimation of the problem in Python using the classical / frequentist approach.\n",
    "\n",
    "<!-- TEASER_END -->"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import numpy as np\n",
    "import pandas as pd\n",
    "import statsmodels.api as sm\n",
    "import sympy as sp\n",
    "import pymc\n",
    "import matplotlib.pyplot as plt\n",
    "import matplotlib.gridspec as gridspec\n",
    "from mpl_toolkits.mplot3d import Axes3D\n",
    "from scipy import stats\n",
    "from scipy.special import gamma\n",
    "\n",
    "from sympy.interactive import printing\n",
    "printing.init_printing()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Setup\n",
    "\n",
    "Let $y$ be a Bernoulli trial:\n",
    "\n",
    "$y \\sim \\text{Bernoulli}(\\theta) = \\text{Binomial}(1, \\theta)$\n",
    "\n",
    "The probability density function, or marginal likelihood function, is:\n",
    "\n",
    "$$p(y|\\theta) = \\theta^{y} (1-\\theta)^{1-y} = \\begin{cases}\n",
    "\\theta & y = 1 \\\\\n",
    "1 - \\theta & y = 0\n",
    "\\end{cases}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# Simulate data\n",
    "np.random.seed(123)\n",
    "\n",
    "nobs = 100\n",
    "theta = 0.3\n",
    "Y = np.random.binomial(1, theta, nobs)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
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2zMxOTNNfdvd3AFOBDzRmcvcvuft+PWhTBF4OzHD3fVIC3wA4Fdgf2Af4mJlt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\npeAAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1112a1b10>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Plot the data\n",
    "fig = plt.figure(figsize=(7,3))\n",
    "gs = gridspec.GridSpec(1, 2, width_ratios=[5, 1]) \n",
    "ax1 = fig.add_subplot(gs[0])\n",
    "ax2 = fig.add_subplot(gs[1])\n",
    "\n",
    "ax1.plot(range(nobs), Y, 'x')\n",
    "ax2.hist(-Y, bins=2)\n",
    "\n",
    "ax1.yaxis.set(ticks=(0,1), ticklabels=('Failure', 'Success'))\n",
    "ax2.xaxis.set(ticks=(-1,0), ticklabels=('Success', 'Failure'));\n",
    "\n",
    "ax1.set(title=r'Bernoulli Trial Outcomes $(\\theta=0.3)$', xlabel='Trial', ylim=(-0.2, 1.2))\n",
    "ax2.set(ylabel='Frequency')\n",
    "\n",
    "fig.tight_layout()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Bayesian Estimation\n",
    "\n",
    "Using Bayes' rule:\n",
    "\n",
    "$$p(\\theta|Y) = \\frac{p(Y|\\theta) p(\\theta)}{p(Y)}$$\n",
    "\n",
    "To perform Bayesian estimation, we need to construct the __posterior__ $p(\\theta|Y)$ given:\n",
    "\n",
    "- the (joint) __likelihood__ $P(Y|\\theta)$\n",
    "- the __prior__ $p(\\theta)$\n",
    "- the __marginal probability density function__ $P(Y)$\n",
    "\n",
    "to perform the estimation, we need to specify the functional forms of the likelihood and the prior. The marginal pdf of $Y$ is a constant with respect to $\\theta$, so it does not need to specified for our purposes."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Likelihood function\n",
    "\n",
    "Consider a sample of $T$ draws from the random variable $y$. The joint likelihood of observing any specific sample $Y = (y_1, ..., y_T)'$ is given by:\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "p(Y|\\theta) & = \\prod_{i=1}^T \\theta^{y_i} (1-\\theta)^{1-y_i} \\\\\n",
    "& = \\theta^{s} (1 - \\theta)^{T-s}\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "where $s = \\sum_i y_i$ is the number of observed \"successes\", and $T-s$ is the number of observed \"failures\"."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "t, T, s = sp.symbols('theta, T, s')\n",
    "\n",
    "# Create the function symbolically\n",
    "likelihood = (t**s)*(1-t)**(T-s)\n",
    "\n",
    "# Convert it to a Numpy-callable function\n",
    "_likelihood = sp.lambdify((t,T,s), likelihood, modules='numpy')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Prior\n",
    "\n",
    "Since $\\theta$ is a probability value, our prior must respect $\\theta \\in (0,1)$. We will use the (conjugate) Beta prior:\n",
    "\n",
    "$\\theta \\sim \\text{Beta}(\\alpha_1, \\alpha_2)$\n",
    "\n",
    "so that $(\\alpha_1, \\alpha_2)$ are the model's hyperparameters. Then the prior is specified as:\n",
    "\n",
    "$$\n",
    "p(\\theta;\\alpha_1,\\alpha_2) = \\frac{1}{B(\\alpha_1, \\alpha_2)} \\theta^{\\alpha_1-1} (1 - \\theta)^{\\alpha_2 - 1}\n",
    "$$\n",
    "\n",
    "where $B(\\alpha_1, \\alpha_2)$ is the Beta function. Note that to have a fully specified prior, we need to also specify the hyperparameters."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Prior mean: 0\n"
     ]
    },
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x114ff2910>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# For alpha_1 = alpha_2 = 1, the Beta distribution\n",
    "# degenerates to a uniform distribution\n",
    "a1 = 1\n",
    "a2 = 1\n",
    "\n",
    "# Prior Mean\n",
    "prior_mean = a1 / (a1 + a2)\n",
    "print 'Prior mean:', prior_mean\n",
    "\n",
    "# Plot the prior\n",
    "fig = plt.figure(figsize=(10,4))\n",
    "ax = fig.add_subplot(111)\n",
    "X = np.linspace(0,1, 1000)\n",
    "ax.plot(X, stats.beta(a1, a2).pdf(X), 'g');\n",
    "\n",
    "# Cleanup\n",
    "ax.set(title='Prior Distribution', ylim=(0,12))\n",
    "ax.legend(['Prior']);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Posterior\n",
    "\n",
    "#### Analytically\n",
    "\n",
    "Given the prior and the likelihood function, we can try to find the kernel of the posterior analytically. In this case, it will be possible:\n",
    "\n",
    "$$\n",
    "\\begin{align}\n",
    "p(\\theta|Y;\\alpha_1,\\alpha_2) & = \\frac{P(Y|\\theta) P(\\theta)}{P(Y)} \\\\\n",
    "& \\propto P(Y|\\theta) P(\\theta) \\\\\n",
    "& = \\theta^s (1-\\theta)^{T-s} \\frac{1}{B(\\alpha_1, \\alpha_2)} \\theta^{\\alpha_1-1} (1 - \\theta)^{\\alpha_2 - 1} \\\\\n",
    "& \\propto \\theta^{s+\\alpha_1-1} (1 - \\theta)^{T-s+\\alpha_2 - 1} \\\\\n",
    "\\end{align}\n",
    "$$\n",
    "\n",
    "The last line is identifiable as the kernel of a beta distribution with parameters $(\\hat \\alpha_1, \\hat \\alpha_2) = (s+\\alpha_1, T-s+\\alpha_2)$\n",
    "\n",
    "Thus the posterior is given by\n",
    "\n",
    "$$\n",
    "P(\\theta|Y;\\alpha_1,\\alpha_2) = \\frac{1}{B(\\hat \\alpha_1, \\hat \\alpha_2)} \\theta^{\\hat \\alpha_1 - 1} (1-\\theta)^{\\hat \\alpha_2 -1}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Posterior Mean (Analytic): 0\n"
     ]
    },
    {
     "data": {
      "image/png": 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av4iISEVUUAkUON1Xtz4A93eAJ0huZSRVoB6OsCl/YVP+skkFlUCehvQU6LSf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\nUiEVVCIiIiIV+v9uIINVKrTpYwAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x114ff2b50>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Find the hyperparameters of the posterior\n",
    "a1_hat = a1 + Y.sum()\n",
    "a2_hat = a2 + nobs - Y.sum()\n",
    "\n",
    "# Posterior Mean\n",
    "post_mean = a1_hat / (a1_hat + a2_hat)\n",
    "print 'Posterior Mean (Analytic):', post_mean\n",
    "\n",
    "# Plot the analytic posterior\n",
    "fig = plt.figure(figsize=(10,4))\n",
    "ax = fig.add_subplot(111)\n",
    "X = np.linspace(0,1, 1000)\n",
    "ax.plot(X, stats.beta(a1_hat, a2_hat).pdf(X), 'r');\n",
    "\n",
    "# Plot the prior\n",
    "ax.plot(X, stats.beta(a1, a2).pdf(X), 'g');\n",
    "\n",
    "# Cleanup\n",
    "ax.set(title='Posterior Distribution (Analytic)', ylim=(0,12))\n",
    "ax.legend(['Posterior (Analytic)', 'Prior']);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Metropolis-Hastings: Pure Python\n",
    "\n",
    "Although since in this case the posterior can be found analytically for the conjugate Beta prior, we can also arrive at it as the stationary distribution of a Markov chain with Metropolis-Hastings transition kernel.\n",
    "\n",
    "To do this, we need a proposal distribution $q(\\theta|\\theta^{[g]})$, and here we will use a random walk proposal: $\\theta^* = \\theta^{[g]} + \\eta_t$ where $\\eta_t \\sim \\text{Normal}(0,\\sigma^2)$ where $\\sigma^2$ will be set to get a desired acceptance ratio."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Timing: 1 loops, best of 3: 356 ms per loop\n"
     ]
    }
   ],
   "source": [
    "#%%timeit\n",
    "print 'Timing: 1 loops, best of 3: 356 ms per loop'\n",
    "\n",
    "# Metropolis-Hastings parameters\n",
    "G1 = 1000 # burn-in period\n",
    "G = 10000 # draws from the (converged) posterior\n",
    "\n",
    "# Model parameters\n",
    "sigma = 0.1\n",
    "thetas = [0.5]             # initial value for theta\n",
    "etas = np.random.normal(0, sigma, G1+G) # random walk errors\n",
    "unif = np.random.uniform(size=G1+G)     # comparators for accept_probs\n",
    "\n",
    "# Callable functions for likelihood and prior\n",
    "prior_const = gamma(a1) * gamma(a2) / gamma(a1 + a2)\n",
    "mh_ll = lambda theta: _likelihood(theta, nobs, Y.sum())\n",
    "def mh_prior(theta):\n",
    "    prior = 0\n",
    "    if theta >= 0 and theta <= 1:\n",
    "        prior = prior_const*(theta**(a1-1))*((1-theta)**(a2-1))\n",
    "    return prior\n",
    "mh_accept = lambda theta: mh_ll(theta) * mh_prior(theta)\n",
    "\n",
    "theta_prob = mh_accept(thetas[-1])\n",
    "\n",
    "# Metropolis-Hastings iterations\n",
    "for i in range(G1+G):\n",
    "    # Draw theta\n",
    "    \n",
    "    # Generate the proposal\n",
    "    theta = thetas[-1]\n",
    "    theta_star = theta + etas[i]\n",
    "    theta_star_prob = mh_accept(theta_star)\n",
    "    # Calculate the acceptance probability\n",
    "    accept_prob = theta_star_prob / theta_prob\n",
    "    \n",
    "    # Append the new draw\n",
    "    if accept_prob > unif[i]:\n",
    "        theta = theta_star\n",
    "        theta_prob = theta_star_prob\n",
    "    thetas.append(theta)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can describe the posterior using the draws after the chain has converged (i.e. following the burn-in period):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Posterior Mean (MH): 0.22532888147\n"
     ]
    },
    {
     "data": {
      "image/png": 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nktw5m+TPN6UU999/P4888gh5eXkMGzaMnJwcGjZsWOW6MTEx51w5XIhQkh4q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\nbHcYtW4xMBJYXdWC0UAplSG/NHIuyZ+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      "text/plain": [
       "<matplotlib.figure.Figure at 0x1112e7710>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Posterior Mean\n",
    "print 'Posterior Mean (MH):', np.mean(thetas[G1:])\n",
    "\n",
    "# Plot the posterior\n",
    "fig = plt.figure(figsize=(10,4))\n",
    "ax = fig.add_subplot(111)\n",
    "# Plot MH draws\n",
    "ax.hist(thetas[G1:], bins=50, normed=True);\n",
    "# Plot analytic posterior\n",
    "X = np.linspace(0,1, 1000)\n",
    "ax.plot(X, stats.beta(a1_hat, a2_hat).pdf(X), 'r');\n",
    "# Plot prior\n",
    "ax.plot(X, stats.beta(a1, a2).pdf(X), 'g')\n",
    "\n",
    "# Cleanup\n",
    "ax.set(title='Metropolis-Hastings via pure Python (10,000 Draws; 1,000 Burned)', ylim=(0,12))\n",
    "ax.legend(['Posterior (Analytic)', 'Prior', 'Posterior Draws (MH)']);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Metropolis-Hastings: Cython\n",
    "\n",
    "The runtime of 356ms is not bad, by we may be able to improve matters by writing it in Cython, a pseudo-language which is then compiled into a C extension that we can call from our Python code. In the right circumstances, this can speed up code dramatically.\n",
    "\n",
    "Although I am not an expert in MATLAB, a pretty much direct port of this code to MATLAB (almost identical to the Cython code below) runs in about 400ms, so pure Python and MATLAB appear to be reasonably similar."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "The Cython magic has been move to the Cython package, hence \n",
      "`%load_ext cythonmagic` is deprecated; Please use `%load_ext Cython` instead.\n",
      "\n",
      "Though, because I am nice, I'll still try to load it for you this time.\n"
     ]
    }
   ],
   "source": [
    "%load_ext cythonmagic"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "%%cython\n",
    "\n",
    "import numpy as np\n",
    "from scipy.special import gamma\n",
    "cimport numpy as np\n",
    "cimport cython\n",
    "\n",
    "from libc.math cimport pow\n",
    "\n",
    "cdef double likelihood(double theta, int T, int s):\n",
    "    return pow(theta, s)*pow(1-theta, T-s)\n",
    "\n",
    "cdef double prior(double theta, double a1, double a2, double prior_const):\n",
    "    if theta < 0 or theta > 1:\n",
    "        return 0\n",
    "    return prior_const*pow(theta, a1-1)*pow(1-theta, a2-1)\n",
    "\n",
    "cdef np.ndarray[np.float64_t, ndim=1] draw_posterior(np.ndarray[np.float64_t, ndim=1] theta, double eta, double unif, int T, int s, double a1, double a2, double prior_const):\n",
    "    cdef double theta_star, theta_star_prob, accept_prob\n",
    "    \n",
    "    theta_star = theta[0] + eta\n",
    "    theta_star_prob = likelihood(theta_star, T, s) * prior(theta_star, a1, a2, prior_const)\n",
    "    \n",
    "    accept_prob = theta_star_prob / theta[1]\n",
    "    \n",
    "    if accept_prob > unif:\n",
    "        theta[0] = theta_star\n",
    "        theta[1] = theta_star_prob\n",
    "        \n",
    "    return theta\n",
    "\n",
    "def mh(double theta_init, int T, int s, double sigma, double a1, double a2, int G1, int G):\n",
    "    \n",
    "    cdef np.ndarray[np.float64_t, ndim = 1] theta, thetas, etas, unif\n",
    "    cdef double prior_const, theta_prob\n",
    "    cdef int t\n",
    "    \n",
    "    prior_const = gamma(a1) * gamma(a2) / gamma(a1 + a2)\n",
    "    theta_prob = likelihood(theta_init, T, s) * prior(theta_init, a1, a2, prior_const)\n",
    "    \n",
    "    theta = np.array([theta_init, theta_prob])\n",
    "    \n",
    "    thetas = np.zeros((G1+G,))\n",
    "    etas = np.random.normal(0, sigma, G1+G)\n",
    "    unif = np.random.uniform(size=G1+G)\n",
    "    \n",
    "    for t in range(G1+G):\n",
    "        theta = draw_posterior(theta, etas[t], unif[t], T, s, a1, a2, prior_const)\n",
    "        thetas[t] = theta[0]\n",
    "        \n",
    "    return thetas"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Timing: 10 loops, best of 3: 20.7 ms per loop\n"
     ]
    }
   ],
   "source": [
    "#%%timeit\n",
    "print 'Timing: 10 loops, best of 3: 20.7 ms per loop'\n",
    "thetas = mh(0.5, nobs, Y.sum(), sigma, a1, a2, G1, G)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Notice that using Cython, we've sped up the code by a factor of about 17-20 from pure Python or MATLAB."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Posterior Mean (MH): 0.22553298482\n"
     ]
    },
    {
     "data": {
      "image/png": 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bC8CVV17JXXfdxYkTJ2jatCkAvXr14uqrr/ZruSI8VVpQGYaRBMwHVpmm+aD9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\nhgQVAMCQoAIAGBJUAABDggoAYOgBXAvfSQpN9GMAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1152a76d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Posterior Mean\n",
    "print 'Posterior Mean (MH):', np.mean(thetas[G1:])\n",
    "\n",
    "# Plot the posterior\n",
    "fig = plt.figure(figsize=(10,4))\n",
    "ax = fig.add_subplot(111)\n",
    "# Plot MH draws\n",
    "ax.hist(thetas[G1:], bins=50, normed=True);\n",
    "# Plot analytic posterior\n",
    "X = np.linspace(0,1, 1000)\n",
    "ax.plot(X, stats.beta(a1_hat, a2_hat).pdf(X), 'r');\n",
    "# Plot prior\n",
    "ax.plot(X, stats.beta(a1, a2).pdf(X), 'g')\n",
    "\n",
    "# Cleanup\n",
    "ax.set(title='Metropolis-Hastings via Cython (10,000 Draws; 1,000 Burned)', ylim=(0,12))\n",
    "ax.legend(['Posterior (Analytic)', 'Prior', 'Posterior Draws (MH)']);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now that we've improved the performance of our Metropolis-Hastings draws, we can increase the burn in period (although that is not necessary to ensure convergence in this case) and increase the number of draws from the converged posterior. Here we'll increase the burn in period and the post-convergence draws by a factor of 100 each. The total increase in runtime will almost exclusively be a result of the 100x increase in the post-convergence draws, so the runtime will likely increase by a factor of about 100).\n",
    "\n",
    "Notice that this would be inconvenient using the pure Python or MATLAB code, since it would take about $100 \\times 0.4 \\text{s} \\approx 40\\text{s}$. Fortunately, our Cython implementation can run it in about $100 \\times 0.02 \\text{s} \\approx 2\\text{s}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "G1 = 100000\n",
    "G = 1000000"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Timing: 1 loops, best of 3: 2.09 s per loop\n"
     ]
    }
   ],
   "source": [
    "#%%timeit\n",
    "print 'Timing: 1 loops, best of 3: 2.09 s per loop'\n",
    "thetas = mh(0.5, nobs, Y.sum(), sigma, a1, a2, G1, G)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Posterior Mean (MH): 0.225454350316\n"
     ]
    },
    {
     "data": {
      "image/png": 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TadeuHQBXXXUV9957LydPnqRhw4YAxMfHc/311/vVrghP5RZUhmGkAM8D35im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\ncra37owqKlRAD2BTlS8wzMi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3AQzestLpUIQQQoSZ8P32rX2sgirMLe8xiMTNUlBJHw53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\nYlOFldFWAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x115d54890>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Posterior Mean\n",
    "print 'Posterior Mean (MH):', np.mean(thetas[G1:])\n",
    "\n",
    "# Plot the posterior\n",
    "fig = plt.figure(figsize=(10,4))\n",
    "ax = fig.add_subplot(111)\n",
    "# Plot MH draws\n",
    "ax.hist(thetas[G1:], bins=50, normed=True);\n",
    "# Plot analytic posterior\n",
    "X = np.linspace(0,1, 1000)\n",
    "ax.plot(X, stats.beta(a1_hat, a2_hat).pdf(X), 'r');\n",
    "# Plot prior\n",
    "ax.plot(X, stats.beta(a1, a2).pdf(X), 'g')\n",
    "\n",
    "# Cleanup\n",
    "ax.set(title='Metropolis-Hastings via Cython (1,000,000 Draws; 100,000 Burned)', ylim=(0,12))\n",
    "ax.legend(['Posterior (Analytic)', 'Prior', 'Posterior Draws (MH)']);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And with this many post-convergence draws, we can match the analytic posterior mean to 4 decimal places."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Metropolis-Hastings: PyMC\n",
    "\n",
    "We can also make use of the [PyMC](https://github.com/pymc-devs/pymc) package to do Metropolis-Hastings runs for us. It is about twice as slow as the custom pure Python approach we employed above (and so ~40 times slower than the Cython implementation), but it is certainly much less work to set up!\n",
    "\n",
    "(Note: I am not well-versed in PyMC, so it is certainly possible - likely, even - that there is a more performant way to do this)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "G1 = 1000\n",
    "G = 10000"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Timing: 1 loops, best of 3: 590 ms per loop\n",
      "\n",
      "pymc_theta:\n",
      " \n",
      "\tMean             SD               MC Error        95% HPD interval\n",
      "\t------------------------------------------------------------------\n",
      "\t0.224            0.042            0.001            [ 0.146  0.302]\n",
      "\t\n",
      "\t\n",
      "\tPosterior quantiles:\n",
      "\t\n",
      "\t2.5             25              50              75             97.5\n",
      "\t |---------------|===============|===============|---------------|\n",
      "\t0.148            0.194           0.222          0.25          0.307\n",
      "\t\n"
     ]
    }
   ],
   "source": [
    "#%%timeit\n",
    "print 'Timing: 1 loops, best of 3: 590 ms per loop'\n",
    "\n",
    "pymc_theta = pymc.Beta('pymc_theta', a1, a2, value=0.5)\n",
    "pymc_Y = pymc.Bernoulli('pymc_Y', p=pymc_theta, value=Y, observed=True)\n",
    "\n",
    "model = pymc.MCMC([pymc_theta, pymc_Y])\n",
    "model.sample(iter=G+G1, burn=G1, progress_bar=False)\n",
    "\n",
    "model.summary()\n",
    "thetas = model.trace('pymc_theta')[:]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Posterior Mean (MH): 0.223732963864\n"
     ]
    },
    {
     "data": {
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wYUMAEhISuOmmmwJqV4SnMgsql8uVAkwHlhqG8aR120jgGWuTZwzDWFS1IQq7\nyDdk55LcOVtNz1+ghVCoKKW4//77eeyxx0LW3vnz5z1u01qjVGgKRhFeyptDFQW8ULzgcrnqAM8C\nl1v/prpcLnnmCSGEqBahvNBvUlISy5cvL5mHNXv2bLp160aDBg1CdgwRPsosqAzDWAgcdbspAdhu\nGMZpwzBOA1lA1yqMT9hIzoXjXJI7Z5P8+eev96givUre27Ro0YKZM2cyduxYhgwZwqxZs3j33XcD\nblcICHwOVXPgmMvlesVazgNaAJkhjUoIIYTw8swzz/hdt3PnznL3z8/PL3Xb8OHDWbFihc/t4+Pj\n2bhxY8UDFGEt0ILqFyAGuB9QwD+AI6EOStQMNX0eh/BPcudskj8hnKci56Fy7+/MArq5LScYhrGj\nzJ3duq6VUmmy7Nzl/GO5MbixOx5ZlmVZrrplIWqzY8eOxYT6+a/KmuDncrn+AIwC2mD+0u9el8t1\nOTDF2uRZwzAW+Ns/PT1djxgxQgagHUopz3PhzF23c8m0RdmpAE8Nj1967cWd08prY1VmzqwD+QXx\nAG2aRGUPTIgbVyXBCg/euRPOYnf+MjIyliQmJqbadXwhqlpGRsbSxMTENO/bK1O3lDnkZxjGi8CL\nXrd9A3wTzMFE+DmQXxDvVoTZHI0QQghRNeTEnsJvL5L0cDiX5M7ZJH9COI8UVEJ6kYQQQohKkosj\nC7/KnaCnVEuUegWlNqPUDyj1FEo1qp7oRFlkcrGzSf6EcB7poRJBabHu+0bAj8Bs4HagPvAQ8D1K\nXY3W2TaGJ4QIoZ0HctmXf7rK2o9tUp/ObZpVWftCVAcpqIRf/uZxxP2yjwHPP5QI3ILWn5SsUGoF\n8CjwNUoNRutfqiVQUYrMwXG2mpa/ffmneXTu1ipr/5Vre0hBJRxPhvxEQCKKinjugz+z66bb9ngU\nU2BeZEvrl4EvgJkouWaDECK0pk6dSufOnUlOTmbAgAHMnz8/ZG3PnTuXLVu2hKy9YkVFRYwfP57C\nwsKQt+1tyZIlNG/enKSkJAYOHMhnn31W5ccMlRkzZvDhhx+WLKelpTFmzBgAzp49S4sWLXj22WcB\n835ec801Jdu+9NJLJesAZs6cyfvvv19NkZukoBJ++ZrHMXb5JxTUjWLrhIf3lrHr/wDxwG1VFJoo\nh8zBcTbJn39KKR588EGWLVuGYRiMHz+e06dDMxw5Z84cfvrpp5C05S4iIoKZM2cSGVk9g0JDhgxh\n5cqVzJ1z5QDyAAAgAElEQVQ7l6lTp7Jq1apqOW5lzJs3jw0bNnDzzTd73L5jxw4OHTrEZ599RvPm\nzf1+T/e+ffz48SxYsMDvZYWqghRUosLqHT0SOW7xB7x4/SMQEeF/Q63PAvcAf4k8nl/GhkIIEbji\nE1LHx8fTvn179uzZA8CpU6eYMGECgwcPZsCAAbz22mse+7388ssMGDCAwYMHc/nll3usu+eee5g/\nfz5PP/00KSkpHj07hw8f5sYbbyQtLY3U1FTWrVvnsW98fDxvv/02gwYNom/fvmRnZ5esmzBhAikp\nKTRu3LjU/UhPT2fIkCEkJyczevRodu/eXbIuOzub3r17M2XKFAYOHMjw4cMDfpzatGnDiy++yLRp\n00pumzVrFnfddRcul4ukpCQef/zxknUZGRlcf/31DBs2jO7duzNnzhwAzp07R5cuXTzazsvLo2vX\nriXLZT22FTFp0iT+9re/edymlOLWW2/l/fff56OPPmLs2LH4Oxm5r9unT5/OE088EXAswZI5VMJD\nTP3IHnPX7VwC8OnaLFZl5pScl6rnjFfaL7kwmZ9bdSi/Ia3XoNTSHv/8exL976jKkIUPNW0OjgiM\n5K9itm3bRl5eHp07dwbg+eefp1mzZqxYsYIzZ84wbNgwLrzwQoYPH86xY8f4y1/+wr59+3z2FL39\n9tvcddddXHPNNdxwww0e6x566CHuvvtuRo0axc8//8yYMWPYsGFDyXqlFFu2bOH7778v1e5bb70F\nUKqgOnLkCPfccw/Lly8nNjaWTz/9lNtuu41vv/22ZJsdO3aQmJjIc889F/RjdOmll5KRkeFx24IF\nC1iwYAE9e/b0uL1Tp0589NFH1KtXjx9//JGxY8dy/fXXU7duXerWrYvWmpycHOrXr8/+/fvp1s28\nEl15j215fvjhB2JjY2nUqPSPxC+77DKefPJJ2rVrR7NmzThx4kTJupUrVzJs2DAA9uzZwx13eH7W\nXHDBBdStW5cdO3Z4FH9VRQoq4SH3dGH089Y5qcDtvFRKNYlr2Cj24Yf+GUhzT3f44pOtDXv9hpPR\nDUMapxAiPGmtefPNN/n8889p2rQps2fPLvkQ//rrr/noo48AiI6OZvz48cybN4/hw4cTExPDqFGj\nuOqqqxgzZgw333wzF1xwgc/2vS1cuJADBw7w17/+FYCCggJyc3Np1uzXifSTJ08O6H6sXLmS5ORk\nYmNjAbjuuut48MEHOXnyJA0bmu+XCQkJ3HTTTQG16837/iiluOGGG0oVUwCNGjVi9+7drF69mp9/\n/pn9+/eXrOvUqRM5OTnceOONtG/fnrvvvpuEhASACj+2/mRlZREXF+dzXUREBFOmTKFt27bMmzfP\nY11SUhKff/45YPZGuRdbxeLi4qqtoJIhP1FRdx/t0z93f/M2Fd9D66yjffofvf77L6ouKuGTzMFx\nNsmff0op7r//fpYsWcLcuXNJTEz0WH/+/HmPv93n1rzzzju89957REZGMnDgQHbt2uWzfW+RkZF8\n/vnnLF68mMWLF7N161aPYirY++EeK5jFT6h/y7N69epSj5G/YbOZM2dy++23U1RURGpqqsd2Xbt2\nZebMmQwdOpTCwkK+++67koIKKvbY+tOoUSO/8+CUUowYMYJevXpVuD13p0+f9jncWhWkoBLlM1/h\nD2TdOr6sieg+bR9//56bl39CRFFRFQQmhAhH/gqCUaNGMWPGDMCcTzVz5kyuuuqqkvVFRUW0bt2a\niRMn0q1bN7Zu9TwVRHR0NAcPHgQ8C7Nrr72WKVOmlCx7F0LBSEpKYvny5SXzv2bPnk23bt1o0KBB\npdsuduDAAf74xz969J75e+zA/JXj5MmTGTt2LJmZmaUKqldeeYUHHniA++67j1dffZXu3buXrC/v\nsS3LJZdcEtD2gcjKyqJv375V0rY3GfITFZEMFBweMCSfJbvL3djd0X6XnDgY04ohW1fBZV3K30GE\nhMzBcbaalr/YJvV55doeVdp+IPz14kyaNIlHHnmEpKQkCgsLufPOO0lLSwPMQmLkyJEUFhZy5swZ\n0tLSuPLKKz32v/XWWxk3bhyGYdC7d++SSe3Tp0/n8ccfZ8CAAURFRZGQkMDMmTPLjaesmFu0aMHM\nmTMZO3YsSiliYmJ49913K3Q/yzvOihUrGDRoEEVFRTzzzDMMGjTIY72/dh999FHuvfde2rZtyxVX\nXEHz5s1LhiC7detGamoqHTt2pEOHDrRv376kh6oij21ZWrZsSc+ePfn+++89YvV3//zdD+/l9PR0\nUlJSfM7NqgqqrGq1stLT0/WIESPkXEQ13Nx1O5cUX8tv8vD4vOcXZTctXvfU8Pil117SJQvYOndt\n1mi3a/4tvfbizmkVafvHv85IHbHxW355/z8V2kcIYa+MjIwliYmJqeVvKURoHD16lFtuuYX33nuP\nVq1aVbq9vXv3MmHCBGbPnu2zoMrIyFiamJiY5n17ZeoWGfITZYo8cbwOcAMQ9BnSFvRJo/fun2iU\nnRUVushEWWQOjrNJ/kS4ad68Of/+97/ZtGlTSNrbvHkz7733XrX1ToEUVKIcHT7/uCWwHK33l7ux\nHwX1ovm63wg6zX6/bQhDE0IIUYvExsYGdb4tX6644oqAfmkYClJQiTK1WbKgJfBRZduZ138kbZYu\nbCWXo6keNW0OjgiM3fnTWld+1rUQNVhVPMeloBJ+RZ89TbOfNsRgXpuvUja374EqKlRAv8pHJoSo\nSgUFBRnHjx8/a3ccQlSF48ePny0oKMgof8vAyK/8hF9J29aS36VbfvOMH3Mr3ZhSHEgZcajTJx/c\nAqyvfHSiLEqpNLt7OUTw7M5ffn7+o1u3biUqKipRKSVfvAN07NixmJiYmGN2xyFK01qfLygoyMjP\nz3801G1LQSX8Stu8jIMpw480D1F7e0bfcKjTJx/cjFJ/RIYUhKixRowYcR54xO44nMrugljYQwoq\n4VNEUSFDtqxi9ZRJR0pfoMDzmn9tGtdrd+D42b0AbZpElVz7z1tuYr+TwAlgAFD6olciZOTN3Nkk\nf84m+QtPUlAJn/r8/BM5zdtyql1Hn/Mo3K/5Z527qgu4XfvPlzp1AD4FrkUKKiGEELWIjI0LnwZt\nX8P33S8JaZsx9SN7rHjjnStOxrZ7eFVmzqyQNi48yHmMnE3y52ySv/AkBZXwadC2tXzf7dKQtpl7\nujD6d8dj+58+fqrBqbU/+hpJFEIIIRxJCipRSsyJY7T/JYeNHYO7undZdJ06fNczibgFX7YIeeOi\nhMzhcDbJn7NJ/sKTFFSilAGZP7CuS1+KIqpmit3SCwfTasXS6j2FrRBCCFGFpKASpSRtXxPy4T53\na7teRKPsnY1Qqmn5W4tgyBwOZ5P8OZvkLzxJQSU8ac0lWT+yputFVXaIgrpR5HfrmQ+MqLKDCCGE\nENVICirhoUHO7jqRRYXsvqBdlR7n8KVJR4ErqvQgYUzmcDib5M/ZJH/hSQoq4eGCdasifujcB6r4\nGsYHho40Cyq5WLIQQohaIOhZxy6X6w7gAaAQeMowjMUhi0rY5oIfVkcu6NQn6P3dz6AOEFFH9fC1\n3bFeiacwn3/dgG1BH1D4JJe+cDbJn7NJ/sJTZX7G9QRwEdAQ+BpICklEwlYt1q+J/OHmq0uW3Qsk\nf8WRO/czqIN5FnWfG5pnTf8ac9hPCiohhBCOVpkhv5+AVOBq5DIitUKrY4eJPHlS7WzdseS23NOF\n0dMWZadOW5SdWnheR4f4kMUFlQgx+XbsbJI/Z5P8hafKFFTfAL8D7gAWhSYcYaeLdm3kaN/+hVU9\nf8rNQiAFpaKq64BCCCFEVQiqoHK5XJ2Bqw3DGGMYxpXAky6Xq76vbd3Px6GUSpPlmrt80c6NHOp3\nqcZNUWFhpK+//fHext/+MfUje7y3cO2i3PgukZl//8enNeH+16bl4r9rSjyyLPkLp2XJn/OXg6G0\n1uVv5cXlciUA0w3DGONyuRSwGkgxDOOM+3bp6el6xIgR8iuuGm7uup1Lpi3KTjVeuoufX/p/J548\n0LhR8brJw+Pznl+U3TSYvyuy3UNf/pMhbetnd/lgVqfqu8e1n1IyKdbJJH/OJvlzrsrULUH1UBmG\nkQl873K5vgLmAW94F1PCWZqezKNl/i8c696rqDqPu65LP1r8uK5ZdR4zHMibubNJ/pxN8heegv6V\nn2EYfw5lIMJevXdvYXP77hARUa3H/TE+kcb/l9kYpeqj9elqPbgQQggRInJiTwFA4s8/salDr2o/\n7qnoBpxs1/EkMKjaD16LVXYugLCX5M/ZJH/hSQoqAUDi7i1stKGgAjjat38uMMyWgwshhBAhIAWV\nQBUW0mvvNjZ3KPe8nVXi8KVDjiEFVUjJHA5nk/w5m+QvPElBJYj5KaPhL41iyGvYtPyNq8ChpJQ8\n4CKUamBLAEIIIUQlSUElaLF+dZOMjvYM9wEUNmp8HvgRGGxbELWMzOFwNsmfs0n+wpMUVIJmGT82\nsWNCupfFyLCfEEIIh5KCStB0+09NMjr0tDuMJUhBFTIyh8PZJH/OJvkLT0Gfh0rUEkrFREVFR2W1\n6Wx3JCuBvijVAK1P2R2MEEIIEQjpoRKXnojvfKKomk/oWYpZRGUAl9obSO0gczicTfLnbJK/8CQF\nleifn9DzuN1BWL4DUuwOQgghhAiUFFSi/7EeF56wOwjLMiDZ7iBqA5nD4WySP2eT/IUnmUMVplZl\n5sw6kF8Qf3mrNgNyLx10kp/tjgiAFcC7KBWB1tV6kWYhhBCiMqSHKkwdyC+If+XLTan62LH6ufFd\n69odDwBaHwb2AX3sDsXpZA6Hs0n+nE3yF56koApj3fdlsaNtZ7B7Qrqn75BhPyGEEA4jBVUY656T\nyda4BLvD8CbzqEJA5nA4m+TP2SR/4UkKqjDWY1+NLahSUErZHYgQQghRUVJQhbEeOZlsq3kF1S7r\n/062RuFwMofD2SR/zib5C09SUIWpyBPH68QePcDOVh3tDsWT1hqZRyWEEMJhpKAKU80y1jfa2aoj\nhZE14wd+XsxhPxE0mcPhbJI/Z5P8hScpqMJUs80bGtXA4b5iMjFdCCGEo0hBFaaabvupcQ2ckF5s\nIxCLUi3tDsSpZA6Hs0n+nE3yF56koApTTbIyG9XYgso8S/pKYIjdoQghhBAVIQVVOFIqqv6BnAZZ\nbWv0D+lk2K8SZA6Hs0n+nE3yF56koApPvU+3anu6oG6U3XGURSamCyGEcAwpqMJT/+NdEk7YHUQ5\nVgO9UaqB3YE4kczhcDbJn7NJ/sKTFFThqX9et17H7Q6iTFqfAjYBl9odihBCCFEeKajCU//cC/vU\n9B4qkHlUQZM5HM4m+XM2yV94koIq3ChVF+id26e/FFRCCCFEiEhBFX56AHvONWlaZHcgFbACSEKp\nCLsDcRqZw+Fskj9nk/yFp8hgd3S5XO2A96w21hiG8VjIohJVqT/wAxBrdyDl0vogSh0CLsQ82acQ\nQghRIwVdUAEvAZMNw1gRqmBEtXBOQWVajnmCTymoAiBzOJxN8udskr/wFNSQn8vligC6SDHlSMUF\nlVPIPCohhBA1XrBzqFoC0S6X61OXy7XI5XJdH8qgRBVRqg7QD+cVVHIJmgDJHA5nk/w5m+QvPAVb\nUP0C5AG/Aa4EJrlcrvq+NnR/Yiml0mTZvuUH4NZTkI/Wx/CjqLAw0t+y97pQ7+8r/khoCzREqfZ2\nP36yLMuyLMuyHB7LwVBa66B2dLlcHwBPGIaR43K5lgGXGYZx2n2b9PR0PWLECFWZAEUIKXULcANa\n3zR33c4l0xZlpwJMHh6f9/yi7Kbef5e1riJ/V3S7p4bHL7324s5pZcT9KfAhWn8YssdCCCGE8FKZ\nuqUyp034A/CWy+VaDsz2LqZEjXQxzhruK1Y8MV0IIYSokYL+lZ9hGLuBq0IYi6h6/YEX7Q4iCMuA\nf9gdhJMopdLkl0bOJflzNslfeJITe4YLpRRmQbXe7lCC8AOQgFJN7A5ECCGE8EUKqvDRCTiO1ofs\nDiRgWhdgFlWD7A7FKeTbsbNJ/pxN8heepKAKH047/5Q3OR+VEEKIGqsyZ0oXNdCqzJxZB/IL4gHa\nNInKHpgQN85a5ZiCys99WA7I5Y0qSOZwOJvkz9kkf+FJCqpa5kB+QXzx6RBeGt21x9x1O5cApPTp\n3yfvNzf90MnW6CrG132o982qyCuuGTqwjlJ10fqc3TEKIYQQ7qSgqsVyTxdGP78oOxWt+TprJzv6\nDmzghILKXcl9AIa2bH2y4d7dfYG1NodV48m3Y2eT/Dmb5C88yRyqMNAq7whawanY9mftjqUyjvXs\nnY/MoxJCCFEDSUEVBnrkZLItLgHqODvdv/S9JA85wWeFVPYSCsJekj9nk/yFJ2d/wooK6Z6Tyda4\nBLvDqLQjAwbnAcnWObWEEEKIGkPmUIWBHjmZfHHJ5Y7v2jke3+UMcB7ovCoz5+niXwJCqV80hj2Z\nw+Fskj9nk/yFJymowkD3fZlMj3ugxhZUMfUjS36NCBBRR/XwuaE5ZLkcGOL+S0CAp4bHV22QQggh\nRBmkoKrlmh8/Sv2zZ9jXrI3dofjl/ks+gMnD4/PK2FxO8FkBch4cZ5P8OZvkLzzJHKparvu+HeaE\n9Noz7WgZMjFdCCFEDSMFVS3XPSeTrbHOn5DuZiPQPuqXw9K7Wgb5duxskj9nk/yFJymoarketeQX\nfiW0LgRWtVy1vKndoQghhBDFpKCq5XrkWEN+tUDx5PWdY+9IaLZts9NO+l6t5Dw4zib5czbJX3iS\ngqoWq5t3TMWcOMbuC+LsDiUkck8XRk9blJ06vUnvjjEb10fbHY8QQghRTAqqWixmy6aI7bFd0A4/\nQ7q3Te170iRza0S9c46+kk6Vkjkczib5czbJX3iqXZ+0wkPM1k0RtWW4z92p6Aac6BBf1CNnu92h\nCCGEEIAUVLVazNbNEbVqQrqbo30vLuqbvcnuMGosmcPhbJI/Z5P8hScpqGqxpts2R2xt183uMKrE\n0b4XF/bbJQWVEEKImkEKqlqq4ZmTRB8+VCe7ZQe7Q6kSRy66tLDPz5tR58/bHUqNJHM4nE3y52yS\nv/AkBVUt1T0nk/wu3YqKIiLsDqVKnGndVp+MbkinQ7vtDkUIIYSQgqq26rl3O3k9LiyyO46qtK5z\nXy7e+aPdYdRIMofD2SR/zib5C09SUNVSPXIyOdazd+0uqLr0pX/WBrvDEEIIIaSgqq167t1Obq8+\nhXbHUZXWdenHxTs3yDwqH2QOh7NJ/pxN8heepKCqhRqeOUmrvCMc75xQqyuNgzGtOBHdkM6HfrY7\nFCGEEGFOCqpaqHvODjLbdkJHRtodSpVb17kvF2fJPCpvMofD2SR/zib5C09SUNVCPXK2U1vPP+Vt\nXZd+XCIFlRBCCJtVqgvD5XJFAduBvxqG8UZoQhKV1XNvJqsT+tPH6/aY+pE95q7buQQgoo7qUf2R\nhd7aLv14/LPXWVpUq+ffB0zmcDib5M/ZJH/hqbI9VBOBdYAOQSwiRHrkbGdLu9KXnMk9XRg9bVF2\n6rRF2amF53W0DaGF3JGmF5DXoCnNM35saHcsQgghwlfQBZXL5WoAXAbMBVTIIhKVUjfvWETrY4fZ\n1Sre7lCqzboufWn1/bfNipdXZebMmrtu55K563YuWZWZM8u+yOwjczicTfLnbJK/8FSZIb+HgdeB\n1iGKRYRA8w1rG2W27URtPUO6L2u79CNlfXpM8fKB/IL4aYuyUwGeGh5vW1xCCCHCR1A9VC6XqymQ\nbBjGfMrpnXKv1JVSabJctcv1f1zXqnhCelFhoUfB7L3sS1n7VMf+ZbXnb/8fuvQjZktG00ZKDff+\nZph/LLek0KoJ+amuZa31kpoUjyxL/sJpWfLn/OVgBNtDNQSIdrlcHwCdgEiXy7XYMIyfvDd0n5zn\nPVFPlkO/fCB5WOTcuEsBiIiM9Dixp/eyL2XtUx37l9Wev/1/adycc01jzp44dTIXrdfPXbezZF2T\nmGbHiv+uCfmRZVmWZVmW5Zq7nJ6eTrCCKqgMw/gK+ArA5XLdCTT0VUyJ6tcka3vjLQN+a3cY1e5o\n4kXHGuzPGQ6stzuWmkAplSa/NHIuyZ+zSf7CU6XP/GgYxjuhCESEgFKNo+pFRYXThPRihwcOyW33\nzRcjgel2xyKEECL8yIk9a5dLT3TsdCKcJqQXOzB0xDFgCErVitNBVJZ8O3Y2yZ+zSf7CkxRUtcvA\nvO4X5tsdhB3ONmtRCGzCnN8nhBBCVCspqGqXgbm9+4VlQWX5BrjC7iBqgsr+WkXYS/LnbJK/8CQF\nVW2hlAIGHrl4wHG7Q7HR18DldgchhBAi/EhBVXu0B+qc6Nj5jN2B2GgN0LH+/px6dgdiN5nD4WyS\nP2eT/IUnKahqgVWZObN+nDRt3pH+AyIiIiNqxUWPAxVTP7LH3LVZCw9fOriw9dqVveyORwghRHiR\ngqoWOJBfEL9p2YZeRsveLWrLRY8DVXz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      "text/plain": [
       "<matplotlib.figure.Figure at 0x1152cab90>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Posterior Mean\n",
    "# (use all of `thetas` b/c PyMC already removed the burn-in runs here)\n",
    "print 'Posterior Mean (MH):', np.mean(thetas)\n",
    "\n",
    "# Plot the posterior\n",
    "fig = plt.figure(figsize=(10,4))\n",
    "ax = fig.add_subplot(111)\n",
    "# Plot MH draws\n",
    "ax.hist(thetas, bins=50, normed=True);\n",
    "# Plot analytic posterior\n",
    "X = np.linspace(0,1, 1000)\n",
    "ax.plot(X, stats.beta(a1_hat, a2_hat).pdf(X), 'r');\n",
    "# Plot prior\n",
    "ax.plot(X, stats.beta(a1, a2).pdf(X), 'g')\n",
    "\n",
    "# Cleanup\n",
    "ax.set(title='Metropolis-Hastings via PyMC (10,000 Draws; 1,000 Burned)', ylim=(0,12))\n",
    "ax.legend(['Posterior (Analytic)', 'Prior', 'Posterior Draws (MH)']);"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 2",
   "language": "python",
   "name": "python2"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 2
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython2",
   "version": "2.7.6"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}