bayes_stats_en_percentage.ipynb

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    "#### Bayesian Statistics Interactive\n",
    "\n",
    "Bayes' theorem is shown here using an example. Individual probabilities can be changed interactively so that the changes to the end result are immediately visible.\n",
    "\n",
    "Bayes' theorem:\n",
    "For two events A and B, for B ≠ 0, Bayes' theorem is:\n",
    "\n",
    "$$ {P(A|B)} = \\frac{P(B|A) \\bullet P(A)}{P(B)}$$\n",
    "\n",
    "P(A | B) is the (conditional) probability of event A under the condition that B has occurred. P(B | A) is the (conditional) probability of event B under the condition that A has occurred\n",
    "P(A) is the probability (initial probability) that event A will occur\n",
    "P(B) is the probability (initial probability) that event B will occur\n",
    "Initial probability means that one event is viewed independently of another."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "lang": "de"
   },
   "outputs": [],
   "source": [
    "#resourcen\n",
    "\n",
    "import ipywidgets as widgets\n",
    "import numpy as np\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "lang": "de"
   },
   "outputs": [],
   "source": [
    "# interactive switches\n",
    "\n",
    "# layout limitations\n",
    "layout_right_text = {'width': '20%'}\n",
    "layout_left_text = {'width': '40%'}\n",
    "\n",
    "# Reliability of detection if patient falls ill\n",
    "pB_A = widgets.BoundedFloatText(value=99, min=0.01, max=100.0, step=0.01, layout=layout_right_text)\n",
    "\n",
    "# Percentage of positive tests in healthy patients\n",
    "pB_Anot = widgets.BoundedFloatText(value=0.05, min=0.01, max=100.0, step=0.01, layout=layout_right_text)\n",
    "\n",
    "#\n",
    "pA = widgets.BoundedFloatText(value=0.01, min=0.01, max=100.0, step=0.01, layout=layout_right_text)\n",
    "\n",
    "#\n",
    "p_spreading = widgets.HBox(\n",
    "    [\n",
    "        widgets.Label(value=\"spread of the disease (%):\", layout=layout_left_text),\n",
    "        pA\n",
    "    ]\n",
    ")\n",
    "#\n",
    "p_verify = widgets.HBox(\n",
    "    [\n",
    "        widgets.Label(\n",
    "            value='Verifiability of the test when the patient is ill (%):', layout=layout_left_text),\n",
    "        pB_A        \n",
    "    ]\n",
    ")\n",
    "#\n",
    "p_error = widgets.HBox(\n",
    "    [\n",
    "        widgets.Label(value='Error rate of the test if the patient is not ill  (%):', layout=layout_left_text),\n",
    "        pB_Anot\n",
    "    ]\n",
    ")\n",
    "#\n",
    "style = {'description_width': 'initial'}\n",
    "#\n",
    "outpriori = widgets.VBox(\n",
    "[\n",
    "    # P (ill) = pA\n",
    "    widgets.Box([widgets.Label(value=\"P(ill) (%)\", layout=layout_left_text),\n",
    "                widgets.FloatText(description = \"\", value = pA.value ,disabled = True,\\\n",
    "                                  layout=layout_right_text, style=style)]\n",
    "                ),\n",
    "    # P (not ill) = 1-pA\n",
    "    widgets.Box([widgets.Label(value=\"P(not ill) (%)\", layout=layout_left_text),\n",
    "                 widgets.FloatText(value = (100 - pA.value), layout=layout_right_text,\\\n",
    "                                   disabled = True, style=style)]\n",
    "               ),\n",
    "    # P (test _positive | ill) = pB_ A (p detection reliability)\n",
    "    widgets.Box([widgets.Label(value= \"P(test positive | ill) (%)\", layout=layout_left_text),\n",
    "                 widgets.FloatText(value = '%2.4f' % (pB_A.value),\\\n",
    "                                  layout=layout_right_text,disabled = True, style=style)]\n",
    "               ),\n",
    "    # P (test_positive | not ill) = pB_Anot\n",
    "    widgets.Box([widgets.Label(value=\"P(test positive | not ill) (%)\", layout=layout_left_text),\n",
    "                 widgets.FloatText(value = '%2.4f' % (pB_Anot.value),\\\n",
    "                                   layout=layout_right_text, disabled = True, style=style)]\n",
    "               )\n",
    "])\n",
    "# P(ill|test_positive) = [pA * pB_A] / [pA * pB_A + pA_not * pB_Anot]\n",
    "outposteriori = widgets.VBox(\n",
    "[\n",
    "    widgets.Label(value=\"Probability that a patient is ill if the test is positive  (%):\"),\n",
    "    widgets.FloatText(value = '%2.4f' % ((pB_A.value * pA.value) / \\\n",
    "                                                (pB_A.value * pA.value + pB_Anot.value * (100 - pA.value))*100),\\\n",
    "                      description=\"\", disabled = True,layout={'width': '30%'})\n",
    "])\n",
    "#\n",
    "BtnReset = widgets.Button(description=\"reset\", button_style='warning')\n",
    "#"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "lang": "en"
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   "outputs": [],
   "source": [
    "# update function that is executed when a switch is changed\n",
    "#\n",
    "def update_view(*args):\n",
    "    \"\"\" Probability that a patient is ill,\\\n",
    "        if the test is positive, recalculate with the given parameters.\n",
    "        \n",
    "        P (diseased) = pA\n",
    "        P (not diseased) = 1-pA\n",
    "        P (test_ positive | sick) = PA *pB_A (p detection security)\n",
    "        P (test_positiv |not diseased) = P (not diseased)* perror rate (pB_Anot)\n",
    "        P (diseased | test_positive) = [pA *pDetection security] / [pA* pDetection security + pA_not *perror rate]\n",
    "                                                                {=> P (B) calculated over total probability}\n",
    "        \"\"\"\n",
    "    outpriori.children[0].children[1].value = pA.value\n",
    "    outpriori.children[1].children[1].value = (100 - pA.value)\n",
    "    outpriori.children[2].children[1].value = '%3.4f' % (pB_A.value)\n",
    "    outpriori.children[3].children[1].value = '%3.4f' % (pB_Anot.value)\n",
    "    #\n",
    "    # Intercept extreme values\n",
    "    if (pB_Anot.value == 0):\n",
    "        outposteriori.children[1].value = 0.0\n",
    "        return\n",
    "    if (pA.value == 100):\n",
    "        outposteriori.children[1].value = 100\n",
    "        return\n",
    "    outposteriori.children[1].value = '%2.4f' % ((pB_A.value * pA.value) / \\\n",
    "                                                 (pB_A.value * pA.value + pB_Anot.value * (100-pA.value))*100)\n",
    "    #"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "lang": "en"
   },
   "outputs": [],
   "source": [
    "# Reset settings to default values\n",
    "#\n",
    "def reset (* btn):\n",
    "    pA.value = 0.01\n",
    "    pB_A.value = 99\n",
    "    pB_Anot.value= 0.05\n",
    "    update_view()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Bayes' theorem is shown here with interactive widgets.<br>Individual probabilities can be changed interactively so that the changes to the end result are immediately visible.\n",
    "\n",
    "\n",
    "In this example, a virtual medical test is taken to demonstrate Bayes' thoerem with the question:\n",
    " **Is a patient ill when the test is positive?**\n",
    "\n",
    "Interactive inputs:\n",
    "- If the patient is diseased, the test has a probability of 0.xx. (default: 0.99)\n",
    "- in x cases the test is positive, even if the patient is not diseased (default: 0.005)\n",
    "- Prior information, i.e. how widespread the disease is (default: 0.0001)\n",
    "\n",
    "The conditional probabilities are first calculated from this:\n",
    "- P(diseased)\n",
    "- P(not diseased)\n",
    "- P(test positive | diseased)\n",
    "- P(test positive | not diseased)\n",
    "\n",
    "And then by means of the Bayesian theorem P(diseased | test positive) - what we want to know.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "lang": "de"
   },
   "outputs": [
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       "VBox(children=(HTML(value='<hr>'), VBox(children=(HBox(children=(Label(value='spread of the disease (%):', lay…"
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   "source": [
    "# Add update function\n",
    "#\n",
    "pA.observe(update_view, 'value')\n",
    "pB_A.observe(update_view, 'value')\n",
    "pB_Anot.observe(update_view, 'value')\n",
    "BtnReset.on_click(reset)\n",
    "#\n",
    "# Arrange switch\n",
    "#\n",
    "widgets.VBox([widgets.HTML(value=\"<hr>\"),\n",
    "              widgets.VBox([p_spreading, p_verify, p_error]),\n",
    "              widgets.HTML(value=\"<hr>\"),\n",
    "              outpriori,\n",
    "              widgets.HTML(value=\"<hr>\"),\n",
    "              outposteriori,\n",
    "              widgets.HTML(value=\"<hr>\"),\n",
    "              BtnReset])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Copyright © 2020 IUBH Internationale Hochschule"
   ]
  }
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