{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "4248ac61",
   "metadata": {},
   "source": [
    "### Exercise 3"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c9785b17",
   "metadata": {},
   "source": [
    "#### Import"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "067e76c0",
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from scipy.stats import gamma\n",
    "from scipy.special import gamma as gammafunc\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "412f4818",
   "metadata": {},
   "source": [
    "#### Data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "eedfc0b6",
   "metadata": {},
   "outputs": [],
   "source": [
    "y = np.array([6, 8, 7, 6, 7, 4, 11, 8, 6, 3])\n",
    "n = len(y)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1bd7a079",
   "metadata": {},
   "source": [
    "### (d)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ed801420",
   "metadata": {},
   "source": [
    "#### Parameters"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "7537a58f",
   "metadata": {},
   "outputs": [],
   "source": [
    "a = .1\n",
    "b = .1"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8061f922",
   "metadata": {},
   "source": [
    "#### Plots"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "4a94a3b8",
   "metadata": {},
   "outputs": [],
   "source": [
    "space = np.linspace(1, 12, 200)\n",
    "\n",
    "# Prior\n",
    "plt.plot(space, gamma(a=a, scale=1/b).pdf(space), linestyle='dotted', color='black', label='prior')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "02f845da",
   "metadata": {},
   "outputs": [],
   "source": [
    "space = np.linspace(1, 12, 200)\n",
    "\n",
    "# Prior\n",
    "plt.plot(space, gamma(a=a, scale=1/b).pdf(space), linestyle='dotted', color='black', label='prior')\n",
    "\n",
    "# Posteriors\n",
    "for j in range(1, 10+1):\n",
    "    a_new = a + np.sum(y[:j])\n",
    "    b_new = b + j\n",
    "    if j < 9:\n",
    "        plt.plot(space, gamma(a=a_new, scale=1/b_new).pdf(space), linestyle='dashed', \n",
    "                 color='black', alpha=.5)\n",
    "    elif j == 9:\n",
    "        plt.plot(space, gamma(a=a_new, scale=1/b_new).pdf(space), linestyle='dashed', \n",
    "                 color='black', alpha=.5, label='intermediate posteriors')\n",
    "    else:\n",
    "        plt.plot(space, gamma(a=a_new, scale=1/b_new).pdf(space), color='black', \n",
    "                 label='posterior')\n",
    "        \n",
    "plt.legend()\n",
    "plt.xlabel(r'$\\theta$')\n",
    "plt.ylabel('density')\n",
    "        \n",
    "        \n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ed2a42ce",
   "metadata": {},
   "source": [
    "#### Bayes estimators and posterior standard deviations"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "149cfa14",
   "metadata": {},
   "outputs": [],
   "source": [
    "for j in range(1, 10+1):\n",
    "    a_new = a + np.sum(y[:j])\n",
    "    b_new = b + j\n",
    "    \n",
    "    Bayes = a_new/b_new\n",
    "    std = np.sqrt(a_new / b_new**2)\n",
    "    \n",
    "    print(f'After {j} observation(s):')\n",
    "    print(f'Bayes estimate: {np.round(Bayes, 3)}')\n",
    "    print(f'Posterior standard deviation: {np.round(std, 3)}')\n",
    "    print('')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5446c936",
   "metadata": {},
   "source": [
    "### (e)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "00c17b5b",
   "metadata": {},
   "source": [
    "#### Parameters"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "aee480c0",
   "metadata": {},
   "outputs": [],
   "source": [
    "a = .5\n",
    "b = 50"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "00fa5326",
   "metadata": {},
   "source": [
    "#### Plots"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "97a4c7c8",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Prior\n",
    "plt.plot(space, np.ones(200)/(b-a), linestyle='dotted', color='black', label='prior')\n",
    "\n",
    "# Posteriors\n",
    "for j in range(1, 10+1):\n",
    "    ybar = np.mean(y[:j])\n",
    "    I = gamma(a=1 + j*ybar, scale=1/j).cdf(b) - gamma(a=1 + j*ybar, scale=1/j).cdf(a)\n",
    "    \n",
    "    posterior = j**(1 + j*ybar)/(gammafunc(1 + j*ybar)) * space**(j*ybar) * np.exp(-j*space) / I\n",
    "    \n",
    "    if j < 9:\n",
    "        plt.plot(space, posterior, linestyle='dashed', \n",
    "                 color='black', alpha=.5)\n",
    "    elif j == 9:\n",
    "        plt.plot(space, posterior, linestyle='dashed', \n",
    "                 color='black', alpha=.5, label='intermediate posteriors')\n",
    "    else:\n",
    "        plt.plot(space, posterior, color='black', \n",
    "                 label='posterior')\n",
    "        \n",
    "plt.legend()\n",
    "plt.xlabel(r'$\\theta$')\n",
    "plt.ylabel('density')\n",
    "        \n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "d6301ca1",
   "metadata": {},
   "outputs": [],
   "source": [
    "for j in range(1, 10+1):\n",
    "    ybar = np.mean(y[:j])\n",
    "    I1 = gamma(a=1 + j*ybar, scale=1/j).cdf(b) - gamma(a=1 + j*ybar, scale=1/j).cdf(a)\n",
    "    I2 = gamma(a=2 + j*ybar, scale=1/j).cdf(b) - gamma(a=2 + j*ybar, scale=1/j).cdf(a)\n",
    "    I3 = gamma(a=3 + j*ybar, scale=1/j).cdf(b) - gamma(a=3 + j*ybar, scale=1/j).cdf(a)\n",
    "    \n",
    "    Bayes = (1 + j*ybar)/j * I2/I1\n",
    "    Var = (2+j*ybar)*(1+j*ybar)/j**2 * I3/I1 - Bayes**2\n",
    "    std = np.sqrt(Var)\n",
    "    \n",
    "    \n",
    "    print(f'After {j} observation(s):')\n",
    "    print(f'Bayes estimate: {np.round(Bayes, 3)}')\n",
    "    print(f'Posterior standard deviation: {np.round(std, 3)}')\n",
    "    print('')"
   ]
  }
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