{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Graphical model assessment for univariate models\n",
    "\n",
    "[Dataset download](https://s3.amazonaws.com/bebi103.caltech.edu/data/singer_transcript_counts.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 iqplot 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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       "  const EXEC_MIME_TYPE = 'application/vnd.bokehjs_exec.v0+json';\n",
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       "\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",
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       "      delete root._bokeh_onload_callbacks\n",
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       "    console.debug(\"Bokeh: all callbacks have finished\");\n",
       "  }\n",
       "\n",
       "  function load_libs(css_urls, js_urls, callback) {\n",
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       "      console.debug(\"Bokeh: BokehJS is being loaded, scheduling callback at\", now());\n",
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       "    if (js_urls == null || js_urls.length === 0) {\n",
       "      run_callbacks();\n",
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       "    }\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",
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       "      if (root._bokeh_is_loading === 0) {\n",
       "        console.debug(\"Bokeh: all BokehJS libraries/stylesheets loaded\");\n",
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       "\n",
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       "      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-3.2.1.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-gl-3.2.1.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-widgets-3.2.1.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-tables-3.2.1.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-mathjax-3.2.1.min.js\", \"https://unpkg.com/@holoviz/panel@1.2.1/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",
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       "          for (let i = 0; i < inline_js.length; i++) {\n",
       "      inline_js[i].call(root, root.Bokeh);\n",
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       "if (force === true) {\n",
       "        display_loaded();\n",
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       "      setTimeout(run_inline_js, 100);\n",
       "    } else if (!root._bokeh_failed_load) {\n",
       "      console.log(\"Bokeh: BokehJS failed to load within specified timeout.\");\n",
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       "    } else if (force !== true) {\n",
       "      const cell = $(document.getElementById(\"ae21fe54-efa1-4b6f-b97f-7793de4208b5\")).parents('.cell').data().cell;\n",
       "      cell.output_area.append_execute_result(NB_LOAD_WARNING)\n",
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       "  }\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(\"ae21fe54-efa1-4b6f-b97f-7793de4208b5\");\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-3.2.1.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-gl-3.2.1.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-widgets-3.2.1.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-tables-3.2.1.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-mathjax-3.2.1.min.js\", \"https://unpkg.com/@holoviz/panel@1.2.1/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(\"ae21fe54-efa1-4b6f-b97f-7793de4208b5\")).parents('.cell').data().cell;\n      cell.output_area.append_execute_result(NB_LOAD_WARNING)\n    }\n  }\n\n  if (root._bokeh_is_loading === 0) {\n    console.debug(\"Bokeh: BokehJS loaded, going straight to plotting\");\n    run_inline_js();\n  } else {\n    load_libs(css_urls, js_urls, function() {\n      console.debug(\"Bokeh: BokehJS plotting callback run at\", now());\n      run_inline_js();\n    });\n  }\n}(window));"
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import warnings\n",
    "\n",
    "import pandas as pd\n",
    "import numpy as np\n",
    "import scipy.optimize\n",
    "import scipy.stats as st\n",
    "\n",
    "import numba\n",
    "\n",
    "import bebi103\n",
    "\n",
    "import iqplot\n",
    "\n",
    "import bokeh.io\n",
    "bokeh.io.output_notebook()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<hr >"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Graphical model assessment\n",
    "\n",
    "We have already [discussed model assessment](../24/graphical_model_assessment.rst). In this lesson, we will discuss implementation of graphical model assessment.\n",
    "\n",
    "We will use the transcript counts from Singer, et al., this time focusing on the Rex1 gene, which shows clear biomodality. As a reminder, here is the ECDF for the transcript counts."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
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   "source": [
    "df = pd.read_csv(os.path.join(data_path, \"singer_transcript_counts.csv\"), comment=\"#\")\n",
    "\n",
    "p = iqplot.ecdf(\n",
    "    data=df, q=\"Rex1\", x_axis_label=\"Rex1 mRNA count\", conf_int=True\n",
    ")\n",
    "\n",
    "bokeh.io.show(p)"
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   "source": [
    "Before we can proceed to the model assessments, we need to write functions to calculate the MLE."
   ]
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Likelihood calculators\n",
    "\n",
    "In this and the following parts of this lesson, we will need the functions we have written to perform maximum likelihood estimation for a Negative Binomial distribution and for a mixture of two Negative Binomials. These are copied directly from the [lesson on mixture models](../26/mixture_models.ipynb), so you do not need to focus on this rather long code cell."
   ]
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   "source": [
    "def log_like_iid_nbinom(params, n):\n",
    "    \"\"\"Log likelihood for i.i.d. NBinom measurements, parametrized\n",
    "    by alpha, b=1/beta.\"\"\"\n",
    "    alpha, b = params\n",
    "\n",
    "    if alpha <= 0 or b <= 0:\n",
    "        return -np.inf\n",
    "\n",
    "    return np.sum(st.nbinom.logpmf(n, alpha, 1 / (1 + b)))\n",
    "\n",
    "\n",
    "def mle_iid_nbinom(n):\n",
    "    \"\"\"Perform maximum likelihood estimates for parameters for i.i.d.\n",
    "    NBinom measurements, parametrized by alpha, b=1/beta\"\"\"\n",
    "    with warnings.catch_warnings():\n",
    "        warnings.simplefilter(\"ignore\")\n",
    "\n",
    "        res = scipy.optimize.minimize(\n",
    "            fun=lambda params, n: -log_like_iid_nbinom(params, n),\n",
    "            x0=np.array([3, 3]),\n",
    "            args=(n,),\n",
    "            method=\"Powell\",\n",
    "        )\n",
    "\n",
    "    if res.success:\n",
    "        return res.x\n",
    "    else:\n",
    "        raise RuntimeError(\"Convergence failed with message\", res.message)\n",
    "\n",
    "\n",
    "def initial_guess_mix(n, w_guess):\n",
    "    \"\"\"Generate initial guess for mixture model.\"\"\"\n",
    "    n_low = n[n < np.percentile(n, 100 * w_guess)]\n",
    "    n_high = n[n >= np.percentile(n, 100 * w_guess)]\n",
    "\n",
    "    alpha1, b1 = mle_iid_nbinom(n_low)\n",
    "    alpha2, b2 = mle_iid_nbinom(n_high)\n",
    "\n",
    "    return alpha1, b1, alpha2, b2\n",
    "\n",
    "\n",
    "def log_like_mix(alpha1, b1, alpha2, b2, w, n):\n",
    "    \"\"\"Log-likeihood of binary Negative Binomial mixture model.\"\"\"\n",
    "    # Fix nonidentifieability be enforcing values of w\n",
    "    if w < 0 or w > 1:\n",
    "        return -np.inf\n",
    "\n",
    "    # Physical bounds on parameters\n",
    "    if alpha1 < 0 or alpha2 < 0 or b1 < 0 or b2 < 0:\n",
    "        return -np.inf\n",
    "\n",
    "    logx1 = st.nbinom.logpmf(n, alpha1, 1 / (1 + b1))\n",
    "    logx2 = st.nbinom.logpmf(n, alpha2, 1 / (1 + b2))\n",
    "\n",
    "    # Multipliers for log-sum-exp\n",
    "    lse_coeffs = np.tile([w, 1 - w], [len(n), 1]).transpose()\n",
    "\n",
    "    # log-likelihood for each measurement\n",
    "    log_likes = scipy.special.logsumexp(np.vstack([logx1, logx2]), axis=0, b=lse_coeffs)\n",
    "\n",
    "    return np.sum(log_likes)\n",
    "\n",
    "\n",
    "def mle_mix(n, w_guess):\n",
    "    \"\"\"Obtain MLE estimate for parameters for binary mixture \n",
    "    of Negative Binomials.\"\"\"\n",
    "    with warnings.catch_warnings():\n",
    "        warnings.simplefilter(\"ignore\")\n",
    "\n",
    "        res = scipy.optimize.minimize(\n",
    "            fun=lambda params, n: -log_like_mix(*params, n),\n",
    "            x0=[*initial_guess_mix(n, w_guess), w_guess],\n",
    "            args=(n,),\n",
    "            method=\"Powell\",\n",
    "            tol=1e-6,\n",
    "        )\n",
    "\n",
    "    if res.success:\n",
    "        return res.x\n",
    "    else:\n",
    "        raise RuntimeError(\"Convergence failed with message\", res.message)"
   ]
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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can now carry out the MLE for both models."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Load in data\n",
    "df = pd.read_csv(\"../data/singer_transcript_counts.csv\", comment=\"#\")\n",
    "n = df['Rex1'].values\n",
    "\n",
    "# Single Negative Binomial MLE\n",
    "alpha, b = mle_iid_nbinom(n)\n",
    "\n",
    "# Mixture model MLE\n",
    "alpha1, b1, alpha2, b2, w = mle_mix(n, 0.2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Constructing a Q-Q plot\n",
    "\n",
    "We will start by making a [Q-Q plot](graphical_model_assessment.rst#Q-Q-plots) for the single Negative-Binomial case. The function `bebi103.viz.qqplot()` generates Q-Q plots. In order to provide the function with samples out of the generative distribution, let's draw out of a single Negative Binomial with our $\\alpha$, $b$ parametrization."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "rng = np.random.default_rng()\n",
    "\n",
    "single_samples = np.array(\n",
    "    [rng.negative_binomial(alpha, 1 / (1 + b), size=len(n)) for _ in range(100000)]\n",
    ")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "With that in place, we can call `bebi103.viz.qqplot()` to generate the plot."
   ]
  },
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   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
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       "  const render_items = [{\"docid\":\"936a7b40-c1c0-4c25-bee6-e40f3685534a\",\"roots\":{\"p1068\":\"fe61d6c6-c6a2-4c6b-b330-1003187f67b0\"},\"root_ids\":[\"p1068\"]}];\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",
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       "        attempts++;\n",
       "        if (attempts > 100) {\n",
       "          clearInterval(timer);\n",
       "          console.log(\"Bokeh: ERROR: Unable to run BokehJS code because BokehJS library is missing\");\n",
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   ],
   "source": [
    "p = bebi103.viz.qqplot(\n",
    "    data=n,\n",
    "    samples=single_samples,\n",
    "    x_axis_label=\"transcript counts\",\n",
    "    y_axis_label=\"transcript counts\",\n",
    ")\n",
    "\n",
    "bokeh.io.show(p)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Clearly, the generative model cannot produce the observed transcript bounds, since the Q-Q plot shows strong separation of the generative quantiles from the observed quantiles."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Q-Q plot for the mixture model\n",
    "\n",
    "We can also make a Q-Q plot for the mixture model. We need to write a function to generate data out of the mixture model. As we did in a previous lesson, we first randomly choose a cell type with weight $w$, and then draw out of a Negative Binomial distribution with parameters corresponding to that cell type."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "def draw_mix(alpha1, b1, alpha2, b2, w, size):\n",
    "    \"\"\"Draw samples from Negative Binomial binary mixture model.\"\"\"\n",
    "    n = np.empty(size)\n",
    "    for i in range(size):\n",
    "        low_cell_type = rng.random() < w\n",
    "\n",
    "        if low_cell_type:\n",
    "            n[i] = rng.negative_binomial(alpha1, 1 / (1 + b1))\n",
    "        else:\n",
    "            n[i] = rng.negative_binomial(alpha2, 1 / (1 + b2))\n",
    "\n",
    "    return n\n",
    "\n",
    "\n",
    "mixture_samples = np.array(\n",
    "    [draw_mix(alpha1, b1, alpha2, b2, w, size=len(n)) for _ in range(100_000)]\n",
    ")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, we can make the Q-Q plot!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
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       "  root.Bokeh.embed.embed_items_notebook(docs_json, render_items);\n",
       "  }\n",
       "  if (root.Bokeh !== undefined) {\n",
       "    embed_document(root);\n",
       "  } else {\n",
       "    let attempts = 0;\n",
       "    const timer = setInterval(function(root) {\n",
       "      if (root.Bokeh !== undefined) {\n",
       "        clearInterval(timer);\n",
       "        embed_document(root);\n",
       "      } else {\n",
       "        attempts++;\n",
       "        if (attempts > 100) {\n",
       "          clearInterval(timer);\n",
       "          console.log(\"Bokeh: ERROR: Unable to run BokehJS code because BokehJS library is missing\");\n",
       "        }\n",
       "      }\n",
       "    }, 10, root)\n",
       "  }\n",
       "})(window);"
      ],
      "application/vnd.bokehjs_exec.v0+json": ""
     },
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      "application/vnd.bokehjs_exec.v0+json": {
       "id": "p1139"
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "p = bebi103.viz.qqplot(\n",
    "    data=n,\n",
    "    samples=mixture_samples,\n",
    "    x_axis_label=\"transcript counts\",\n",
    "    y_axis_label=\"transcript counts\",\n",
    ")\n",
    "\n",
    "bokeh.io.show(p)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The Q-Q plot shows a much stronger agreement between the theoretical distribution and the observed."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Predictive ECDFs\n",
    "\n",
    "If we wish to accept a model with the MLE for the parameters as a good representation of the true generative model, we would like to see how data generated from that model would look. We can do a parametric bootstrap to generate data sets and then plot confidence intervals of the resulting ECDFs. We call ECDFs that come from the generative model **predictive ECDFs**. We can try that for the single Negative Binomial model.\n",
    "\n",
    "Given the samples (which we already computed and stored in `single_samples`), we can then compute the predictive ECDFs for each of these samples. We want the values of the predictive ECDF at each possible value of $n$, which requires us to compute the value of the ECDF at an arbitrary value. The function below accomplishes this."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "@numba.njit\n",
    "def ecdf(x, data):\n",
    "    \"\"\"Give the value of an ECDF at arbitrary points x.\"\"\"\n",
    "    y = np.arange(len(data) + 1) / len(data)\n",
    "    return y[np.searchsorted(np.sort(data), x, side=\"right\")]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we can compute the value of the ECDF for the respective values of $n$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "n_theor = np.arange(0, single_samples.max() + 1)\n",
    "\n",
    "ecdfs = np.array([ecdf(n_theor, sample) for sample in single_samples])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now that we have the ECDF values, we can compute a 95th-percentile of the value of the ECDF."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "ecdf_low, ecdf_high = np.percentile(ecdfs, [2.5, 97.5], axis=0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And now, we can make a plot."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "  const render_items = [{\"docid\":\"a4d08ac5-06fc-47a9-98f7-354e1135525a\",\"roots\":{\"p1210\":\"caa876cc-39be-49e8-b00e-1dd51edcaa7e\"},\"root_ids\":[\"p1210\"]}];\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": "p1210"
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "p = bebi103.viz.fill_between(\n",
    "    x1=n_theor,\n",
    "    y1=ecdf_high,\n",
    "    x2=n_theor,\n",
    "    y2=ecdf_low,\n",
    "    patch_kwargs={\"fill_alpha\": 0.5},\n",
    "    x_axis_label=\"n\",\n",
    "    y_axis_label=\"ECDF\",\n",
    ")\n",
    "\n",
    "bokeh.io.show(p)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This is a predictive ECDF. Is is the envelope in which we would expect 95% of the data points to lie if the generative model were true."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Comparing with data\n",
    "\n",
    "We can overlay the ECDF of the data to see if it falls within this envelope."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "  <div id=\"b2c064c5-2902-4701-b251-a0d9557de91a\" data-root-id=\"p1210\" style=\"display: contents;\"></div>\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "application/javascript": [
       "(function(root) {\n",
       "  function embed_document(root) {\n",
       "  const docs_json = {\"a446360f-3b74-4669-9918-7f6bb606eaed\":{\"version\":\"3.2.1\",\"title\":\"Bokeh 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       "  const render_items = [{\"docid\":\"a446360f-3b74-4669-9918-7f6bb606eaed\",\"roots\":{\"p1210\":\"b2c064c5-2902-4701-b251-a0d9557de91a\"},\"root_ids\":[\"p1210\"]}];\n",
       "  root.Bokeh.embed.embed_items_notebook(docs_json, render_items);\n",
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       "  if (root.Bokeh !== undefined) {\n",
       "    embed_document(root);\n",
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       "    let attempts = 0;\n",
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       "        if (attempts > 100) {\n",
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       "          console.log(\"Bokeh: ERROR: Unable to run BokehJS code because BokehJS library is missing\");\n",
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       "    }, 10, root)\n",
       "  }\n",
       "})(window);"
      ],
      "application/vnd.bokehjs_exec.v0+json": ""
     },
     "metadata": {
      "application/vnd.bokehjs_exec.v0+json": {
       "id": "p1210"
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "p = iqplot.ecdf(data=df, q=\"Rex1\", palette=[\"orange\"], p=p)\n",
    "\n",
    "bokeh.io.show(p)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The Negative Binomial model generated some counts that were quite large, which is why the plot extends past counts of 1000. If you zoom in on the region where the data lie, the deviation of the measured ECDF from the ECDF generated by the model is clear. \n",
    "\n",
    "We can also compare with the mixture model."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Using predictive ECDFs in the bebi103 package\n",
    "\n",
    "The `bebi103` package enables automated generation of predictive ECDFs. To demonstrate its usage, we can generate samples from the mixture model and plot a predictive ECDF."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "  function embed_document(root) {\n",
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fa-exclamation-triangle\"],[\"tagName\",\"i\"],[\"color\",\"white\"]]}]]},{\"type\":\"map\",\"entries\":[[\"type\",\"info\"],[\"background\",\"#007bff\"],[\"icon\",{\"type\":\"map\",\"entries\":[[\"className\",\"fas 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       "  const render_items = [{\"docid\":\"180150ad-4030-4605-ab93-b07bf8b2b44b\",\"roots\":{\"p1306\":\"dc38259f-27d4-4ae7-81fd-f745060f14ea\"},\"root_ids\":[\"p1306\"]}];\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": "p1306"
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Use discrete=True for discrete data\n",
    "p = bebi103.viz.predictive_ecdf(\n",
    "    samples=mixture_samples, data=n, discrete=True, x_axis_label=\"n\"\n",
    ")\n",
    "\n",
    "bokeh.io.show(p)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here we see that the predictive ECDF captures the measured data much better. If you zoom into the plot, you will notice that the `bebi103.viz.predictive_ecdf()` function plots quantiles in various shades of blue. By default, the outermost region (lightest blue) contains the middle 95% of the generated ECDFs, and the next one in captures the middle 68%. The median ECDF from the generative model is a dark blue line in the middle."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### ECDF difference\n",
    "\n",
    "While the predictive ECDF plot is useful, it is perhaps more illuminating to plot the *difference* between the predictive ECDF and the measured ECDF. We can do this with the `diff='ecdf'` kwarg."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "  <div id=\"e32cdbb5-44b4-4e71-9702-d615ae32987d\" data-root-id=\"p1414\" style=\"display: contents;\"></div>\n"
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     },
     "metadata": {},
     "output_type": "display_data"
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       "  function embed_document(root) {\n",
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       "  const render_items = [{\"docid\":\"952df816-22c0-4d8b-b11e-5530fef7353b\",\"roots\":{\"p1414\":\"e32cdbb5-44b4-4e71-9702-d615ae32987d\"},\"root_ids\":[\"p1414\"]}];\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": "p1414"
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "p = bebi103.viz.predictive_ecdf(\n",
    "    samples=mixture_samples, data=n, diff=\"ecdf\", discrete=True, x_axis_label=\"n\"\n",
    ")\n",
    "\n",
    "bokeh.io.show(p)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It is now clear that very few of the 279 data points lie outside the 95% envelope of the predictive ECDF. We might expect more to lie outside; it is conceivable that the model is too flexible."
   ]
  },
  {
   "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       : 3.11.4\n",
      "IPython version      : 8.12.0\n",
      "\n",
      "numpy     : 1.24.3\n",
      "pandas    : 1.5.3\n",
      "scipy     : 1.10.1\n",
      "bokeh     : 3.2.1\n",
      "iqplot    : 0.3.6\n",
      "bebi103   : 0.1.17\n",
      "jupyterlab: 4.0.4\n",
      "\n"
     ]
    }
   ],
   "source": [
    "%load_ext watermark\n",
    "%watermark -v -p numpy,pandas,scipy,bokeh,iqplot,bebi103,jupyterlab"
   ]
  }
 ],
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  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
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