Winter School Exercises (Answers).ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "time: 827 ms\n"
     ]
    }
   ],
   "source": [
    "import subprocess\n",
    "\n",
    "from sklearn.gaussian_process import GaussianProcessRegressor as GPR\n",
    "from sklearn.gaussian_process import kernels\n",
    "from sklearn.decomposition import PCA\n",
    "from sklearn.preprocessing import StandardScaler\n",
    "\n",
    "import matplotlib.cm as cm\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "from scipy import stats\n",
    "import emcee\n",
    "import numpy as np\n",
    "\n",
    "\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise 1 - Latin Hypercube Design"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "time: 8.9 ms\n"
     ]
    }
   ],
   "source": [
    "def generate_lhs(npoints, ndim, seed):\n",
    "    \"\"\"\n",
    "    Generate a maximin Latin-hypercube sample (LHS) with the given number of\n",
    "    points, dimensions, and random seed.\n",
    "\n",
    "    \"\"\"\n",
    "\n",
    "    proc = subprocess.run(\n",
    "        ['R', '--slave'],\n",
    "        input=\"\"\"\n",
    "        library('lhs')\n",
    "        set.seed({})\n",
    "        write.table(maximinLHS({}, {}), col.names=FALSE, row.names=FALSE)\n",
    "        \"\"\".format(seed, npoints, ndim).encode(),\n",
    "        stdout=subprocess.PIPE,\n",
    "        check=True\n",
    "    )\n",
    "\n",
    "    lhs = np.array(\n",
    "        [l.split() for l in proc.stdout.splitlines()],\n",
    "        dtype=float\n",
    "    )\n",
    "\n",
    "\n",
    "    return lhs\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Create and plot design matrix below\n",
    "##### 20 points, 2 dimensions, set the seed to 80"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise**: Fill in the parameters for generate_lhs(), and for plt.scatter"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "time: 336 ms\n"
     ]
    }
   ],
   "source": [
    "##Fill in the parameters to generate_lhs\n",
    "design = generate_lhs(npoints = 20,\n",
    "                      ndim = 2,\n",
    "                      seed = 80)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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+Yqs29bP20XPnOPe8JY5DUp8WesRW2lPzdh8lORF4NnBIkt6ZyCuBG9oOTNLcfMRWbVpo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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10e180630>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "time: 346 ms\n"
     ]
    }
   ],
   "source": [
    "###Fill in parameters to plt.scatter\n",
    "plt.scatter(x = design[:,0],\n",
    "           y = design[:,1]) \n",
    "plt.title('Latin Hypercube Design')\n",
    "plt.xlabel('Input 1')\n",
    "plt.ylabel('Input 2')\n",
    "pass"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise 2 - Toy GP Example"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Part a - Mean and Variance Estimate"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Run the two cells below"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "time: 2.7 ms\n"
     ]
    }
   ],
   "source": [
    "def truth(x):\n",
    "    return(3*x+np.cos(5*x))    \n",
    "design = np.linspace(start =-1,stop=1,num=5)\n",
    "model_data = truth(design)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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dhHTGUM+ZhdmgNmveElavWdelbPWadcyat8RnCbZRqmfH/s3C47XA74HjmxJNNyRNB6YD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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1114b4240>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "time: 244 ms\n"
     ]
    }
   ],
   "source": [
    "plt.scatter(x=design,y=model_data)\n",
    "plt.title('Computer Model Output at Design Points')\n",
    "plt.xlabel('Design')\n",
    "plt.ylabel('Model Output')\n",
    "pass"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Run the cell below, training the GP"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "time: 56.8 ms\n"
     ]
    }
   ],
   "source": [
    "ptp = 2\n",
    "kernel = (\n",
    "    1. * kernels.RBF(\n",
    "        length_scale=ptp,\n",
    "        length_scale_bounds=np.outer(ptp, (.1, 10))\n",
    "    ) \n",
    ")\n",
    "gp = GPR(kernel=kernel,\n",
    "    #alpha=0,\n",
    "    n_restarts_optimizer=0,\n",
    "    copy_X_train=False).fit(design.reshape(-1,1), model_data)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise**: Create the vector of points X on which we will predict. X should be a (n x 1) numpy array (you choose n)\n",
    "\n",
    "    Hint: We only want to predict within the bounds of our design"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "time: 1.35 ms\n"
     ]
    }
   ],
   "source": [
    "X = np.linspace(-1,1,101)\n",
    "X = X.reshape(-1,1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "time: 8.23 ms\n"
     ]
    }
   ],
   "source": [
    "#This returns the predictive mean and covariance at all the points in X\n",
    "#mean is a (n,) numpy array, and cov is a (n,n) numpy array\n",
    "mean, cov = gp.predict(return_cov=True,X=X)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise**: Fill in the lines for plt.plot, top_var, and bot_var"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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GhuLm9pfeHThwIEFBQXh7e/Ptt99Wb1+6dClBQUF06dKF4uLiBucCxowZw4YNGygoKGDn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\nFQQREfGoIIiICAD/B0SV5ovXq/tIAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1115c3a90>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "time: 281 ms\n"
     ]
    }
   ],
   "source": [
    "#Set up the figure by first plotting the output at the design points\n",
    "plt.scatter(x = design,y = model_data,color = 'black',label = 'Design Output')\n",
    "plt.title('Computer Model Output at Design Points')\n",
    "plt.xlabel('Input')\n",
    "plt.ylabel('Model Output')\n",
    "\n",
    "\n",
    "#Add the mean 95% uncertainty interval of the GP predictions at all the in-between points\n",
    "##Fill in parameters to plt.plot\n",
    "##Find expressions for top_var and bot_var\n",
    "###### Hint: For a normal distribution, how many standard deviations away from the mean is the 97.5% quantile?\n",
    "###### Hint: Use the diagonal of predictive covariance matrix`\n",
    "plt.plot( X, mean,color= 'blue',label = 'GP Mean')\n",
    "top_var = mean + 1.96*np.sqrt(np.diag(cov))\n",
    "bot_var = mean - 1.96*np.sqrt(np.diag(cov))\n",
    "plt.fill_between(X[:,0], bot_var, top_var, where=top_var >= bot_var, facecolor='lightgray', interpolate=True)\n",
    "\n",
    "plt.plot(X,truth(X),color='black',label = 'Truth')\n",
    "plt.legend(loc='best', fontsize=12)\n",
    "\n",
    "pass"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Part b - Random Draws"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The prior exercises displayed the mean and variance of our function at all the points in X. But what about actual samples of the function itself? Here we visualize what a random draw of the function would look like."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercse**: Fill in the lines for top_var, bot_var, and rand_draw"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "    Hint: For rand_draw, examine np.random.multivariate_normal()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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I1SB5KK2MLoHau1C520zZ7xbcZQoBKUayr3uAMhnJG3NTGTakgnEXtDz4yQv0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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x11147e080>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "time: 320 ms\n"
     ]
    }
   ],
   "source": [
    "#Get the upper 95% quantile, and lower 95% quantile of the GP predictions at all the in-between points\n",
    "##Fill in same values as previous exercise\n",
    "top_var = mean + 1.96*np.sqrt(np.diag(cov))\n",
    "bot_var = mean - 1.96*np.sqrt(np.diag(cov))\n",
    "\n",
    "plt.scatter(design,model_data,color = 'black',label = 'Design Output')\n",
    "plt.title('Computer Model Output at Design Points')\n",
    "plt.xlabel('Input')\n",
    "plt.ylabel('Model Output')\n",
    "plt.fill_between(X[:,0], bot_var, top_var, where=top_var >= bot_var, facecolor='lightgray', interpolate=True)\n",
    "plt.plot(X,truth(X),color='black',label = 'Truth')\n",
    "\n",
    "ndraws = 10\n",
    "colors = cm.rainbow(np.linspace(0, 1, ndraws))\n",
    "\n",
    "#Get [ndraws] random draws from the predictive distribution of the GP at all of the in-between points \n",
    "#Use the predictive mean and covariance\n",
    "rand_draw = np.random.multivariate_normal(mean,cov,10)\n",
    "for i in range(ndraws):\n",
    "    plt.plot(X,rand_draw[i,:],color = colors[i],linestyle = \":\")\n",
    "\n",
    "plt.legend(loc='best', fontsize=14)\n",
    "\n",
    "pass\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise 3 - Principal Component Analysis"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The dataset below contains 20 developmental indices for 132 countries around the world. The country labels (rows) can be found in 'countries.txt,' and the index descriptions can be found in 'indices.txt.' Here we look to explore some of the practical uses of PCA for the purposes of Computer Emulation."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "time: 7.9 ms\n"
     ]
    }
   ],
   "source": [
    "developmental_indices = np.loadtxt('dev_indices.txt')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "time: 11.6 ms\n"
     ]
    }
   ],
   "source": [
    "scaler = StandardScaler(copy=False)\n",
    "pca = PCA(copy=False, whiten=True, svd_solver='full')\n",
    "Z = pca.fit_transform(scaler.fit_transform(developmental_indices))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise**: Plot the cummulative fraction of variance explained. How many PCs would you recommend using?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "    Hint: Examine attributes of the object pca, as well as np.cumsum()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "time: 875 µs\n"
     ]
    }
   ],
   "source": [
    "F_r = np.cumsum(pca.explained_variance_ratio_)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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ARo8e3d0xtrJjbx0P/LOMj00bzuQjBqY0FhGRrpbsM4ZyYFTE+MhwWqTNwEJ3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\n/x80JTz8rX2XowAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1129010f0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "time: 226 ms\n"
     ]
    }
   ],
   "source": [
    "plt.plot(range(len(F_r)),F_r,'-o')\n",
    "plt.title('Fraction of Variance Explained')\n",
    "plt.xlabel('Number of Components')\n",
    "plt.ylabel('F_r')\n",
    "pass"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise**: Find the correlation between the first two principal component vectors"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([  2.75335310e-14])"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "time: 4.4 ms\n"
     ]
    }
   ],
   "source": [
    "corr = np.correlate(Z[:,0],Z[:,1])\n",
    "corr"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise**: Plot the second principal component against the first"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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WG1R5AsITwAZJ/87kye3+qGup6rCyZlpVL4n2Q0BLM8jzGpX1t2K9kWfY6ZeBPwe+S+2O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\nt4Fttaph1a4AAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x11289dcc0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "time: 202 ms\n"
     ]
    }
   ],
   "source": [
    "plt.scatter(Z[:,0],Z[:,1])\n",
    "plt.title('Principle Components')\n",
    "plt.xlabel('Component 1')\n",
    "plt.ylabel('Component 2')\n",
    "pass"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercise 4 - Bayes Rule"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here, we're going to explore the mechanics of Bayesian analysis, and how prior choices can affect the posterior outcome. Let's revisit the coin example from the slides - we flipped 10 coins, and 7 came up heads. Our goal to make inference on the true probability of heads $\\theta$, using Bayesian analysis. To do this, we need a likelihood and prior. \n",
    "\n",
    "\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "time: 3.31 ms\n"
     ]
    }
   ],
   "source": [
    "coin_tosses = np.loadtxt('coin_tosses.txt')\n",
    "num_heads = sum(coin_tosses)\n",
    "N = len(coin_tosses)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Likelihood**:\n",
    "A common model for independent coin flips (or anything with a 0/1 outcome) is Binomial. So our likelihood is Binomial with size N = 10 and probability $\\theta$:\n",
    "\\begin{align}\n",
    "    y\\sim \\text{Binom}(N,\\theta)\n",
    "\\end{align}\n",
    "\n",
    "The mean of a Binomial random variable $Y$ with size $N$ (flips) and probability $\\theta$ (probability of heads) is\n",
    "\n",
    "\\begin{align}\n",
    "    \\mathsf{E}(Y) = N\\theta.\n",
    "\\end{align}\n",
    "\n",
    "The variance of $Y$ is\n",
    "\n",
    "\\begin{align}\n",
    "    \\mathsf{V}(Y) = N\\theta(1-\\theta)\n",
    "\\end{align}\n",
    "\n",
    "More information on Binomial distribution can found here: https://en.wikipedia.org/wiki/Binomial_distribution. \n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Prior**:\n",
    "The most widely used prior for Binomial data and unknown probability is a Beta distribution. A Beta random variable can take value in (0,1).  \n",
    "\n",
    "\\begin{align}\n",
    "    \\theta\\sim \\text{Beta}(a, b)\n",
    "\\end{align}\n",
    "\n",
    "The mean of a Beta random variable $\\theta$ with first parameter $a$ and second paramter $b$ is\n",
    "\n",
    "\\begin{align}\n",
    "    \\mathsf{E}(\\theta) = \\frac{a}{a+b}\n",
    "\\end{align}\n",
    "\n",
    "The variance of $\\theta$ is\n",
    "\n",
    "\\begin{align}\n",
    "    \\mathsf{V}(\\theta) = \\frac{a + b}{(a+b)^2(a+b+1)}\n",
    "\\end{align}\n",
    "\n",
    "More information on the Beta distribution can found here: https://en.wikipedia.org/wiki/Beta_distribution"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Goal: Explore different priors for $\\theta$ to see how the posterior of $\\theta$ responds. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Method 1: MCMC via python package emcee"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Fill in missing pieces to calc_lnprior and calc_lnlike. These functions calculate the log of the prior and likelihood, respectively. For example, for the prior, you need to calculate the log of the pdf at theta for given parameters $a$ and $b$. Use the Wikipedia pages to find the pdfs.\n",
    "\n",
    "    Hint 1: For the Beta(a,b) prior, you will need to calculate the Beta function of parameters a and b (see the Wikipedia page). The cell below imports betln, which calculates the natural log of the Beta function.\n",
    "\n",
    "    Hint 2: For the Binomial likelihood, you'll need to calculate \n",
    "$$\\left(\\begin{matrix}N\\\\y\\end{matrix}\\right)$$\n",
    "    \n",
    "    This is the number of ways to get y heads in N flips. The function nCr(N,y) in the cell below calculates this.\n",
    "\n",
    "\n",
    "    Hint 3: If done correctly:\n",
    "\n",
    "    prior_ln_pdf(theta=0.4, a = 2,b = 3) = 0.5470\n",
    "\n",
    "    likelihood_ln_pdf(theta=0.5,y=num_heads,N=10) = -2.1440\n",
    "    \n",
    "    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "6\n",
      "time: 5.38 ms\n"
     ]
    }
   ],
   "source": [
    "from scipy.special import betaln\n",
    "import math\n",
    "\n",
    "def nCr(n,r):\n",
    "    f = math.factorial\n",
    "    return f(n) // f(r) // f(n-r)\n",
    "\n",
    "print(nCr(4,2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "time: 29.8 ms\n"
     ]
    }
   ],
   "source": [
    "#Calculate the log pdf of the prior of theta\n",
    "def prior_ln_pdf(theta,a,b):\n",
    "    #Ensure theta is within (0,1)\n",
    "    if theta <=0 or theta >=1:\n",
    "        return -np.inf\n",
    "    \n",
    "    ######\n",
    "    ##Add function return\n",
    "    ######\n",
    "    return (a-1)*np.log(theta) + (b-1)*np.log(1-theta) - betaln(a,b)\n",
    "\n",
    "#Calculate the log pdf of the likelihood of y successes (heads) in N trials (flips)\n",
    "def likelihood_ln_pdf(y, theta, N):\n",
    "    ######\n",
    "    ##Add function return\n",
    "    ######\n",
    "    return np.log(nCr(N,y)) + y*np.log(theta) + (N-y)*np.log(1-theta)\n",
    "\n",
    "def posterior_ln_pdf(theta,a,b,y, N):\n",
    "    ln_pr = prior_ln_pdf(theta=theta,a=a,b=b)\n",
    "    \n",
    "    if not np.isfinite(ln_pr):\n",
    "        return -np.inf\n",
    "        \n",
    "    ln_like = likelihood_ln_pdf(y=y, theta=theta,  N = N)\n",
    "    return ln_pr + ln_like"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The code below sets up the sampler. We provide the number of walkers nwalkers (chosen), the number of parameters ndim (1 in our case, since we just have $\\theta$), and the posterior function which takes as argument the parameters we're inferrin on. \"args\" is a tuple of all of arguments to the probability function"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "time: 1.15 ms\n"
     ]
    }
   ],
   "source": [
    "a = 1\n",
    "b = 10"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "time: 753 ms\n"
     ]
    }
   ],
   "source": [
    "nsamples = 100\n",
    "nburnin= 50\n",
    "ndim, nwalkers = 1, 200\n",
    "sampler = emcee.EnsembleSampler(nwalkers, ndim, posterior_ln_pdf, args=(a, b, num_heads,N))\n",
    "p0 = np.random.rand(nwalkers,ndim)\n",
    "out_post = sampler.run_mcmc(p0,nsamples)\n",
    "samples = sampler.chain[:, nburnin:, :].reshape((-1, ndim))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$nsteps$ is the number of steps each walker will run, while $nburnin$ is the number of \"burn-in\" or \"warmup\" steps for each walker.  Our total number of samples will be $nwalkers\\times (nsteps-nburnin)$ (recall we want 10,000 total samples). After running each walker for $nsteps$ and discarding the first $nburnin$, we reshape the output chain so each row is a draw from the posterior"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "time: 722 ms\n"
     ]
    }
   ],
   "source": [
    "nsamples = 100\n",
    "nburnin= 50\n",
    "out_post = sampler.run_mcmc(p0,nsamples)\n",
    "samples = sampler.chain[:, nburnin:, :].reshape((-1, ndim))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Use the samples to find posterior means and variances"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "time: 13 ms\n"
     ]
    }
   ],
   "source": [
    "#Compare the prior density to the posterior samples\n",
    "def plot_comparison_mcmc(samples,prior_a, prior_b):\n",
    "    theta_grid = np.linspace(0.01,0.99,100)\n",
    "    \n",
    "    #Calculate the posterior density at the grid of theta values\n",
    "    prior_pdf_vals = stats.beta(a,b).pdf(theta_grid)\n",
    "    \n",
    "    print('Prior mean is')\n",
    "    print(prior_a/(prior_b+prior_b))\n",
    "    print('Posterior mean is')\n",
    "    print(samples.mean())\n",
    "    \n",
    "    \n",
    "    plt.plot(theta_grid,prior_pdf_vals,label = 'Prior Density' )\n",
    "    plt.hist(samples,normed = True,label = 'Posterior Samples')\n",
    "    plt.legend(loc='best', fontsize=12)\n",
    "    pass"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise**: In the above code, change a and b to plot a comparison of the prior density to the histogram of posterior samples for each set of priors. Also report the prior mean versus the mean of the posterior samples\n",
    "\n",
    "a = 1, b = 1\n",
    "\n",
    "a = 1, b = 10\n",
    "\n",
    "a = 10, b = 1\n",
    "\n",
    "a = 30, b = 30"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Prior mean is\n",
      "0.05\n",
      "Posterior mean is\n",
      "0.389830738237\n"
     ]
    },
    {
     "data": {
      "image/png": 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OlyMiQUDB7WeR4WE8euUZlB/08F9/XI3XqzsIisipUXC3g36Z8dzz3X58uLFE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\nAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x11277df98>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "time: 225 ms\n"
     ]
    }
   ],
   "source": [
    "plot_comparison_mcmc(samples,a,b)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Method 2: Direct computation of the posterior"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can find the analytics posterior in this case"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "First, recall the likelihood and prior"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\\begin{align}\n",
    "\ty\\mid\\theta&\\sim \\mathsf{Binom}(N,\\theta)\\\\\n",
    "\t\\theta&\\sim \\mathsf{Beta}(a,b)\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Then, we apply Bayes Rule and do a little algebra"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\\begin{align}\n",
    "\tp(\\theta\\mid y)&\\propto p(y\\mid\\theta)p(\\theta)\\\\\n",
    "\t &= \\left[\\begin{pmatrix}N \\\\ y\\end{pmatrix}\\theta^y(1-\\theta)^{N-y}\\right]\\frac{\\theta^{a-1}(1-\\theta)^{b-1}}{B(a,b)}\\\\\n",
    "\t &\\propto \\theta^{a+y-1}(1-\\theta)^{b+N-y-1}\n",
    "\\end{align}\n",
    "Note that when doing this algebra, we can disregard the constants $\\begin{pmatrix}N \\\\ y\\end{pmatrix}$ and $B(a,b)$ because they fall away in the proportionality.\n",
    "\n",
    "At the end, we recognize the kernel of a $\\mathsf{Beta}(a^*,b^*)$ distribution, where\n",
    "\\begin{align}\n",
    "\ta^* &= a + y\\\\\n",
    "\tb^* &= b + N - y\n",
    "\\end{align}\n",
    "This is posterior distribution of $\\theta$.\n",
    "\n",
    "Note that this math was very simple only because of the choice of prior and likelihood; a Beta distribution is a so-called \"conjugate prior\" for a Binomial likelihood. For more comlicated models the posterior will not be available analytically, and we must resort to sampling methods. We only used sampling methods in this model for pedagogical purposes."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise**: Find the parameters for the posterior distribution of $\\theta$ in our model. Compare the prior density to posterior density for the above choices of $a$ and $b$. Do the posterior densities align with the histogram of your posterior samples?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "time: 17.3 ms\n"
     ]
    }
   ],
   "source": [
    "a = 1\n",
    "b = 10\n",
    "N = len(coin_tosses)\n",
    "y = sum(coin_tosses)\n",
    "a_star = a + y\n",
    "b_star = b + N -y \n",
    "\n",
    "def plot_comparison_analytic(prior_a,prior_b, posterior_a, posterior_b):\n",
    "\n",
    "    theta_grid = np.linspace(0.01,0.99,100)\n",
    "    \n",
    "    #Calculate the prior pdf at the grid of theta values\n",
    "    prior_pdf_vals = stats.beta(prior_a,prior_b).pdf(theta_grid)\n",
    "    plt.plot(theta_grid, prior_pdf_vals)\n",
    "    plt.xlabel('Theta')\n",
    "    plt.ylabel('Pdf')\n",
    "    \n",
    "    #Calculate the posterior pdf at the grid of theta values\n",
    "    posterior_pdf_vals = stats.beta(posterior_a,posterior_b).pdf(theta_grid)\n",
    "    plt.plot(theta_grid,posterior_pdf_vals, label = 'Posterior')\n",
    "    plt.legend(loc='best', fontsize=12)\n",
    "    pass"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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Yj8cao+8PGP764Sp21zdwx457kKZ6GPs4eLxOl+heHg9c8CTsLremU07KtMbs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\nAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x112755908>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "time: 213 ms\n"
     ]
    }
   ],
   "source": [
    "plot_comparison_analytic(a,b,a_star,b_star)"
   ]
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   "source": []
  }
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