{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Parameter estimation by optimization case study: Normal likelihood\n",
    "\n",
    "[Data set download](https://s3.amazonaws.com/bebi103.caltech.edu/data/good_invitro_droplet_data.csv)\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": [
    {
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       "  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",
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       "\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",
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       "\n",
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       "      );\n",
       "      this.keyboard_manager.register_events(toinsert);\n",
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       "      const props = {data: data, metadata: metadata[EXEC_MIME_TYPE]};\n",
       "      render(props, toinsert[toinsert.length - 1]);\n",
       "      element.append(toinsert);\n",
       "      return toinsert\n",
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       "\n",
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       "    events.on('delete.Cell', handleClearOutput);\n",
       "\n",
       "    /* Handle when a new output is added */\n",
       "    events.on('output_added.OutputArea', handleAddOutput);\n",
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       "     */\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",
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       "\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",
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       "  }\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(\"a809643d-e832-412f-89b8-16c7dc1a703b\");\n",
       "    if (el != null) {\n",
       "      el.textContent = \"BokehJS is loading...\";\n",
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       "    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(\"a809643d-e832-412f-89b8-16c7dc1a703b\")).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));"
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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(\"a809643d-e832-412f-89b8-16c7dc1a703b\");\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(\"a809643d-e832-412f-89b8-16c7dc1a703b\")).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 numpy as np\n",
    "import pandas as pd\n",
    "import scipy.optimize\n",
    "import scipy.stats as st\n",
    "import statsmodels.tools.numdiff as smnd\n",
    "\n",
    "import tqdm\n",
    "\n",
    "import bebi103\n",
    "\n",
    "import bokeh.io\n",
    "bokeh.io.output_notebook()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<hr>\n",
    "\n",
    "We will begin our exploration of methods of parameter estimation by optimization again using the data set from [Good, et al., \n",
    " *Science*, **342**, 856-860, 2013](https://doi.org/10.1126/science.1243147). You should [refamiliarize yourself](https://bebi103a.github.io/lessons/20/model_building.html) with the data set if you need to."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exploratory data analysis\n",
    "\n",
    "To start, let's make a quick plot of the spindle length versus droplet data, which you can download here: [https://s3.amazonaws.com/bebi103.caltech.edu/data/good_invitro_droplet_data.csv](https://s3.amazonaws.com/bebi103.caltech.edu/data/good_invitro_droplet_data.csv)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
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    "# Load in Data Frame\n",
    "df = pd.read_csv(os.path.join(data_path, \"good_invitro_droplet_data.csv\"), comment=\"#\")\n",
    "\n",
    "# Pull out numpy arrays\n",
    "ell = df[\"Spindle Length (um)\"].values\n",
    "d = df[\"Droplet Diameter (um)\"].values\n",
    "\n",
    "# Make a plot\n",
    "p = bokeh.plotting.figure(\n",
    "    frame_height=200,\n",
    "    frame_width=300,\n",
    "    x_axis_label=\"droplet diameter (µm)\",\n",
    "    y_axis_label=\"spindle length (µm)\",\n",
    "    x_range=[0, 250],\n",
    "    y_range=[0, 50],\n",
    ")\n",
    "\n",
    "p.circle(\n",
    "    source=df,\n",
    "    x=\"Droplet Diameter (um)\",\n",
    "    y=\"Spindle Length (um)\",\n",
    "    alpha=0.3,\n",
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   "source": [
    "Our goal is to get parameter estimates for models describing how spindle length varies with droplet size. We will start with the independent size model, in which we assume that spindle length is independent of droplet diameter."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Independent size model\n",
    "\n",
    "We choose a Normal likelihood for the spindle size. We assume all spindles are distributed about the same location parameter $\\phi$. In our updated model, we will again use a Normal likelihood.\n",
    "\n",
    "\\begin{align}\n",
    "l_i \\sim \\text{Norm}(\\phi, \\sigma) \\;\\forall i.\n",
    "\\end{align}\n",
    "\n",
    "In specifying this likelihood, we see that there are two parameters to estimate, $\\phi$ and $\\sigma$. Now that we have thought a bit more about Bayesian modeling, we will abandon using Jeffreys priors and instead use weakly informative priors. The parameter $\\phi$ is necessarily positive and is the characteristic length of a spindle. Using the bet-the-farm approach, I do not think the spindle would be smaller than 100 nm, and not bigger than 1 mm (the size scale of a *Xenopus* egg). Thus, for my prior for $\\phi$ in units of microns, I have\n",
    "\n",
    "\\begin{align}\n",
    "&\\log_{10} \\phi \\sim \\text{Norm}(1, 1), \\\\[1em]\n",
    "&\\phi = 10^{\\log_{10}\\phi}.\n",
    "\\end{align}\n",
    "\n",
    "Alternatively, we could write this as\n",
    "\n",
    "\\begin{align}\n",
    "&\\phi \\sim \\text{LogNorm}(2.3, 2.3).\n",
    "\\end{align}\n",
    "\n",
    "Note that for brevity, I do not include the units of the parameters in the likelihood definition, and units of microns are assumed.\n",
    "\n",
    "To understand the prior, here is a plot of its CDF."
   ]
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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\":\"597678f2-400f-46d4-8917-f6fc7d916065\",\"roots\":{\"p1050\":\"f00e7aaa-a797-4e75-a96e-ec93263ef284\"},\"root_ids\":[\"p1050\"]}];\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": "p1050"
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "phi = np.logspace(-3, 5, 200)\n",
    "cdf = st.lognorm.cdf(phi, s=2.3, loc=0, scale=np.exp(2.3))\n",
    "\n",
    "p = bokeh.plotting.figure(\n",
    "    frame_height=200,\n",
    "    frame_width=300,\n",
    "    x_axis_label=\"'ϕ [µm]'\",\n",
    "    y_axis_label=\"g(ϕ)\",\n",
    "    x_range=[1e-3, 1e5],\n",
    "    x_axis_type='log',\n",
    ")\n",
    "\n",
    "p.line(phi, cdf, line_width=2);\n",
    "bokeh.io.show(p)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now on to our next prior. The parameter $\\sigma$ describes how the data vary off of the characteristic value of $\\phi$. I am not really sure how variable spindle length is, so I will choose a weakly informative Half-Normal prior.\n",
    "\n",
    "\\begin{align}\n",
    "\\sigma \\sim \\text{HalfNorm}(20).\n",
    "\\end{align}\n",
    "\n",
    "Here is a plot of this prior PDF."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "  <div id=\"b25fb88a-4b24-4bb0-a49d-030262b23ed9\" data-root-id=\"p1090\" style=\"display: contents;\"></div>\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "application/javascript": [
       "(function(root) {\n",
       "  function embed_document(root) {\n",
       "  const docs_json = {\"d3c28ba9-5f50-4ffc-969b-b6be93bccf4b\":{\"version\":\"3.3.0\",\"title\":\"Bokeh 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     "output_type": "display_data"
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   ],
   "source": [
    "sigma = np.linspace(0, 100, 200)\n",
    "pdf = st.halfnorm.pdf(sigma, 0, 20)\n",
    "\n",
    "p = bokeh.plotting.figure(\n",
    "    frame_height=200,\n",
    "    frame_width=300,\n",
    "    x_axis_label=\"'ϕ [µm]'\",\n",
    "    y_axis_label=\"g(ϕ)\",\n",
    "    x_range=[0, 100],\n",
    ")\n",
    "\n",
    "p.line(sigma, pdf, line_width=2);\n",
    "bokeh.io.show(p)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note that zero is the most probable value of $\\sigma$, and I don't really believe this. Nonetheless, there is not much probability *mass* near zero, and the prior is very broad, so these details will have little effect on the posterior.\n",
    "\n",
    "Thus, my complete generative model for the independent spindle size model is\n",
    "\n",
    "\\begin{align}\n",
    "&\\phi \\sim \\text{Norm}(2.3, 2.3), \\\\[1em]\n",
    "&\\sigma \\sim \\text{HalfNorm}(20),\\\\[1em]\n",
    "&l_i \\sim \\text{Norm}(\\phi, \\sigma) \\;\\forall i.\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Estimation of the MAP parameters\n",
    "\n",
    "We could plot the entire posterior distribution (and in fact we did, albeit with different priors, in a previous lesson) to get a full understanding of the posterior. Instead, we will compute the MAP and approximate the posterior as Normal near the MAP.\n",
    "\n",
    "Finding the MAP amounts to an **optimization problem** in which we find the arguments (parameters) for which a function (the posterior distribution) is maximal. It is almost always much easier to find the maximum for the *logarithm* of the posterior. In fact, in the fast majority of Bayesian inference problems, we work with log posteriors instead of the posteriors themselves. Since the logarithm function is monotonic, a maximum of the log posterior is also a maximum of the posterior. So, our first step is code up a function that returns the log posterior. \n",
    "\n",
    "In considering a log posterior, it is useful to know that\n",
    "\n",
    "\\begin{align}\n",
    "\\ln g(\\theta \\mid y) = \\text{constant} + \\ln f(y \\mid \\theta) + \\ln g(\\theta).\n",
    "\\end{align}\n",
    "\n",
    "Furthermore, if the priors for each parameter are independent, such that\n",
    "\n",
    "\\begin{align}\n",
    "g(\\theta) = g(\\theta_1)\\, g(\\theta_2)\\cdots,\n",
    "\\end{align}\n",
    "\n",
    "and the measurements are independent and identically distributed (i.i.d.), such that\n",
    "\n",
    "\\begin{align}\n",
    "f(y\\mid \\theta) = f(y_1 \\mid \\theta)\\,f(y_2 \\mid \\theta)\\,f(y_3 \\mid \\theta)\\cdots,\n",
    "\\end{align}\n",
    "\n",
    "then\n",
    "\n",
    "\\begin{align}\n",
    "\\ln g(\\theta \\mid y) = \\text{constant} + \\sum_i \\ln f(y_i \\mid \\theta) + \\sum_j \\ln g(\\theta_j).\n",
    "\\end{align}\n",
    "\n",
    "This means that we can construct the log posterior by summing up all of the terms in the log likelihood and log prior.\n",
    "\n",
    "Fortunately, we do not have to hand-code any of the mathematical expressions for the log PDFs; we can use the built-in `scipy.stats` functions."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "def log_prior(phi, sigma):\n",
    "    \"\"\"Log prior for independent size model\"\"\"\n",
    "    lp = st.lognorm.logpdf(phi, 2.3, loc=0, scale=np.exp(2.3))\n",
    "    lp += st.halfnorm.logpdf(sigma, 0, 20)\n",
    "\n",
    "    return lp\n",
    "\n",
    "\n",
    "def log_likelihood(ell, phi, sigma):\n",
    "    \"\"\"Log likelihood for independent size model\"\"\"\n",
    "    return np.sum(st.norm.logpdf(ell, phi, sigma))\n",
    "\n",
    "\n",
    "def log_posterior(params, ell):\n",
    "    \"\"\"Log posterior of indpendent size model.\"\"\"\n",
    "    phi, sigma = params\n",
    "\n",
    "    lp = log_prior(phi, sigma)\n",
    "    if lp == -np.inf:\n",
    "        return lp\n",
    "    \n",
    "    return lp + log_likelihood(ell, phi, sigma)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now that we have the log posterior, we can perform numerical optimization. The function `scipy.optimize.minimize()` offers many options for algorithms, which you can read about in the [documentation](http://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.minimize.html).  I find that [Powell's method](https://en.wikipedia.org/wiki/Powell%27s_method) tends to work well. It does not rely on derivative information, so discontinuities don't hurt, nor do the $-\\infty$ values we get in the log posterior for parameter values that are out of bounds. That is particularly important for the present example because we have hard bounds on $\\sigma$ and $\\phi$; they must be positive. We could use constrained optimizers such as L-BFGS-B or COBYLA, but Powell's method generally suffices. It is sometimes slow, though.\n",
    "\n",
    "The optimizers offered by `scipy.optimize.minimize()` find *minimizers*, so we need to define our objective function as the *negative* log posterior."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "def neg_log_posterior(params, ell):\n",
    "    return -log_posterior(params, ell)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Next, we have to provide an initial guess for parameter values. The optimization routine only finds a *local* maximum and is not in general guaranteed to converge. Therefore, the initial guess can be very important. Looking at the data, we would expect $\\phi$ to be somewhere around 35 µm and for $\\sigma$ to be 7 µm."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "params_0 = np.array([35, 7])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We are now ready to use `scipy.optimize.minimze()` to compute the MAP. We use the `args` kwarg to pass in the other arguments to the `neg_log_posterior()` function. In our case, these arguments are the data points. The `scipy.optimzie.minimize()` function returns an object that has several attributes about the resulting optimization. Most important is `x`, the optimal parameter values."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "Most probable parameters\n",
      "------------------------\n",
      "φ = 32.86 µm\n",
      "σ = 4.784 µm\n",
      "\n"
     ]
    }
   ],
   "source": [
    "# Specify extra arguments into posterior function\n",
    "args = (ell,)\n",
    "\n",
    "# Compute the MAP\n",
    "res = scipy.optimize.minimize(\n",
    "    neg_log_posterior, params_0, args=args, method=\"powell\"\n",
    ")\n",
    "\n",
    "# Extract optimal parameters\n",
    "popt = res.x\n",
    "\n",
    "# For convenience...\n",
    "phi_MAP, sigma_MAP = popt\n",
    "\n",
    "# Print results\n",
    "print(\n",
    "    \"\"\"\n",
    "Most probable parameters\n",
    "------------------------\n",
    "φ = {0:.2f} µm\n",
    "σ = {1:.3f} µm\n",
    "\"\"\".format(\n",
    "        phi_MAP, sigma_MAP\n",
    "    )\n",
    ")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We have successfully found the MAP which provides just that; the maximally probable parameter values. This tells us only one piece of information about the posterior; where it is maximal. So we know where all of the first derivatives of the posterior are zero. We want to learn more about it, at least near the MAP. To that end, we can compute all of the second derivatives of the log posterior near the MAP. Using the second derivatives (the Hessian), we can start to understand the posterior, at least locally, by approximating it as Normal."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Normal approximation of the posterior\n",
    "\n",
    "To compute the Normal approximation of the posterior, our task is two-fold. First, find the parameter values that give the maximal posterior probability, which we have already done. Second, compute the Hessian of the log posterior at the MAP and invert it to get your covariance matrix. Note that the covariance matrix is **not** the inverse of every element of the Hessian; it is the inverse of the Hessian *matrix*. The credible interval for each parameter can then be calculated from the diagonal elements of the covariance matrix.\n",
    "\n",
    "To compute the covariance matrix, we need to compute the Hessian of the log of the posterior at the MAP. We will do this numerically using the `statsmodels.tools.numdiff` module, which we imported as `smnd`. We use the function `smnd.approx_hess()` to compute the Hessian at the optimal parameter values. Note that since we already have the log posterior coded up and the MAP parameter values, we can just directly shove these into the numerical Hessian calculator."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "hes = smnd.approx_hess(popt, log_posterior, args=args)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now that we have the Hessian, we take its negative inverse to get the covariance matrix."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 3.41668904e-02, -9.09388085e-06],\n",
       "       [-9.09388085e-06,  1.70799615e-02]])"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Compute the covariance matrix\n",
    "cov = -np.linalg.inv(hes)\n",
    "\n",
    "# Look at it\n",
    "cov"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The diagonal terms give the approximate variance in the parameters in the posterior. The off-diagonal terms give the covariance, which describes how the parameters relate to each other. Nonzero covariance indicates that they are not completely independent."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Credible intervals\n",
    "\n",
    "We can compute the approximate 95% credible intervals for the parameters by multiplying the diagonal terms of the posterior covariance matrix $\\mathsf{\\Sigma}$ by 1.96."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "Results for conserved tubulin model (≈ 95% of total probability density)\n",
      "------------------------------------------------------------------------\n",
      "ϕ = 32.86 ± 0.36 µm\n",
      "σ = 4.784 ± 0.256 µm\n",
      "\n"
     ]
    }
   ],
   "source": [
    "# Report results\n",
    "print(\"\"\"\n",
    "Results for conserved tubulin model (≈ 95% of total probability density)\n",
    "------------------------------------------------------------------------\n",
    "ϕ = {0:.2f} ± {1:.2f} µm\n",
    "σ = {2:.3f} ± {3:.3f} µm\n",
    "\"\"\".format(phi_MAP, 1.96*np.sqrt(cov[0,0]), sigma_MAP, 1.96*np.sqrt(cov[1,1])))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### How good is the approximation?\n",
    "\n",
    "To assess how close the Normal approximation is to the posterior, let's plot the Normal approximation and the exact marginalized posterior on the same contour plot. We can use `scipy.stats.multivariate_normal.pdf()` function to generate the PDF for the Normal approximation.\n",
    "\n",
    "The calculation will take a while, since we have to compute the whole posterior in the neighborhood of the MAP to plot it. The whole reason we used optimization in the first place was to avoid this calculation! But we are doing it here to visualize the Normal approximation."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "100%|█████████████████████████████████████████| 200/200 [00:24<00:00,  8.22it/s]\n"
     ]
    }
   ],
   "source": [
    "# Set up ranges for plots; should be near MAP\n",
    "phi = np.linspace(32.4, 33.5, 200)\n",
    "sigma = np.linspace(4.4, 5.2, 200)\n",
    "\n",
    "# Compute posteriors\n",
    "post_norm = np.empty((len(phi), len(sigma)))\n",
    "log_post_exact = np.empty((len(phi), len(sigma)))\n",
    "for i in tqdm.tqdm(range(len(phi))):\n",
    "    for j in range(len(sigma)):\n",
    "        post_norm[i, j] = st.multivariate_normal.pdf(\n",
    "            (phi[i], sigma[j]), popt, cov\n",
    "        )\n",
    "        log_post_exact[i, j] = log_posterior((phi[i], sigma[j]), args)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now let's plot the contours, overlaying the exact posterior with the Normal approximation."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
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AUvFeZZPMSQLb+9sBG7xJAWkGW6CjrEkD9gzUQC+cSQKDG1Dft4hJAQI1S74bgEkBECXRfz94SQOhLE4ex2hJAio6yrpPWEkBvcctN+dQSQC3RUdZ10hJA0BPx/VfOEkBU+8D5Tc0SQHNWkCU6yhJAYzzec2fHEkAWmS9NHMYSQH/zvxOUwhJAudvOdP7BEkCwmKNSj74SQFwebpzgvRJAHVLqSiq7EkAAYQ3EwrkSQMiK2h1EuBJAjh7dLce1EkCjo6zrpLUSQNJe0jy4sxJAYjAKefexEkBG5ksTh7ESQA==\"},\"shape\":[131],\"dtype\":\"float64\",\"order\":\"little\"}]]]}}},\"view\":{\"type\":\"object\",\"name\":\"CDSView\",\"id\":\"p1174\",\"attributes\":{\"filter\":{\"type\":\"object\",\"name\":\"AllIndices\",\"id\":\"p1175\"}}},\"glyph\":{\"type\":\"object\",\"name\":\"MultiLine\",\"id\":\"p1170\",\"attributes\":{\"xs\":{\"type\":\"field\",\"field\":\"xs\"},\"ys\":{\"type\":\"field\",\"field\":\"ys\"},\"line_color\":{\"type\":\"value\",\"value\":\"orange\"},\"line_width\":{\"type\":\"value\",\"value\":2}}},\"nonselection_glyph\":{\"type\":\"object\",\"name\":\"MultiLine\",\"id\":\"p1171\",\"attributes\":{\"xs\":{\"type\":\"field\",\"field\":\"xs\"},\"ys\":{\"type\":\"field\",\"field\":\"ys\"},\"line_color\":{\"type\":\"value\",\"value\":\"orange\"},\"line_alpha\":{\"type\":\"value\",\"value\":0.1},\"line_width\":{\"type\":\"value\",\"value\":2}}},\"muted_glyph\":{\"type\":\"object\",\"name\":\"MultiLine\",\"id\":\"p1172\",\"attributes\":{\"xs\":{\"type\":\"field\",\"field\":\"xs\"},\"ys\":{\"type\":\"field\",\"field\":\"ys\"},\"line_color\":{\"type\":\"value\",\"value\":\"orange\"},\"line_alpha\":{\"type\":\"value\",\"value\":0.2},\"line_width\":{\"type\":\"value\",\"value\":2}}}}}],\"toolbar\":{\"type\":\"object\",\"name\":\"Toolbar\",\"id\":\"p1138\",\"attributes\":{\"tools\":[{\"type\":\"object\",\"name\":\"PanTool\",\"id\":\"p1151\"},{\"type\":\"object\",\"name\":\"WheelZoomTool\",\"id\":\"p1152\",\"attributes\":{\"renderers\":\"auto\"}},{\"type\":\"object\",\"name\":\"BoxZoomTool\",\"id\":\"p1153\",\"attributes\":{\"overlay\":{\"type\":\"object\",\"name\":\"BoxAnnotation\",\"id\":\"p1154\",\"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\":\"p1155\"},{\"type\":\"object\",\"name\":\"ResetTool\",\"id\":\"p1156\"},{\"type\":\"object\",\"name\":\"HelpTool\",\"id\":\"p1157\"}]}},\"left\":[{\"type\":\"object\",\"name\":\"LinearAxis\",\"id\":\"p1146\",\"attributes\":{\"ticker\":{\"type\":\"object\",\"name\":\"BasicTicker\",\"id\":\"p1147\",\"attributes\":{\"mantissas\":[1,2,5]}},\"formatter\":{\"type\":\"object\",\"name\":\"BasicTickFormatter\",\"id\":\"p1148\"},\"axis_label\":\"\\u03c3 [\\u00b5m]\",\"major_label_policy\":{\"type\":\"object\",\"name\":\"AllLabels\",\"id\":\"p1149\"}}}],\"below\":[{\"type\":\"object\",\"name\":\"LinearAxis\",\"id\":\"p1141\",\"attributes\":{\"ticker\":{\"type\":\"object\",\"name\":\"BasicTicker\",\"id\":\"p1142\",\"attributes\":{\"mantissas\":[1,2,5]}},\"formatter\":{\"type\":\"object\",\"name\":\"BasicTickFormatter\",\"id\":\"p1143\"},\"axis_label\":\"\\u03d5 [\\u00b5m]\",\"major_label_policy\":{\"type\":\"object\",\"name\":\"AllLabels\",\"id\":\"p1144\"}}}],\"center\":[{\"type\":\"object\",\"name\":\"Grid\",\"id\":\"p1145\",\"attributes\":{\"axis\":{\"id\":\"p1141\"}}},{\"type\":\"object\",\"name\":\"Grid\",\"id\":\"p1150\",\"attributes\":{\"dimension\":1,\"axis\":{\"id\":\"p1146\"}}}],\"frame_width\":300,\"frame_height\":300}}],\"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\":\"17b3e7c4-4ed7-420e-ad40-b9abbb77a9d2\",\"roots\":{\"p1130\":\"d62696a6-5a02-43e7-8d1e-08197b41028d\"},\"root_ids\":[\"p1130\"]}];\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": "p1130"
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Scale exact to about underflow\n",
    "post_exact = np.exp(log_post_exact - log_post_exact.max())\n",
    "\n",
    "# Make contours\n",
    "p = bebi103.viz.contour(\n",
    "    phi, sigma, post_exact, x_axis_label=\"ϕ [µm]\", y_axis_label=\"σ [µm]\"\n",
    ")\n",
    "p = bebi103.viz.contour(\n",
    "    phi, sigma, post_norm, line_kwargs=dict(line_color=\"orange\"), p=p\n",
    ")\n",
    "\n",
    "bokeh.io.show(p)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The Normal approximation, shown in orange, is not that far off from the posterior, especially at the peak. **This is not always the case!** This is one of the main dangers of using optimization, you are using *very* local information to summarize a posterior that may have relevant features away from the MAP."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Computing environment"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "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      : 1.26.2\n",
      "pandas     : 2.1.4\n",
      "scipy      : 1.11.4\n",
      "tqdm       : 4.65.0\n",
      "statsmodels: 0.14.0\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,pandas,scipy,tqdm,statsmodels,bokeh,bebi103,jupyterlab"
   ]
  }
 ],
 "metadata": {
  "anaconda-cloud": {},
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.11.5"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}