{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Parameter estimation by optimization: A variate-covariate model\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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       "        <span id=\"a72b4236-9776-4755-b031-751e84ddcb75\">Loading BokehJS ...</span>\n",
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       "    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",
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       "\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",
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       "      // 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",
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       "\n",
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       "\n",
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       "\n",
       "    const toinsert = output_area.element.find(\".\" + CLASS_NAME.split(' ')[0]);\n",
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       "      toinsert[toinsert.length - 1].firstChild.textContent = output.data[JS_MIME_TYPE];\n",
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       "      }\n",
       "      // store reference to server id on output_area\n",
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       "  }\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(\"a72b4236-9776-4755-b031-751e84ddcb75\");\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(\"a72b4236-9776-4755-b031-751e84ddcb75\")).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(\"a72b4236-9776-4755-b031-751e84ddcb75\");\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(\"a72b4236-9776-4755-b031-751e84ddcb75\")).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": [
    "We continue our analysis of the data set from [Good, et al., \n",
    " *Science*, **342**, 856-860, 2013](https://doi.org/10.1126/science.1243147), now considering the tubulin conservation model. Before we continue, we'll again get a quick plot."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
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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_data = 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_data.circle(\n",
    "    source=df,\n",
    "    x=\"Droplet Diameter (um)\",\n",
    "    y=\"Spindle Length (um)\",\n",
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    "\n",
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   "source": [
    "## The tubulin conservation model\n",
    "\n",
    "In the tubulin conservation model, recall that the theoretical relationship between spindle length and droplet diameter is\n",
    "\n",
    "\\begin{align}\n",
    "l(d) = \\frac{\\gamma d}{\\left(1+(\\gamma d/\\phi)^3\\right)^{\\frac{1}{3}}}\n",
    "\\end{align}\n",
    "\n",
    "We will assume homoscedastic variation about this theoretical curve, giving us a likelihood of\n",
    "\n",
    "\\begin{align}\n",
    "&\\mu_i =  \\frac{\\gamma d_i}{\\left(1+(\\gamma d_i/\\phi)^3\\right)^{\\frac{1}{3}}}, \\\\[1em]\n",
    "&l_i \\sim \\text{Norm}(\\mu_i, \\sigma) \\;\\forall i.\n",
    "\\end{align}\n",
    "\n",
    "We therefore need to provide priors for $\\gamma$, $\\phi$, and $\\sigma$. We will use the same priors as we did for the independent size model for $\\phi$ and $\\sigma$ because they have the same meaning in the context of this model. \n",
    "\n",
    "Physically, $0 < \\gamma \\le 1$, so we need to provide a prior for $\\gamma$ that is defined between zero and one. We can use a Beta distribution. I think $\\gamma$ having a values close to zero is very unlikely, since that sets the spindle lengths to zero. Similarly, I do not think values of $\\gamma$ close to one are likely, since this sets the spindle length to be the same as the droplet diameter, meaning that the spindle would always press against the sides of the droplet. So, I will choose a prior for $\\gamma$ that is fairly flat for intermediate values of $\\gamma$, but goes toward zero at $\\gamma = 0, 1$. A Beta distribution with $\\alpha = \\beta = 3/2$ accomplishes this. The prior is plotted below."
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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 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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\":\"d405b748-42a5-4da5-a036-371f5a621fe8\",\"roots\":{\"p1050\":\"a5299434-2b71-4237-9f65-bba4bfb8d26e\"},\"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": [
    "gamma = np.linspace(0, 1, 200)\n",
    "pdf = st.beta.pdf(gamma, 1.5, 1.5)\n",
    "\n",
    "# Make a plot\n",
    "p = bokeh.plotting.figure(\n",
    "    frame_height=200,\n",
    "    frame_width=300,\n",
    "    x_axis_label=\"γ\",\n",
    "    y_axis_label=\"g(γ)\",\n",
    "    x_range=[0, 1],\n",
    ")\n",
    "\n",
    "p.line(gamma, pdf, line_width=2)\n",
    "\n",
    "bokeh.io.show(p)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So, our complete generative model is\n",
    "\n",
    "\\begin{align}\n",
    "&\\phi \\sim \\text{LogNorm}(2.3, 2.3),\\\\[1em]\n",
    "&\\gamma \\sim \\text{Beta}(1.5, 1.5), \\\\[1em]\n",
    "&\\sigma \\sim \\text{HalfNorm}(20),\\\\[1em]\n",
    "&\\mu_i =  \\frac{\\gamma d_i}{\\left(1+(\\gamma d_i/\\phi)^3\\right)^{\\frac{1}{3}}}\\;\\forall i, \\\\[1em]\n",
    "&l_i \\sim \\text{Norm}(\\mu_i, \\sigma) \\;\\forall i.\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Parameter estimation\n",
    "\n",
    "We will now proceed to compute the MAP and credible intervals. Before embarking on this calculation, I write down the steps for making sense of the posterior using optimization and a multivariate Normal approximation. After defining your model, do the following.\n",
    "\n",
    "1. Write a function to compute the log posterior.\n",
    "2. Define the negative log posterior function.\n",
    "3. Set up the `args` that need to be passed to the log posterior function.\n",
    "4. Provide an initial guess for the solver.\n",
    "5. Use `scipy.optimize.minimize()` to find the MAP.\n",
    "6. Extract the optimal parameters from the output.\n",
    "7. Compute the Hessian evaluated at the MAP of the log posterior by numerical differentiation.\n",
    "8. Compute the covariance by inverting the Hessian.\n",
    "9. Report the results (usually by reporting credible intervals).\n",
    "\n",
    "We will step through this procedure for this model, starting with coding up the posterior."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "def theoretical_spindle_length(d, phi, gamma):\n",
    "    \"\"\"Theoretical spindle length for tubulin conservation model.\"\"\"\n",
    "    return gamma * d / np.cbrt(1 + (gamma * d / phi)**3)\n",
    "\n",
    "\n",
    "def log_prior(phi, gamma, sigma):\n",
    "    \"\"\"Log prior for tubulin conservation model\"\"\"\n",
    "    lp = st.lognorm.logpdf(phi, 2.3, loc=0, scale=np.exp(2.3))\n",
    "    lp += st.beta.logpdf(gamma, 1.5, 1.5)\n",
    "    lp += st.halfnorm.logpdf(sigma, 0, 20)\n",
    "\n",
    "    return lp\n",
    "\n",
    "\n",
    "def log_likelihood(d, ell, phi, gamma, sigma):\n",
    "    \"\"\"Log likelihood for independent size model\"\"\"\n",
    "    mu = theoretical_spindle_length(d, phi, gamma)\n",
    "\n",
    "    return np.sum(st.norm.logpdf(ell, mu, sigma))\n",
    "\n",
    "\n",
    "def log_posterior(params, d, ell):\n",
    "    \"\"\"Log posterior of indpendent size model.\"\"\"\n",
    "    phi, gamma, sigma = params\n",
    "\n",
    "    lp = log_prior(phi, gamma, sigma)\n",
    "    if lp == -np.inf:\n",
    "        return lp\n",
    "    \n",
    "    return lp + log_likelihood(d, ell, phi, gamma, sigma)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Next, we define the negative log posterior, which we need for minimization."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "def neg_log_posterior(params, d, ell):\n",
    "    return -log_posterior(params, d, ell)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And the arguments we need to pass...."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "args = (d, ell)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And now for the initial guesses for $\\phi$, $\\gamma$, and $\\sigma$, respectively."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "params_0 = np.array([35, 0.8, 4])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, we can find the MAP and extract the parameter values."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Compute the MAP\n",
    "res = scipy.optimize.minimize(neg_log_posterior, params_0, args=args, method=\"powell\")\n",
    "\n",
    "# Extract optimal parameters\n",
    "popt = res.x"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now to compute the Hessian and invert it to get the covariance."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Compute Hessian and covariance matrix\n",
    "hes = smnd.approx_hess(popt, log_posterior, args=args)\n",
    "cov = -np.linalg.inv(hes)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Finally, we can report the results."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "Most probable parameters\n",
      "------------------------\n",
      "φ = 38.26 ± 0.77  µm\n",
      "γ = 0.859 ± 0.034\n",
      "σ = 3.754 ± 0.201 µm\n",
      "\n"
     ]
    }
   ],
   "source": [
    "# For convenience...\n",
    "phi_MAP, gamma_MAP, sigma_MAP = popt\n",
    "\n",
    "# Print results\n",
    "print(\n",
    "    \"\"\"\n",
    "Most probable parameters\n",
    "------------------------\n",
    "φ = {0:.2f} ± {1:.2f}  µm\n",
    "γ = {2:.3f} ± {3:.3f}\n",
    "σ = {4:.3f} ± {5:.3f} µm\n",
    "\"\"\".format(\n",
    "        phi_MAP,\n",
    "        1.96 * np.sqrt(cov[0, 0]),\n",
    "        gamma_MAP,\n",
    "        1.96 * np.sqrt(cov[1, 1]),\n",
    "        sigma_MAP,\n",
    "        1.96 * np.sqrt(cov[2, 2]),\n",
    "    )\n",
    ")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Checking the Normal approximation\n",
    "\n",
    "Plotting this distribution is harder than in the independent size model because we now have three parameters. We therefore have to marginalize out one of the parameters to make a contour plot of two parameters we are interested in. To do so, we first need to grid up the parameter values and compute the log posterior. Unlike in our previous treatment of this model when we could analytically marginalize away $\\sigma$, we cannot do that here because of our choice of prior. We need to compute the posterior for many values of $\\sigma$ and then use numerical quadrature to marginalize it away.\n",
    "\n",
    "Note that this problem of computing the posterior for plotting does not scale well. For the independent size model, we evaluated the posterior at 200 $\\phi$ values and 200 $\\sigma$ values for a total of 40,000 posterior evaluations. It took a few seconds to run. Now, we will evaluate the posterior at 100 $\\phi$ values, 100 $\\gamma$ values, and 100 $\\sigma_0$ values, for a total of a million posterior evaluations, and you can expect the calculation in the next cell to take several minutes. If we had any more parameters, this brute force style of posterior evaluation becomes intractable. Furthermore, notice below that I chose a tight range of parameter values. I was able to do this because I already found the MAP and credible regions first. Without this, the posterior would be mostly a big sea of low probability and finding where it is not is like finding a needle in a haystack."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "100%|█████████████████████████████████████████| 100/100 [09:48<00:00,  5.88s/it]\n"
     ]
    }
   ],
   "source": [
    "# Set up plotting range\n",
    "phi = np.linspace(37, 39.5, 100)\n",
    "gamma = np.linspace(0.8, 0.91, 100)\n",
    "sigma = np.linspace(3.0, 4.5, 100)\n",
    "PHI, GAMMA, SIGMA = np.meshgrid(phi, gamma, sigma)\n",
    "\n",
    "# Compute log posterior\n",
    "LOG_POST = np.empty_like(PHI)\n",
    "for i in tqdm.tqdm(range(PHI.shape[0])):\n",
    "    for j in range(PHI.shape[1]):\n",
    "        for k in range(PHI.shape[2]):\n",
    "            LOG_POST[i, j, k] = log_posterior(\n",
    "                    np.array([PHI[i,j,k], GAMMA[i,j,k], SIGMA[i,j,k]]), d, ell)\n",
    "        \n",
    "# Exponentiate. Ignore normalization, so set max LOG_POST to zero\n",
    "POST_exact = np.exp(LOG_POST - LOG_POST.max())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To plot the marginalized posterior $g(\\phi, \\gamma\\mid d, l)$, which is what we are interested in, we need to marginalize over $\\sigma$. We can do so numerically using `np.trapz()`, integrating over the $\\sigma$ values."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Compute marginalized posterior\n",
    "POST_marginalized = np.trapz(POST_exact, x=sigma, axis=2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can now make the plot."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
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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\":\"63f544ef-4283-4c52-8c7d-1efe316e137e\",\"roots\":{\"p1090\":\"bca717c9-91e6-4cfe-8500-396c628eec99\"},\"root_ids\":[\"p1090\"]}];\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": "p1090"
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Plot\n",
    "p = bebi103.viz.contour(\n",
    "    PHI[:, :, 0],\n",
    "    GAMMA[:, :, 0],\n",
    "    POST_marginalized,\n",
    "    x_axis_label=\"ϕ (µm)\",\n",
    "    y_axis_label=\"γ\",\n",
    "    overlaid=False,\n",
    ")\n",
    "bokeh.io.show(p)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Notice that for both $\\phi$ and $\\gamma$, the posterior is much narrower than the prior. The data have been informative!\n",
    "\n",
    "To see how good the Normal approximation is, we can plot the marginalized Normal distribution as well. In this case, we can marginalize a multivariate Normal simply by ignoring the parameters we are marginalizing out. We simply cut $\\sigma$ out of the MAP and out of the posterior covariance matrix and compute the PDF for a multivariate Normal. This calculation is almost instantaneous compared to the calculation of the full posterior."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "  <div id=\"b8ec8c97-de5a-4333-be4b-887d9ec5a660\" 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 = {\"fd47790b-8166-4240-af27-ba688779f73a\":{\"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\":\"fd47790b-8166-4240-af27-ba688779f73a\",\"roots\":{\"p1090\":\"b8ec8c97-de5a-4333-be4b-887d9ec5a660\"},\"root_ids\":[\"p1090\"]}];\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": "p1090"
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Set up plotting range\n",
    "POST_norm = st.multivariate_normal.pdf(\n",
    "    np.dstack((PHI[:, :, 0], GAMMA[:, :, 0])), popt[:-1], cov[:-1, :-1]\n",
    ")\n",
    "\n",
    "p = bebi103.viz.contour(\n",
    "    PHI[:, :, 0],\n",
    "    GAMMA[:, :, 0],\n",
    "    POST_norm,\n",
    "    line_kwargs=dict(line_color=\"orange\"),\n",
    "    p=p,\n",
    ")\n",
    "bokeh.io.show(p)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We are again fortunate that the Normal approximation is a good one. This is not always the case, though, and this is a **major** danger in using optimization methods to making sense of the posterior."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Displaying the best fit line\n",
    "\n",
    "Researchers typically display the best-fit line with their data points. I generally discourage this; you should instead show results from **posterior predictive checks**, which we will discuss in coming weeks. However, for now we do not know how to do that, so we will just display the best fit line."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "  <div id=\"eec73105-699c-424d-bd15-2272e32a3208\" data-root-id=\"p1002\" style=\"display: contents;\"></div>\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
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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\":\"fecb81d5-1e38-48c6-a5ec-6d13ddfae145\",\"roots\":{\"p1002\":\"eec73105-699c-424d-bd15-2272e32a3208\"},\"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": ""
     },
     "metadata": {
      "application/vnd.bokehjs_exec.v0+json": {
       "id": "p1002"
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Extract parameters\n",
    "phi, gamma = popt[:2]\n",
    "\n",
    "# Make theoretical curve\n",
    "d_theor = np.linspace(0, 250, 250)\n",
    "ell_theor = theoretical_spindle_length(d_theor, phi, gamma)\n",
    "\n",
    "# Add to the plot and show\n",
    "p_data.line(d_theor, ell_theor, line_width=2, color=\"orange\")\n",
    "\n",
    "bokeh.io.show(p_data)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It looks like most of the sampling was in the curved part of the plot, meaning we had few samples in the linear regime where the only relevant parameter is $\\gamma$, and few in the regime where $\\phi$ is the only relevant parameter. Importantly, we caught enough of the dynamic region of the curve to get good parameter estimates."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Computing environment"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "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
}