{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Bayesian approach to parameter estimation by optimization\n",
    "\n",
    "<hr>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "nbsphinx": "hidden",
    "tags": []
   },
   "outputs": [],
   "source": [
    "# Colab setup ------------------\n",
    "import os, sys, subprocess\n",
    "if \"google.colab\" in sys.modules:\n",
    "    cmd = \"pip install --upgrade bebi103 watermark\"\n",
    "    process = subprocess.Popen(cmd.split(), stdout=subprocess.PIPE, stderr=subprocess.PIPE)\n",
    "    stdout, stderr = process.communicate()\n",
    "    data_path = \"https://s3.amazonaws.com/bebi103.caltech.edu/data/\"\n",
    "else:\n",
    "    data_path = \"../data/\"\n",
    "# ------------------------------"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "        }\n",
       "    </style>\n",
       "    <div>\n",
       "        <a href=\"https://bokeh.org\" target=\"_blank\" class=\"bk-notebook-logo\"></a>\n",
       "        <span id=\"eabccb05-b1e7-4844-baf9-a05e274347a1\">Loading BokehJS ...</span>\n",
       "    </div>\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "application/javascript": [
       "(function(root) {\n",
       "  function now() {\n",
       "    return new Date();\n",
       "  }\n",
       "\n",
       "  const force = true;\n",
       "\n",
       "  if (typeof root._bokeh_onload_callbacks === \"undefined\" || force === true) {\n",
       "    root._bokeh_onload_callbacks = [];\n",
       "    root._bokeh_is_loading = undefined;\n",
       "  }\n",
       "\n",
       "const JS_MIME_TYPE = 'application/javascript';\n",
       "  const HTML_MIME_TYPE = 'text/html';\n",
       "  const EXEC_MIME_TYPE = 'application/vnd.bokehjs_exec.v0+json';\n",
       "  const CLASS_NAME = 'output_bokeh rendered_html';\n",
       "\n",
       "  /**\n",
       "   * Render data to the DOM node\n",
       "   */\n",
       "  function render(props, node) {\n",
       "    const script = document.createElement(\"script\");\n",
       "    node.appendChild(script);\n",
       "  }\n",
       "\n",
       "  /**\n",
       "   * Handle when an output is cleared or removed\n",
       "   */\n",
       "  function handleClearOutput(event, handle) {\n",
       "    function drop(id) {\n",
       "      const view = Bokeh.index.get_by_id(id)\n",
       "      if (view != null) {\n",
       "        view.model.document.clear()\n",
       "        Bokeh.index.delete(view)\n",
       "      }\n",
       "    }\n",
       "\n",
       "    const cell = handle.cell;\n",
       "\n",
       "    const id = cell.output_area._bokeh_element_id;\n",
       "    const server_id = cell.output_area._bokeh_server_id;\n",
       "\n",
       "    // Clean up Bokeh references\n",
       "    if (id != null) {\n",
       "      drop(id)\n",
       "    }\n",
       "\n",
       "    if (server_id !== undefined) {\n",
       "      // Clean up Bokeh references\n",
       "      const cmd_clean = \"from bokeh.io.state import curstate; print(curstate().uuid_to_server['\" + server_id + \"'].get_sessions()[0].document.roots[0]._id)\";\n",
       "      cell.notebook.kernel.execute(cmd_clean, {\n",
       "        iopub: {\n",
       "          output: function(msg) {\n",
       "            const id = msg.content.text.trim()\n",
       "            drop(id)\n",
       "          }\n",
       "        }\n",
       "      });\n",
       "      // Destroy server and session\n",
       "      const cmd_destroy = \"import bokeh.io.notebook as ion; ion.destroy_server('\" + server_id + \"')\";\n",
       "      cell.notebook.kernel.execute(cmd_destroy);\n",
       "    }\n",
       "  }\n",
       "\n",
       "  /**\n",
       "   * Handle when a new output is added\n",
       "   */\n",
       "  function handleAddOutput(event, handle) {\n",
       "    const output_area = handle.output_area;\n",
       "    const output = handle.output;\n",
       "\n",
       "    // limit handleAddOutput to display_data with EXEC_MIME_TYPE content only\n",
       "    if ((output.output_type != \"display_data\") || (!Object.prototype.hasOwnProperty.call(output.data, EXEC_MIME_TYPE))) {\n",
       "      return\n",
       "    }\n",
       "\n",
       "    const toinsert = output_area.element.find(\".\" + CLASS_NAME.split(' ')[0]);\n",
       "\n",
       "    if (output.metadata[EXEC_MIME_TYPE][\"id\"] !== undefined) {\n",
       "      toinsert[toinsert.length - 1].firstChild.textContent = output.data[JS_MIME_TYPE];\n",
       "      // store reference to embed id on output_area\n",
       "      output_area._bokeh_element_id = output.metadata[EXEC_MIME_TYPE][\"id\"];\n",
       "    }\n",
       "    if (output.metadata[EXEC_MIME_TYPE][\"server_id\"] !== undefined) {\n",
       "      const bk_div = document.createElement(\"div\");\n",
       "      bk_div.innerHTML = output.data[HTML_MIME_TYPE];\n",
       "      const script_attrs = bk_div.children[0].attributes;\n",
       "      for (let i = 0; i < script_attrs.length; i++) {\n",
       "        toinsert[toinsert.length - 1].firstChild.setAttribute(script_attrs[i].name, script_attrs[i].value);\n",
       "        toinsert[toinsert.length - 1].firstChild.textContent = bk_div.children[0].textContent\n",
       "      }\n",
       "      // store reference to server id on output_area\n",
       "      output_area._bokeh_server_id = output.metadata[EXEC_MIME_TYPE][\"server_id\"];\n",
       "    }\n",
       "  }\n",
       "\n",
       "  function register_renderer(events, OutputArea) {\n",
       "\n",
       "    function append_mime(data, metadata, element) {\n",
       "      // create a DOM node to render to\n",
       "      const toinsert = this.create_output_subarea(\n",
       "        metadata,\n",
       "        CLASS_NAME,\n",
       "        EXEC_MIME_TYPE\n",
       "      );\n",
       "      this.keyboard_manager.register_events(toinsert);\n",
       "      // Render to node\n",
       "      const props = {data: data, metadata: metadata[EXEC_MIME_TYPE]};\n",
       "      render(props, toinsert[toinsert.length - 1]);\n",
       "      element.append(toinsert);\n",
       "      return toinsert\n",
       "    }\n",
       "\n",
       "    /* Handle when an output is cleared or removed */\n",
       "    events.on('clear_output.CodeCell', handleClearOutput);\n",
       "    events.on('delete.Cell', handleClearOutput);\n",
       "\n",
       "    /* Handle when a new output is added */\n",
       "    events.on('output_added.OutputArea', handleAddOutput);\n",
       "\n",
       "    /**\n",
       "     * Register the mime type and append_mime function with output_area\n",
       "     */\n",
       "    OutputArea.prototype.register_mime_type(EXEC_MIME_TYPE, append_mime, {\n",
       "      /* Is output safe? */\n",
       "      safe: true,\n",
       "      /* Index of renderer in `output_area.display_order` */\n",
       "      index: 0\n",
       "    });\n",
       "  }\n",
       "\n",
       "  // register the mime type if in Jupyter Notebook environment and previously unregistered\n",
       "  if (root.Jupyter !== undefined) {\n",
       "    const events = require('base/js/events');\n",
       "    const OutputArea = require('notebook/js/outputarea').OutputArea;\n",
       "\n",
       "    if (OutputArea.prototype.mime_types().indexOf(EXEC_MIME_TYPE) == -1) {\n",
       "      register_renderer(events, OutputArea);\n",
       "    }\n",
       "  }\n",
       "  if (typeof (root._bokeh_timeout) === \"undefined\" || force === true) {\n",
       "    root._bokeh_timeout = Date.now() + 5000;\n",
       "    root._bokeh_failed_load = false;\n",
       "  }\n",
       "\n",
       "  const NB_LOAD_WARNING = {'data': {'text/html':\n",
       "     \"<div style='background-color: #fdd'>\\n\"+\n",
       "     \"<p>\\n\"+\n",
       "     \"BokehJS does not appear to have successfully loaded. If loading BokehJS from CDN, this \\n\"+\n",
       "     \"may be due to a slow or bad network connection. Possible fixes:\\n\"+\n",
       "     \"</p>\\n\"+\n",
       "     \"<ul>\\n\"+\n",
       "     \"<li>re-rerun `output_notebook()` to attempt to load from CDN again, or</li>\\n\"+\n",
       "     \"<li>use INLINE resources instead, as so:</li>\\n\"+\n",
       "     \"</ul>\\n\"+\n",
       "     \"<code>\\n\"+\n",
       "     \"from bokeh.resources import INLINE\\n\"+\n",
       "     \"output_notebook(resources=INLINE)\\n\"+\n",
       "     \"</code>\\n\"+\n",
       "     \"</div>\"}};\n",
       "\n",
       "  function display_loaded() {\n",
       "    const el = document.getElementById(\"eabccb05-b1e7-4844-baf9-a05e274347a1\");\n",
       "    if (el != null) {\n",
       "      el.textContent = \"BokehJS is loading...\";\n",
       "    }\n",
       "    if (root.Bokeh !== undefined) {\n",
       "      if (el != null) {\n",
       "        el.textContent = \"BokehJS \" + root.Bokeh.version + \" successfully loaded.\";\n",
       "      }\n",
       "    } else if (Date.now() < root._bokeh_timeout) {\n",
       "      setTimeout(display_loaded, 100)\n",
       "    }\n",
       "  }\n",
       "\n",
       "  function run_callbacks() {\n",
       "    try {\n",
       "      root._bokeh_onload_callbacks.forEach(function(callback) {\n",
       "        if (callback != null)\n",
       "          callback();\n",
       "      });\n",
       "    } finally {\n",
       "      delete root._bokeh_onload_callbacks\n",
       "    }\n",
       "    console.debug(\"Bokeh: all callbacks have finished\");\n",
       "  }\n",
       "\n",
       "  function load_libs(css_urls, js_urls, callback) {\n",
       "    if (css_urls == null) css_urls = [];\n",
       "    if (js_urls == null) js_urls = [];\n",
       "\n",
       "    root._bokeh_onload_callbacks.push(callback);\n",
       "    if (root._bokeh_is_loading > 0) {\n",
       "      console.debug(\"Bokeh: BokehJS is being loaded, scheduling callback at\", now());\n",
       "      return null;\n",
       "    }\n",
       "    if (js_urls == null || js_urls.length === 0) {\n",
       "      run_callbacks();\n",
       "      return null;\n",
       "    }\n",
       "    console.debug(\"Bokeh: BokehJS not loaded, scheduling load and callback at\", now());\n",
       "    root._bokeh_is_loading = css_urls.length + js_urls.length;\n",
       "\n",
       "    function on_load() {\n",
       "      root._bokeh_is_loading--;\n",
       "      if (root._bokeh_is_loading === 0) {\n",
       "        console.debug(\"Bokeh: all BokehJS libraries/stylesheets loaded\");\n",
       "        run_callbacks()\n",
       "      }\n",
       "    }\n",
       "\n",
       "    function on_error(url) {\n",
       "      console.error(\"failed to load \" + url);\n",
       "    }\n",
       "\n",
       "    for (let i = 0; i < css_urls.length; i++) {\n",
       "      const url = css_urls[i];\n",
       "      const element = document.createElement(\"link\");\n",
       "      element.onload = on_load;\n",
       "      element.onerror = on_error.bind(null, url);\n",
       "      element.rel = \"stylesheet\";\n",
       "      element.type = \"text/css\";\n",
       "      element.href = url;\n",
       "      console.debug(\"Bokeh: injecting link tag for BokehJS stylesheet: \", url);\n",
       "      document.body.appendChild(element);\n",
       "    }\n",
       "\n",
       "    for (let i = 0; i < js_urls.length; i++) {\n",
       "      const url = js_urls[i];\n",
       "      const element = document.createElement('script');\n",
       "      element.onload = on_load;\n",
       "      element.onerror = on_error.bind(null, url);\n",
       "      element.async = false;\n",
       "      element.src = url;\n",
       "      console.debug(\"Bokeh: injecting script tag for BokehJS library: \", url);\n",
       "      document.head.appendChild(element);\n",
       "    }\n",
       "  };\n",
       "\n",
       "  function inject_raw_css(css) {\n",
       "    const element = document.createElement(\"style\");\n",
       "    element.appendChild(document.createTextNode(css));\n",
       "    document.body.appendChild(element);\n",
       "  }\n",
       "\n",
       "  const js_urls = [\"https://cdn.bokeh.org/bokeh/release/bokeh-3.3.0.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-gl-3.3.0.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-widgets-3.3.0.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-tables-3.3.0.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-mathjax-3.3.0.min.js\", \"https://unpkg.com/@holoviz/panel@1.3.1/dist/panel.min.js\"];\n",
       "  const css_urls = [];\n",
       "\n",
       "  const inline_js = [    function(Bokeh) {\n",
       "      Bokeh.set_log_level(\"info\");\n",
       "    },\n",
       "function(Bokeh) {\n",
       "    }\n",
       "  ];\n",
       "\n",
       "  function run_inline_js() {\n",
       "    if (root.Bokeh !== undefined || force === true) {\n",
       "          for (let i = 0; i < inline_js.length; i++) {\n",
       "      inline_js[i].call(root, root.Bokeh);\n",
       "    }\n",
       "if (force === true) {\n",
       "        display_loaded();\n",
       "      }} else if (Date.now() < root._bokeh_timeout) {\n",
       "      setTimeout(run_inline_js, 100);\n",
       "    } else if (!root._bokeh_failed_load) {\n",
       "      console.log(\"Bokeh: BokehJS failed to load within specified timeout.\");\n",
       "      root._bokeh_failed_load = true;\n",
       "    } else if (force !== true) {\n",
       "      const cell = $(document.getElementById(\"eabccb05-b1e7-4844-baf9-a05e274347a1\")).parents('.cell').data().cell;\n",
       "      cell.output_area.append_execute_result(NB_LOAD_WARNING)\n",
       "    }\n",
       "  }\n",
       "\n",
       "  if (root._bokeh_is_loading === 0) {\n",
       "    console.debug(\"Bokeh: BokehJS loaded, going straight to plotting\");\n",
       "    run_inline_js();\n",
       "  } else {\n",
       "    load_libs(css_urls, js_urls, function() {\n",
       "      console.debug(\"Bokeh: BokehJS plotting callback run at\", now());\n",
       "      run_inline_js();\n",
       "    });\n",
       "  }\n",
       "}(window));"
      ],
      "application/vnd.bokehjs_load.v0+json": "(function(root) {\n  function now() {\n    return new Date();\n  }\n\n  const force = true;\n\n  if (typeof root._bokeh_onload_callbacks === \"undefined\" || force === true) {\n    root._bokeh_onload_callbacks = [];\n    root._bokeh_is_loading = undefined;\n  }\n\n\n  if (typeof (root._bokeh_timeout) === \"undefined\" || force === true) {\n    root._bokeh_timeout = Date.now() + 5000;\n    root._bokeh_failed_load = false;\n  }\n\n  const NB_LOAD_WARNING = {'data': {'text/html':\n     \"<div style='background-color: #fdd'>\\n\"+\n     \"<p>\\n\"+\n     \"BokehJS does not appear to have successfully loaded. If loading BokehJS from CDN, this \\n\"+\n     \"may be due to a slow or bad network connection. Possible fixes:\\n\"+\n     \"</p>\\n\"+\n     \"<ul>\\n\"+\n     \"<li>re-rerun `output_notebook()` to attempt to load from CDN again, or</li>\\n\"+\n     \"<li>use INLINE resources instead, as so:</li>\\n\"+\n     \"</ul>\\n\"+\n     \"<code>\\n\"+\n     \"from bokeh.resources import INLINE\\n\"+\n     \"output_notebook(resources=INLINE)\\n\"+\n     \"</code>\\n\"+\n     \"</div>\"}};\n\n  function display_loaded() {\n    const el = document.getElementById(\"eabccb05-b1e7-4844-baf9-a05e274347a1\");\n    if (el != null) {\n      el.textContent = \"BokehJS is loading...\";\n    }\n    if (root.Bokeh !== undefined) {\n      if (el != null) {\n        el.textContent = \"BokehJS \" + root.Bokeh.version + \" successfully loaded.\";\n      }\n    } else if (Date.now() < root._bokeh_timeout) {\n      setTimeout(display_loaded, 100)\n    }\n  }\n\n  function run_callbacks() {\n    try {\n      root._bokeh_onload_callbacks.forEach(function(callback) {\n        if (callback != null)\n          callback();\n      });\n    } finally {\n      delete root._bokeh_onload_callbacks\n    }\n    console.debug(\"Bokeh: all callbacks have finished\");\n  }\n\n  function load_libs(css_urls, js_urls, callback) {\n    if (css_urls == null) css_urls = [];\n    if (js_urls == null) js_urls = [];\n\n    root._bokeh_onload_callbacks.push(callback);\n    if (root._bokeh_is_loading > 0) {\n      console.debug(\"Bokeh: BokehJS is being loaded, scheduling callback at\", now());\n      return null;\n    }\n    if (js_urls == null || js_urls.length === 0) {\n      run_callbacks();\n      return null;\n    }\n    console.debug(\"Bokeh: BokehJS not loaded, scheduling load and callback at\", now());\n    root._bokeh_is_loading = css_urls.length + js_urls.length;\n\n    function on_load() {\n      root._bokeh_is_loading--;\n      if (root._bokeh_is_loading === 0) {\n        console.debug(\"Bokeh: all BokehJS libraries/stylesheets loaded\");\n        run_callbacks()\n      }\n    }\n\n    function on_error(url) {\n      console.error(\"failed to load \" + url);\n    }\n\n    for (let i = 0; i < css_urls.length; i++) {\n      const url = css_urls[i];\n      const element = document.createElement(\"link\");\n      element.onload = on_load;\n      element.onerror = on_error.bind(null, url);\n      element.rel = \"stylesheet\";\n      element.type = \"text/css\";\n      element.href = url;\n      console.debug(\"Bokeh: injecting link tag for BokehJS stylesheet: \", url);\n      document.body.appendChild(element);\n    }\n\n    for (let i = 0; i < js_urls.length; i++) {\n      const url = js_urls[i];\n      const element = document.createElement('script');\n      element.onload = on_load;\n      element.onerror = on_error.bind(null, url);\n      element.async = false;\n      element.src = url;\n      console.debug(\"Bokeh: injecting script tag for BokehJS library: \", url);\n      document.head.appendChild(element);\n    }\n  };\n\n  function inject_raw_css(css) {\n    const element = document.createElement(\"style\");\n    element.appendChild(document.createTextNode(css));\n    document.body.appendChild(element);\n  }\n\n  const js_urls = [\"https://cdn.bokeh.org/bokeh/release/bokeh-3.3.0.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-gl-3.3.0.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-widgets-3.3.0.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-tables-3.3.0.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-mathjax-3.3.0.min.js\", \"https://unpkg.com/@holoviz/panel@1.3.1/dist/panel.min.js\"];\n  const css_urls = [];\n\n  const inline_js = [    function(Bokeh) {\n      Bokeh.set_log_level(\"info\");\n    },\nfunction(Bokeh) {\n    }\n  ];\n\n  function run_inline_js() {\n    if (root.Bokeh !== undefined || force === true) {\n          for (let i = 0; i < inline_js.length; i++) {\n      inline_js[i].call(root, root.Bokeh);\n    }\nif (force === true) {\n        display_loaded();\n      }} else if (Date.now() < root._bokeh_timeout) {\n      setTimeout(run_inline_js, 100);\n    } else if (!root._bokeh_failed_load) {\n      console.log(\"Bokeh: BokehJS failed to load within specified timeout.\");\n      root._bokeh_failed_load = true;\n    } else if (force !== true) {\n      const cell = $(document.getElementById(\"eabccb05-b1e7-4844-baf9-a05e274347a1\")).parents('.cell').data().cell;\n      cell.output_area.append_execute_result(NB_LOAD_WARNING)\n    }\n  }\n\n  if (root._bokeh_is_loading === 0) {\n    console.debug(\"Bokeh: BokehJS loaded, going straight to plotting\");\n    run_inline_js();\n  } else {\n    load_libs(css_urls, js_urls, function() {\n      console.debug(\"Bokeh: BokehJS plotting callback run at\", now());\n      run_inline_js();\n    });\n  }\n}(window));"
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import warnings\n",
    "with warnings.catch_warnings():\n",
    "    warnings.simplefilter('ignore')\n",
    "\n",
    "    import numpy as np\n",
    "    import scipy.stats as st\n",
    "    import bebi103.viz\n",
    "\n",
    "import bokeh.io\n",
    "import bokeh.plotting\n",
    "bokeh.io.output_notebook()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We have learned in previous lessons that the posterior distribution for a set of parameters $\\theta$ given a data set $y$ is given by the likelihood $f(y\\mid \\theta)$ and prior $g(\\theta)$ according to\n",
    "\n",
    "\\begin{align}\n",
    "g(\\theta\\mid y) \\propto f(y\\mid \\theta)\\,g(\\theta).\n",
    "\\end{align}\n",
    "\n",
    "The central goal of Bayesian inference is making sense of this posterior distribution; writing it down is the easy part."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Summarizing the posterior near its maximum\n",
    "\n",
    "For many posteriors, there exists a set of parameters for which the posterior is maximal. This set of parameters, denoted $\\theta^*$, is referred to as the **maximal a posteriori** parameter set, abbreviated **MAP**. Given that $\\theta^*$ maximizes the posterior, it is of some importance (but certainly not paramount importance, as we will learn throughout the course), and may be useful in summarizing a posterior.\n",
    "\n",
    "Finding the MAP is an optimization problem. In some simple, rare cases, the optimization problem may be solved analytically. We will not dwell on these, because they are so seldom encountered. Instead, we will discuss numerical methods for finding $\\theta^*$ for continuous $\\theta$ as we proceed through this lesson. For now, we will proceed assuming we have found $\\theta^*$ and seek to approximate the posterior distribution near the MAP.\n",
    "\n",
    "Near the MAP, we can approximately express the log posterior as a Taylor expansion truncated to second order,\n",
    "\n",
    "\\begin{align}\n",
    "\\ln g(\\theta\\mid y) \\approx \\ln g(\\theta^*\\mid y) + \\frac{1}{2}\\left(\\theta - \\theta^*\\right)^\\mathsf{T} \\cdot \\mathsf{B} \\cdot \\left(\\theta - \\theta^*\\right),\n",
    "\\end{align}\n",
    "\n",
    "where $\\mathsf{B}$ is the symmetric **Hessian matrix** of second derivatives, with entries\n",
    "\n",
    "\\begin{align}\n",
    "B_{ij} = \\left.\\frac{\\partial^2\\,\\ln g(\\theta\\mid y)}{\\partial \\theta_i\\,\\partial \\theta_j}\\right|_{\\theta = \\theta^*}.\n",
    "\\end{align}\n",
    "\n",
    "In one dimension, this is\n",
    "\n",
    "\\begin{align}\n",
    "\\ln g(\\theta\\mid y) \\approx \\ln g(\\theta^*\\mid y) + \\frac{1}{2}\\left(\\left.\\frac{\\partial^2\\,\\ln g(\\theta\\mid y)}{\\partial \\theta^2}\\right|_{\\theta = \\theta^*}\\right) \\left(\\theta - \\theta^*\\right)^2 = \\ln g(\\theta^*\\mid y) - \\frac{\\left(\\theta - \\theta^*\\right)^2}{2\\sigma^2},\n",
    "\\end{align}\n",
    "\n",
    "where we have defined\n",
    "\n",
    "\\begin{align}\n",
    "\\sigma^2 = -\\left(\\left.\\frac{\\partial^2\\,\\ln g(\\theta\\mid y)}{\\partial \\theta^2}\\right|_{\\theta = \\theta^*}\\right)^{-1}.\n",
    "\\end{align}\n",
    "\n",
    "We note that the $-(\\theta-\\theta^*)^2/2\\sigma^2$ term is the $\\theta$-dependence of the logarithm of the PDF of a Normal distribution! We can therefore locally approximate the posterior distribution as Normal near the MAP;\n",
    "\n",
    "\\begin{align}\n",
    "g(\\theta\\mid y) \\approx \\frac{1}{\\sqrt{2\\pi\\sigma^2}}\\,\\exp\\left[-\\frac{\\left(\\theta - \\theta^*\\right)^2}{2\\sigma^2}\\right].\n",
    "\\end{align}\n",
    "\n",
    "This generalizes to multiple dimensions, where the covariance matrix $\\mathsf{\\Sigma}$ of the approximate multivariate Normal distribution is related to the Hessian matrix by\n",
    "\n",
    "\\begin{align}\n",
    "\\mathsf{\\Sigma} = - \\mathsf{B}^{-1}.\n",
    "\\end{align}\n",
    "\n",
    "Note that the covariance matrix is **not** the inverse of every element of the Hessian; it is the inverse of the Hessian *matrix*. Note also that optimization theory tells us that because the Hessian is evaluated at $\\theta^*$, which maximizes the posterior, the Hessian must be negative definite. Therefore, the covariance matrix must be positive definite, which we already know must by the case for a multivariate Normal distribution. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Demonstration of the Normal approximation\n",
    "\n",
    "Actually, the probability density function of *any* continuous, peaked distribution can be approximated as Normal near a peak. We can demonstrate this fact by approximating a Gamma distribution locally as Normal. Recall that the PDF for a Gamma distribution is\n",
    "\n",
    "\\begin{align}\n",
    "f(x ; \\alpha, \\beta) = \\frac{1}{\\Gamma(\\alpha)}\\,\\frac{(\\beta x)^\\alpha}{x}\\,\\mathrm{e}^{-\\beta x}.\n",
    "\\end{align}\n",
    "\n",
    "We can analytically find the mode by setting\n",
    "\n",
    "\\begin{align}\n",
    "\\frac{\\mathrm{d}\\,\\ln f}{\\mathrm{d}x} = \\frac{\\mathrm{d}}{\\mathrm{d}x}\\left((\\alpha-1)\\ln x - \\beta x\\right) = \\frac{\\alpha-1}{x} - \\beta = 0,\n",
    "\\end{align}\n",
    "\n",
    "and solving to get $x^* = (\\alpha - 1) / \\beta$. We can also compute the second derivative as\n",
    "\n",
    "\\begin{align}\n",
    "\\frac{\\mathrm{d}^2\\,\\ln f}{\\mathrm{d}x^2} = -\\frac{\\alpha - 1}{x^2},\n",
    "\\end{align}\n",
    "\n",
    "which evaluated at $x^*$ is $-\\beta^2 / (\\alpha - 1)$. Therefore, the scale parameter for the approximate Normal distribution is $\\sigma = \\sqrt{\\alpha - 1} / \\beta$. We can plot the distribution and its Normal approximation as a visualization. We do this for $\\alpha = 50$ and $\\beta = 20$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
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fa-info-circle\"],[\"tagName\",\"i\"],[\"color\",\"white\"]]}]]}]}]},{\"type\":\"model\",\"name\":\"Notification\",\"properties\":[{\"name\":\"background\",\"kind\":\"Any\",\"default\":null},{\"name\":\"duration\",\"kind\":\"Any\",\"default\":3000},{\"name\":\"icon\",\"kind\":\"Any\",\"default\":null},{\"name\":\"message\",\"kind\":\"Any\",\"default\":\"\"},{\"name\":\"notification_type\",\"kind\":\"Any\",\"default\":null},{\"name\":\"_destroyed\",\"kind\":\"Any\",\"default\":false}]},{\"type\":\"model\",\"name\":\"TemplateActions1\",\"properties\":[{\"name\":\"open_modal\",\"kind\":\"Any\",\"default\":0},{\"name\":\"close_modal\",\"kind\":\"Any\",\"default\":0}]},{\"type\":\"model\",\"name\":\"BootstrapTemplateActions1\",\"properties\":[{\"name\":\"open_modal\",\"kind\":\"Any\",\"default\":0},{\"name\":\"close_modal\",\"kind\":\"Any\",\"default\":0}]},{\"type\":\"model\",\"name\":\"MaterialTemplateActions1\",\"properties\":[{\"name\":\"open_modal\",\"kind\":\"Any\",\"default\":0},{\"name\":\"close_modal\",\"kind\":\"Any\",\"default\":0}]}]}};\n",
       "  const render_items = [{\"docid\":\"d7ad4ed7-65fc-4dbb-a337-4559e7b2bd2a\",\"roots\":{\"p1002\":\"f051ca4d-bfa5-4501-9c86-73b3d584a17e\"},\"root_ids\":[\"p1002\"]}];\n",
       "  root.Bokeh.embed.embed_items_notebook(docs_json, render_items);\n",
       "  }\n",
       "  if (root.Bokeh !== undefined) {\n",
       "    embed_document(root);\n",
       "  } else {\n",
       "    let attempts = 0;\n",
       "    const timer = setInterval(function(root) {\n",
       "      if (root.Bokeh !== undefined) {\n",
       "        clearInterval(timer);\n",
       "        embed_document(root);\n",
       "      } else {\n",
       "        attempts++;\n",
       "        if (attempts > 100) {\n",
       "          clearInterval(timer);\n",
       "          console.log(\"Bokeh: ERROR: Unable to run BokehJS code because BokehJS library is missing\");\n",
       "        }\n",
       "      }\n",
       "    }, 10, root)\n",
       "  }\n",
       "})(window);"
      ],
      "application/vnd.bokehjs_exec.v0+json": ""
     },
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       "id": "p1002"
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = np.linspace(1, 5, 400)\n",
    "alpha = 50\n",
    "beta = 20\n",
    "y_gamma = st.gamma.pdf(x, alpha, loc=0, scale=1 / beta)\n",
    "y_norm = st.norm.pdf(x, (alpha - 1) / beta, np.sqrt(alpha - 1) / beta)\n",
    "\n",
    "p = bokeh.plotting.figure(\n",
    "    frame_width=400,\n",
    "    frame_height=200,\n",
    "    x_range=[1, 5],\n",
    "    x_axis_label=\"x\",\n",
    "    y_axis_label=\"PDF\",\n",
    ")\n",
    "\n",
    "p.line(x, y_gamma, line_width=2, legend_label=\"Gamma\")\n",
    "p.line(x, y_norm, line_width=2, color=\"tomato\", legend_label=\"Normal\")\n",
    "\n",
    "bokeh.io.show(p)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Credible intervals\n",
    "\n",
    "A $\\alpha$-percent Bayesian **credible interval** for a parameter is an an interval that contains $\\alpha$ percent of the posterior probability density. (This is different form a frequentist confidence interval, though a confidence interval is often erroneously interpreted as a credible interval.) For example, if $g(\\theta \\mid y)$ is the marginalized posterior probability density function for a single parameter $\\theta$ and $G(\\theta\\mid y)$ is the cumulative distribution function, then a 95% credible interval is $[G^{-1}(0.025),\\;G^{-1}(0.975)]$.\n",
    "\n",
    "If we approximate the posterior as a multivariate Normal distribution, we have *analytical* results for credible intervals for each parameter. This is the main benefit of using optimization; it affords us an approximate (Normal) expression for the posterior that has convenient, known analytical results. This enables us to make sense of the posterior. \n",
    "\n",
    "Specifically, if $\\Sigma_{ii}$ is a diagonal element in the covariance matrix $\\mathsf{\\Sigma}$, then, e.g., the 95% credible interval for parameter $\\theta_i$ is centered on the MAP estimate, $\\theta_i^*$ and is $[\\theta_i^* - 1.96\\sqrt{\\Sigma_{ii}},\\;\\theta_i^* + 1.96\\sqrt{\\Sigma_{ii}}]$. Because the credible intervals are symmetric under this approximation, we can report the estimates as the $\\theta_i^* \\pm 1.96 \\sqrt{\\Sigma_{ii}}$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Credible intervals may be skewed\n",
    "\n",
    "We are approximating the *PDF* of a distribution close to its mode as Normal. However, for an asymmetric distribution, which posterior distributions can certainly be, the mode may not be all that relevant as far as credible regions go. As an extreme example, consider an Exponential distribution. It's mode is zero, which is also the minimal allowed value. If we were to compute a central 95% credible region for a posterior that happened to be Exponential, the mode would not be included. In cases like these, though the Normal distribution approximately describes the PDF of the posterior near the MAP, the Normal approximation can lead to rather flawed credible regions.\n",
    "\n",
    "To demonstrate this, let us again return to the Gamma example, this time with $\\alpha = 5$ and $\\beta = 2$. The mode of this Gamma distribution is 2, but the median is about 2.34. If we look graphically, we can see right away that the Normal approximation is not as good as the $\\alpha = 50$, $\\beta = 20$ case."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
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",\"id\":\"p1104\"}}},\"glyph\":{\"type\":\"object\",\"name\":\"Line\",\"id\":\"p1099\",\"attributes\":{\"x\":{\"type\":\"field\",\"field\":\"x\"},\"y\":{\"type\":\"field\",\"field\":\"y\"},\"line_color\":\"tomato\",\"line_width\":2}},\"nonselection_glyph\":{\"type\":\"object\",\"name\":\"Line\",\"id\":\"p1100\",\"attributes\":{\"x\":{\"type\":\"field\",\"field\":\"x\"},\"y\":{\"type\":\"field\",\"field\":\"y\"},\"line_color\":\"tomato\",\"line_alpha\":0.1,\"line_width\":2}},\"muted_glyph\":{\"type\":\"object\",\"name\":\"Line\",\"id\":\"p1101\",\"attributes\":{\"x\":{\"type\":\"field\",\"field\":\"x\"},\"y\":{\"type\":\"field\",\"field\":\"y\"},\"line_color\":\"tomato\",\"line_alpha\":0.2,\"line_width\":2}}}}],\"toolbar\":{\"type\":\"object\",\"name\":\"Toolbar\",\"id\":\"p1064\",\"attributes\":{\"tools\":[{\"type\":\"object\",\"name\":\"PanTool\",\"id\":\"p1078\"},{\"type\":\"object\",\"name\":\"WheelZoomTool\",\"id\":\"p1079\",\"attributes\":{\"renderers\":\"auto\"}},{\"type\":\"object\",\"name\":\"BoxZoomTool\",\"id\":\"p1080\",\"attributes\":{\"overlay\":{\"type\":\"object\",\"name\":\"BoxAnnotation\",\"id\":\"p1081\",\"attributes\":{\"syncable\":false,\"level\":\"overlay\",\"visible\":false,\"left_units\":\"canvas\",\"right_units\":\"canvas\",\"top_units\":\"canvas\",\"bottom_units\":\"canvas\",\"line_color\":\"black\",\"line_alpha\":1.0,\"line_width\":2,\"line_dash\":[4,4],\"fill_color\":\"lightgrey\",\"fill_alpha\":0.5}}}},{\"type\":\"object\",\"name\":\"SaveTool\",\"id\":\"p1082\"},{\"type\":\"object\",\"name\":\"ResetTool\",\"id\":\"p1083\"},{\"type\":\"object\",\"name\":\"HelpTool\",\"id\":\"p1084\"}]}},\"left\":[{\"type\":\"object\",\"name\":\"LinearAxis\",\"id\":\"p1073\",\"attributes\":{\"ticker\":{\"type\":\"object\",\"name\":\"BasicTicker\",\"id\":\"p1074\",\"attributes\":{\"mantissas\":[1,2,5]}},\"formatter\":{\"type\":\"object\",\"name\":\"BasicTickFormatter\",\"id\":\"p1075\"},\"axis_label\":\"PDF\",\"major_label_policy\":{\"type\":\"object\",\"name\":\"AllLabels\",\"id\":\"p1076\"}}}],\"below\":[{\"type\":\"object\",\"name\":\"LinearAxis\",\"id\":\"p1068\",\"attributes\":{\"ticker\":{\"type\":\"object\",\"name\":\"BasicTicker\",\"id\":\"p1069\",\"attributes\":{\"mantissas\":[1,2,5]}},\"formatter\":{\"type\":\"object\",\"name\":\"BasicTickFormatter\",\"id\":\"p1070\"},\"axis_label\":\"x\",\"major_label_policy\":{\"type\":\"object\",\"name\":\"AllLabels\",\"id\":\"p1071\"}}}],\"center\":[{\"type\":\"object\",\"name\":\"Grid\",\"id\":\"p1072\",\"attributes\":{\"axis\":{\"id\":\"p1068\"}}},{\"type\":\"object\",\"name\":\"Grid\",\"id\":\"p1077\",\"attributes\":{\"dimension\":1,\"axis\":{\"id\":\"p1073\"}}},{\"type\":\"object\",\"name\":\"Legend\",\"id\":\"p1094\",\"attributes\":{\"items\":[{\"type\":\"object\",\"name\":\"LegendItem\",\"id\":\"p1095\",\"attributes\":{\"label\":{\"type\":\"value\",\"value\":\"Gamma\"},\"renderers\":[{\"id\":\"p1091\"}]}},{\"type\":\"object\",\"name\":\"LegendItem\",\"id\":\"p1105\",\"attributes\":{\"label\":{\"type\":\"value\",\"value\":\"Normal\"},\"renderers\":[{\"id\":\"p1102\"}]}}]}}],\"frame_width\":400,\"frame_height\":200}}],\"defs\":[{\"type\":\"model\",\"name\":\"ReactiveHTML1\"},{\"type\":\"model\",\"name\":\"FlexBox1\",\"properties\":[{\"name\":\"align_content\",\"kind\":\"Any\",\"default\":\"flex-start\"},{\"name\":\"align_items\",\"kind\":\"Any\",\"default\":\"flex-start\"},{\"name\":\"flex_direction\",\"kind\":\"Any\",\"default\":\"row\"},{\"name\":\"flex_wrap\",\"kind\":\"Any\",\"default\":\"wrap\"},{\"name\":\"justify_content\",\"kind\":\"Any\",\"default\":\"flex-start\"}]},{\"type\":\"model\",\"name\":\"FloatPanel1\",\"properties\":[{\"name\":\"config\",\"kind\":\"Any\",\"default\":{\"type\":\"map\"}},{\"name\":\"contained\",\"kind\":\"Any\",\"default\":true},{\"name\":\"position\",\"kind\":\"Any\",\"default\":\"right-top\"},{\"name\":\"offsetx\",\"kind\":\"Any\",\"default\":null},{\"name\":\"offsety\",\"kind\":\"Any\",\"default\":null},{\"name\":\"theme\",\"kind\":\"Any\",\"default\":\"primary\"},{\"name\":\"status\",\"kind\":\"Any\",\"default\":\"normalized\"}]},{\"type\":\"model\",\"name\":\"GridStack1\",\"properties\":[{\"name\":\"mode\",\"kind\":\"Any\",\"default\":\"warn\"},{\"name\":\"ncols\",\"kind\":\"Any\",\"default\":null},{\"name\":\"nrows\",\"kind\":\"Any\",\"default\":null},{\"name\":\"allow_resize\",\"kind\":\"Any\",\"default\":true},{\"name\":\"allow_drag\",\"kind\":\"Any\",\"default\":true},{\"name\":\"state\",\"kind\":\"Any\",\"default\":[]}]},{\"type\":\"model\",\"name\":\"drag1\",\"properties\":[{\"name\":\"slider_width\",\"kind\":\"Any\",\"default\":5},{\"name\":\"slider_color\",\"kind\":\"Any\",\"default\":\"black\"},{\"name\":\"value\",\"kind\":\"Any\",\"default\":50}]},{\"type\":\"model\",\"name\":\"click1\",\"properties\":[{\"name\":\"terminal_output\",\"kind\":\"Any\",\"default\":\"\"},{\"name\":\"debug_name\",\"kind\":\"Any\",\"default\":\"\"},{\"name\":\"clears\",\"kind\":\"Any\",\"default\":0}]},{\"type\":\"model\",\"name\":\"toggle_value1\",\"properties\":[{\"name\":\"active_icons\",\"kind\":\"Any\",\"default\":{\"type\":\"map\"}},{\"name\":\"options\",\"kind\":\"Any\",\"default\":{\"type\":\"map\",\"entries\":[[\"favorite\",\"heart\"]]}},{\"name\":\"value\",\"kind\":\"Any\",\"default\":[]},{\"name\":\"_reactions\",\"kind\":\"Any\",\"default\":[]},{\"name\":\"_base_url\",\"kind\":\"Any\",\"default\":\"https://tabler-icons.io/static/tabler-icons/icons/\"}]},{\"type\":\"model\",\"name\":\"copy_to_clipboard1\",\"properties\":[{\"name\":\"value\",\"kind\":\"Any\",\"default\":null},{\"name\":\"fill\",\"kind\":\"Any\",\"default\":\"none\"}]},{\"type\":\"model\",\"name\":\"FastWrapper1\",\"properties\":[{\"name\":\"object\",\"kind\":\"Any\",\"default\":null},{\"name\":\"style\",\"kind\":\"Any\",\"default\":null}]},{\"type\":\"model\",\"name\":\"NotificationAreaBase1\",\"properties\":[{\"name\":\"js_events\",\"kind\":\"Any\",\"default\":{\"type\":\"map\"}},{\"name\":\"position\",\"kind\":\"Any\",\"default\":\"bottom-right\"},{\"name\":\"_clear\",\"kind\":\"Any\",\"default\":0}]},{\"type\":\"model\",\"name\":\"NotificationArea1\",\"properties\":[{\"name\":\"js_events\",\"kind\":\"Any\",\"default\":{\"type\":\"map\"}},{\"name\":\"notifications\",\"kind\":\"Any\",\"default\":[]},{\"name\":\"position\",\"kind\":\"Any\",\"default\":\"bottom-right\"},{\"name\":\"_clear\",\"kind\":\"Any\",\"default\":0},{\"name\":\"types\",\"kind\":\"Any\",\"default\":[{\"type\":\"map\",\"entries\":[[\"type\",\"warning\"],[\"background\",\"#ffc107\"],[\"icon\",{\"type\":\"map\",\"entries\":[[\"className\",\"fas fa-exclamation-triangle\"],[\"tagName\",\"i\"],[\"color\",\"white\"]]}]]},{\"type\":\"map\",\"entries\":[[\"type\",\"info\"],[\"background\",\"#007bff\"],[\"icon\",{\"type\":\"map\",\"entries\":[[\"className\",\"fas fa-info-circle\"],[\"tagName\",\"i\"],[\"color\",\"white\"]]}]]}]}]},{\"type\":\"model\",\"name\":\"Notification\",\"properties\":[{\"name\":\"background\",\"kind\":\"Any\",\"default\":null},{\"name\":\"duration\",\"kind\":\"Any\",\"default\":3000},{\"name\":\"icon\",\"kind\":\"Any\",\"default\":null},{\"name\":\"message\",\"kind\":\"Any\",\"default\":\"\"},{\"name\":\"notification_type\",\"kind\":\"Any\",\"default\":null},{\"name\":\"_destroyed\",\"kind\":\"Any\",\"default\":false}]},{\"type\":\"model\",\"name\":\"TemplateActions1\",\"properties\":[{\"name\":\"open_modal\",\"kind\":\"Any\",\"default\":0},{\"name\":\"close_modal\",\"kind\":\"Any\",\"default\":0}]},{\"type\":\"model\",\"name\":\"BootstrapTemplateActions1\",\"properties\":[{\"name\":\"open_modal\",\"kind\":\"Any\",\"default\":0},{\"name\":\"close_modal\",\"kind\":\"Any\",\"default\":0}]},{\"type\":\"model\",\"name\":\"MaterialTemplateActions1\",\"properties\":[{\"name\":\"open_modal\",\"kind\":\"Any\",\"default\":0},{\"name\":\"close_modal\",\"kind\":\"Any\",\"default\":0}]}]}};\n",
       "  const render_items = [{\"docid\":\"53ec7d47-851c-4463-bd23-aec77d088bf6\",\"roots\":{\"p1056\":\"f5b80e75-a464-41b5-9677-7dd1ffafef5f\"},\"root_ids\":[\"p1056\"]}];\n",
       "  root.Bokeh.embed.embed_items_notebook(docs_json, render_items);\n",
       "  }\n",
       "  if (root.Bokeh !== undefined) {\n",
       "    embed_document(root);\n",
       "  } else {\n",
       "    let attempts = 0;\n",
       "    const timer = setInterval(function(root) {\n",
       "      if (root.Bokeh !== undefined) {\n",
       "        clearInterval(timer);\n",
       "        embed_document(root);\n",
       "      } else {\n",
       "        attempts++;\n",
       "        if (attempts > 100) {\n",
       "          clearInterval(timer);\n",
       "          console.log(\"Bokeh: ERROR: Unable to run BokehJS code because BokehJS library is missing\");\n",
       "        }\n",
       "      }\n",
       "    }, 10, root)\n",
       "  }\n",
       "})(window);"
      ],
      "application/vnd.bokehjs_exec.v0+json": ""
     },
     "metadata": {
      "application/vnd.bokehjs_exec.v0+json": {
       "id": "p1056"
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = np.linspace(0, 8, 400)\n",
    "alpha = 5\n",
    "beta = 2\n",
    "y_gamma = st.gamma.pdf(x, alpha, loc=0, scale=1 / beta)\n",
    "y_norm = st.norm.pdf(x, (alpha - 1) / beta, np.sqrt(alpha - 1) / beta)\n",
    "\n",
    "p = bokeh.plotting.figure(\n",
    "    frame_width=400,\n",
    "    frame_height=200,\n",
    "    x_range=[0, 8],\n",
    "    x_axis_label=\"x\",\n",
    "    y_axis_label=\"PDF\",\n",
    ")\n",
    "\n",
    "p.line(x, y_gamma, line_width=2, legend_label=\"Gamma\")\n",
    "p.line(x, y_norm, line_width=2, color=\"tomato\", legend_label=\"Normal\")\n",
    "\n",
    "bokeh.io.show(p)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's look at how this will affect the credible intervals. We will compute the lower and upper bound to a credible interval directly from the Gamma distribution and from its Normal approximation. We can compute credible intervals using the `ppf()` method of distributions in `scipy.stats`, which returns the inverse of the CDF. We can then plot the 1%, ..., 99% credible intervals for the Gamma distribution and for its Normal approximation."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "  <div id=\"f4744ad5-4e11-4584-9209-2fc237aa06c5\" data-root-id=\"p1110\" style=\"display: contents;\"></div>\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "application/javascript": [
       "(function(root) {\n",
       "  function embed_document(root) {\n",
       "  const docs_json = {\"43cb59f6-3c8a-45c9-a22c-624daf7c0143\":{\"version\":\"3.3.0\",\"title\":\"Bokeh 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fa-info-circle\"],[\"tagName\",\"i\"],[\"color\",\"white\"]]}]]}]}]},{\"type\":\"model\",\"name\":\"Notification\",\"properties\":[{\"name\":\"background\",\"kind\":\"Any\",\"default\":null},{\"name\":\"duration\",\"kind\":\"Any\",\"default\":3000},{\"name\":\"icon\",\"kind\":\"Any\",\"default\":null},{\"name\":\"message\",\"kind\":\"Any\",\"default\":\"\"},{\"name\":\"notification_type\",\"kind\":\"Any\",\"default\":null},{\"name\":\"_destroyed\",\"kind\":\"Any\",\"default\":false}]},{\"type\":\"model\",\"name\":\"TemplateActions1\",\"properties\":[{\"name\":\"open_modal\",\"kind\":\"Any\",\"default\":0},{\"name\":\"close_modal\",\"kind\":\"Any\",\"default\":0}]},{\"type\":\"model\",\"name\":\"BootstrapTemplateActions1\",\"properties\":[{\"name\":\"open_modal\",\"kind\":\"Any\",\"default\":0},{\"name\":\"close_modal\",\"kind\":\"Any\",\"default\":0}]},{\"type\":\"model\",\"name\":\"MaterialTemplateActions1\",\"properties\":[{\"name\":\"open_modal\",\"kind\":\"Any\",\"default\":0},{\"name\":\"close_modal\",\"kind\":\"Any\",\"default\":0}]}]}};\n",
       "  const render_items = [{\"docid\":\"43cb59f6-3c8a-45c9-a22c-624daf7c0143\",\"roots\":{\"p1110\":\"f4744ad5-4e11-4584-9209-2fc237aa06c5\"},\"root_ids\":[\"p1110\"]}];\n",
       "  root.Bokeh.embed.embed_items_notebook(docs_json, render_items);\n",
       "  }\n",
       "  if (root.Bokeh !== undefined) {\n",
       "    embed_document(root);\n",
       "  } else {\n",
       "    let attempts = 0;\n",
       "    const timer = setInterval(function(root) {\n",
       "      if (root.Bokeh !== undefined) {\n",
       "        clearInterval(timer);\n",
       "        embed_document(root);\n",
       "      } else {\n",
       "        attempts++;\n",
       "        if (attempts > 100) {\n",
       "          clearInterval(timer);\n",
       "          console.log(\"Bokeh: ERROR: Unable to run BokehJS code because BokehJS library is missing\");\n",
       "        }\n",
       "      }\n",
       "    }, 10, root)\n",
       "  }\n",
       "})(window);"
      ],
      "application/vnd.bokehjs_exec.v0+json": ""
     },
     "metadata": {
      "application/vnd.bokehjs_exec.v0+json": {
       "id": "p1110"
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "perc_cred_int = np.linspace(1, 99, 200)\n",
    "mu = (alpha - 1) / beta\n",
    "sigma = np.sqrt(alpha - 1) / beta\n",
    "\n",
    "gamma_low = st.gamma.ppf((50 - perc_cred_int / 2) / 100, alpha, loc=0, scale=1 / beta)\n",
    "gamma_high = st.gamma.ppf((50 + perc_cred_int / 2) / 100, alpha, loc=0, scale=1 / beta)\n",
    "norm_low = st.norm.ppf((50 - perc_cred_int / 2) / 100, mu, sigma)\n",
    "norm_high = st.norm.ppf((50 + perc_cred_int / 2) / 100, mu, sigma)\n",
    "\n",
    "p = bebi103.viz.fill_between(\n",
    "    gamma_low,\n",
    "    perc_cred_int,\n",
    "    gamma_high,\n",
    "    perc_cred_int,\n",
    "    patch_kwargs=dict(alpha=0.5),\n",
    "    x_axis_label=\"x\",\n",
    "    y_axis_label=\"percent credible interval\",\n",
    "    y_range=[0, 100],\n",
    ")\n",
    "p = bebi103.viz.fill_between(\n",
    "    norm_low,\n",
    "    perc_cred_int,\n",
    "    norm_high,\n",
    "    perc_cred_int,\n",
    "    patch_kwargs=dict(alpha=0.5, color=\"tomato\"),\n",
    "    line_kwargs=dict(color=\"tomato\"),\n",
    "    p=p,\n",
    ")\n",
    "\n",
    "bokeh.io.show(p)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For tiny credible intervals, the shift is clear. The median for a Gamma distribution with these parameters is shifted rightward from its mode, whereas the mode and median for a Normal distribution are equal. As we get broader credible intervals, the Normal approximation notably has the 99% credible interval containing negative values (not allowed for a Gamma distribution) and missing important values at the high end."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Why use the MAP for parameter estimation?\n",
    "\n",
    "Despite these drawbacks, using the MAP/Normal approximation for parameter estimation is still useful in Bayesian inference, at least for simple, well-behaved problems. Why? The most important reason is speed. For problems with simple models (which we will encountered in the next section of this lesson), optimization is *much* faster than other methods for characterizing the posterior, such as directly plotting it or using Markov chain Monte Carlo. If you have high-throughput data, you may not have the computational resources to do a full Bayesian treatment with other methods. Provided your model that is tractable by optimization method, finding the MAP and summarizing with an approximate Normal distribution is an attractive method.\n",
    "\n",
    "In many cases, the Normal approximation can also be a good approximation. But this is definitely not guaranteed! In real inference problems, it is very difficult to identify pathologies that would lead to breakdown of the Normal approximation, since comparing to the real posterior would involve computationally intensive methods to compute it.\n",
    "\n",
    "In the next sections of this lesson, we will work though the mechanics of finding the MAP and computing credible regions."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Computing environment"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Python implementation: CPython\n",
      "Python version       : 3.11.5\n",
      "IPython version      : 8.15.0\n",
      "\n",
      "numpy     : 1.26.3\n",
      "scipy     : 1.11.4\n",
      "bokeh     : 3.3.0\n",
      "bebi103   : 0.1.19\n",
      "jupyterlab: 4.0.10\n",
      "\n"
     ]
    }
   ],
   "source": [
    "%load_ext watermark\n",
    "%watermark -v -p numpy,scipy,bokeh,bebi103,jupyterlab"
   ]
  }
 ],
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  "anaconda-cloud": {},
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
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  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
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   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
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