notebooks/marginalized_changepoint_model.ipynb

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   "source": [
    "import pymc as pm\n",
    "from pymc_extras.model.marginal.marginal_model import MarginalModel\n",
    "import pandas as pd\n",
    "import numpy as np\n",
    "import arviz as az"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The original model"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "# TODO: Try to handle np.nan\n",
    "# fmt: off\n",
    "disaster_data = pd.Series(\n",
    "    [4, 5, 4, 0, 1, 4, 3, 4, 0, 6, 3, 3, 4, 0, 2, 6,\n",
    "     3, 3, 5, 4, 5, 3, 1, 4, 4, 1, 5, 5, 3, 4, 2, 5,\n",
    "     2, 2, 3, 4, 2, 1, 3, np.nan, 2, 1, 1, 1, 1, 3, 0, 0,\n",
    "     1, 0, 1, 1, 0, 0, 3, 1, 0, 3, 2, 2, 0, 1, 1, 1,\n",
    "     0, 1, 0, 1, 0, 0, 0, 2, 1, 0, 0, 0, 1, 1, 0, 2,\n",
    "     3, 3, 1, np.nan, 2, 1, 1, 1, 1, 2, 4, 2, 0, 0, 1, 4,\n",
    "     0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1]\n",
    ")\n",
    "# fmt: on\n",
    "\n",
    "years = np.arange(1851, 1962)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/home/ricardo/Documents/Projects/pymc/pymc/model.py:1378: ImputationWarning: Data in disasters contains missing values and will be automatically imputed from the sampling distribution.\n",
      "  warnings.warn(impute_message, ImputationWarning)\n"
     ]
    }
   ],
   "source": [
    "with MarginalModel() as disaster_model:\n",
    "    switchpoint = pm.DiscreteUniform(\"switchpoint\", lower=years.min(), upper=years.max())\n",
    "\n",
    "    early_rate = pm.Exponential(\"early_rate\", 1.0)\n",
    "    late_rate = pm.Exponential(\"late_rate\", 1.0)\n",
    "    rate = pm.math.switch(switchpoint >= years, early_rate, late_rate)\n",
    "\n",
    "    disasters = pm.Poisson(\"disasters\", rate, observed=disaster_data)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
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      "Multiprocess sampling (4 chains in 4 jobs)\n",
      "CompoundStep\n",
      ">CompoundStep\n",
      ">>Metropolis: [switchpoint]\n",
      ">>Metropolis: [disasters_missing]\n",
      ">NUTS: [early_rate, late_rate]\n"
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      "Sampling 4 chains for 1_000 tune and 1_000 draw iterations (4_000 + 4_000 draws total) took 6 seconds.\n"
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    "with disaster_model:\n",
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",
      "text/plain": [
       "<Figure size 2944x552 with 4 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "az.plot_posterior(trace, var_names=[\"~switchpoint\", \"~disasters\"]);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>mean</th>\n",
       "      <th>sd</th>\n",
       "      <th>hdi_3%</th>\n",
       "      <th>hdi_97%</th>\n",
       "      <th>mcse_mean</th>\n",
       "      <th>mcse_sd</th>\n",
       "      <th>ess_bulk</th>\n",
       "      <th>ess_tail</th>\n",
       "      <th>r_hat</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>disasters_missing[0]</th>\n",
       "      <td>2.203</td>\n",
       "      <td>1.869</td>\n",
       "      <td>0.000</td>\n",
       "      <td>5.000</td>\n",
       "      <td>0.095</td>\n",
       "      <td>0.067</td>\n",
       "      <td>373.0</td>\n",
       "      <td>417.0</td>\n",
       "      <td>1.01</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>disasters_missing[1]</th>\n",
       "      <td>0.913</td>\n",
       "      <td>0.967</td>\n",
       "      <td>0.000</td>\n",
       "      <td>3.000</td>\n",
       "      <td>0.037</td>\n",
       "      <td>0.027</td>\n",
       "      <td>723.0</td>\n",
       "      <td>932.0</td>\n",
       "      <td>1.00</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>early_rate</th>\n",
       "      <td>3.081</td>\n",
       "      <td>0.289</td>\n",
       "      <td>2.551</td>\n",
       "      <td>3.646</td>\n",
       "      <td>0.006</td>\n",
       "      <td>0.004</td>\n",
       "      <td>2427.0</td>\n",
       "      <td>2785.0</td>\n",
       "      <td>1.00</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>late_rate</th>\n",
       "      <td>0.930</td>\n",
       "      <td>0.116</td>\n",
       "      <td>0.713</td>\n",
       "      <td>1.149</td>\n",
       "      <td>0.002</td>\n",
       "      <td>0.002</td>\n",
       "      <td>2725.0</td>\n",
       "      <td>2749.0</td>\n",
       "      <td>1.00</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "                       mean     sd  hdi_3%  hdi_97%  mcse_mean  mcse_sd  \\\n",
       "disasters_missing[0]  2.203  1.869   0.000    5.000      0.095    0.067   \n",
       "disasters_missing[1]  0.913  0.967   0.000    3.000      0.037    0.027   \n",
       "early_rate            3.081  0.289   2.551    3.646      0.006    0.004   \n",
       "late_rate             0.930  0.116   0.713    1.149      0.002    0.002   \n",
       "\n",
       "                      ess_bulk  ess_tail  r_hat  \n",
       "disasters_missing[0]     373.0     417.0   1.01  \n",
       "disasters_missing[1]     723.0     932.0   1.00  \n",
       "early_rate              2427.0    2785.0   1.00  \n",
       "late_rate               2725.0    2749.0   1.00  "
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "az.summary(trace, var_names=[\"~switchpoint\", \"~disasters\"])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Marginalized model"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/home/ricardo/Documents/Projects/pymc-experimental/pymc_experimental/marginal_model.py:117: UserWarning: There are multiple dependent variables in a FiniteDiscreteMarginalRV.\n",
      "Their joint logp terms will be assigned to the first RV: disasters_missing\n",
      "  warnings.warn(\n"
     ]
    }
   ],
   "source": [
    "disaster_model.marginalize([switchpoint])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
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     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pm.model_to_graphviz(disaster_model)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Multiprocess sampling (4 chains in 4 jobs)\n",
      "CompoundStep\n",
      ">NUTS: [early_rate, late_rate]\n",
      ">Metropolis: [disasters_missing]\n"
     ]
    },
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       "    /* Turns off some styling */\n",
       "    progress {\n",
       "        /* gets rid of default border in Firefox and Opera. */\n",
       "        border: none;\n",
       "        /* Needs to be in here for Safari polyfill so background images work as expected. */\n",
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       "    progress:not([value]), progress:not([value])::-webkit-progress-bar {\n",
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     "data": {
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       "\n",
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       "      <progress value='8000' class='' max='8000' style='width:300px; height:20px; vertical-align: middle;'></progress>\n",
       "      100.00% [8000/8000 01:16&lt;00:00 Sampling 4 chains, 0 divergences]\n",
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    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Sampling 4 chains for 1_000 tune and 1_000 draw iterations (4_000 + 4_000 draws total) took 77 seconds.\n"
     ]
    }
   ],
   "source": [
    "with disaster_model:\n",
    "    trace = pm.sample()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
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tX75I+TnZGjbhXN328AsKCKz9eQ0AQGMpKqvQHz9Yr6zCMnVvGa6nz+/JQS8AThXk76M3ru6vG4d3kCS9OG+3Hvpyq0or7BYnA+CNCksrdPP76/TpuhTZDOmpKT103wRW+gGA+vKxGbptdEfNvWuEzkxsptIKh/61YI8m/GupZm9Mld3B8eKmxrBuAKfEdDgkSTabjx759wfq3neQgkJC1blnXz367w8UGBSsvds3a/OaZRYnBQAAcF3NW7TS7I2H9dWGFL09d63ueOwl7diwRnddOl77d261Oh4AwIuU2x267aMN2nw4V5HBfnrz6v58MxNAo/CxGXpkcnc9OaWHbIb06boUXfzGKiVnFVodDYAXOZRVpEveXKUluzMU6GfTm1f31zVD462OBQBurWNsmD66abBev6KvWoQH6NDxIt396Sad/cpSzd2aJgcNQE2mXo0/QUEhkqTS4uJa95dUbQ8MYQ444GkCgyuf/70HDTtpnFdkdHP1HzFOkrR9/eomzwYA3oR6DPAMhmEoJq6Vxp9/uf72yrvKyzmu1x+7l9UTAQBNotzu0D2fbqo58PXu1IFq1yzY6lgAPNy1Q+P17tSBigr209bUXE1+bbm+3pRKDQyg0S3ceUyTX1+mHWl5ahbir5k3D9GEHnFWxwIAj2AYhs49o5UW3Tdaf5nYReGBvtqbXqA/fbRBE19ZqplrDqm4jNUeG1u9Gn+at2wtScpMT6t1f1bV9tiq0wHwHDFVz+vYVm1r3R/bqo0kKed4ZpNlAgBvRD0GeJ5OPfqodftEHdy7U8dSD1kdBwDg4UrK7frTh+v13ZY0+fkY+r+r+qlfuyirYwHwEqO7xGrOXSM0MD5K+aUVuuuTTbr5/XVKy639yy0AcDpKyu16+rsdunHGOuWVVKhvu0h99+fh1D4A0AhCAnx1+5iOWvbgWP15XCeFBlQ2AP3tq60a8txCPf7Ndm1LzaXpu5HUq/GnQ+dukqSkXbUvQV+9NH37Tt1OMxYAV5PQpackqSAvp9b9BbmV26tXogAANA7qMcAzhUdWfuiYl51lcRIAgCfLLynXTTPWacHOdAX42vTWtQM0tmsLq2MB8DItI4L08c1DdM/4zvLzMbRgZ7rOenmp3vxpv0rK+TY4AOfYlJKjSa8t0zvLD0iSrh3aXp/eMlQtI4IsTgYAni0iyE/3ntVZK/86Vo9M7q620UHKLS7X9JUHNfn15Rr/8k96ds5OrdyfqXK7w+q4HsO3Pifu2meggkPDdTQlWUm7tyuhS48T9q9a8L0kaeDIs5yXEIBLGDh6gj5+85/atXm9KsrL5evnV7PP4XBo56afJUkduvao6yIAAE5APQZ4nqKCfCXt3i7DMBTbup3VcQAAHupAZqFufn+d9qUXKMTfR+9cN1BDE5tZHQuAl/L1semu8Z10Tq84PfjlFm08lKPn5+7SeysO6Nqh8bpyUDtFhfhbHROAGyous+vfi/fqzZ+SZHeYig0L0PMX9aLZGQCaWHign24c3kFTz4zX8n2Z+nxdiubvOKb9GYXan5Gkt5YmKSzAVwPiozQgPloD46PVu02EAv18rI7uluq14o+fn78mXT5VkvTWc39XSXFRzb6vP3hLB/fuVI/+Q9Sxe2+nhgRgvYQuPXTGkJHKSDusj/7vhROWYfv87Vd1+MA+RUQ319Bxf7AwJQB4PuoxwD39OGumjh5OPml7VnqaXv7bHSouLFD/EeMUGd3cgnQAAE9mmqZmbTis815frn3pBWoRHqCPbxlC0w8Al9C5RZi++OOZ+uclZ6h1ZJCO5ZXqxXm7NeS5hbr3001asOMYqwABOCWmaWre9qMa//JPmrZ4v+wOU1P6tNL8e0bS9AMAFvKxGRrVOUb/vrKffv77eL1+RV9d2K+1moX4K7+0Qot3Z+jFebt16X9Wqcdj8zThXz/prk826j8/7deyvRnKLCi1+ia4BcOsxxC1rzelqqy0RA/ffIn2bN2oqOYt1L3fIGWkHdaerRsVEdVM/3j/G8W1ad+YmQFYJOPoET103XnKSj+qVu0T1L5jVx0+sE8pSXvkHxiov//rPZ0xZISm9GltdVQAcElfb0p1yuVQjwHu5+83Xazt61erbUJntY5PlK+vnzKPHdH+nVtVXlaqdold9PgbMxUdc/ofRlKLAQCqHcws1NPf79CCnemSpP7to/TGVf0UGx5ocTIAOFlphV3fbU7TuysOaPuRvJrtoQG+GhgfpUEdmql/+yh1axmmsEC/37gkAN5mU0qOXpy3Syv2VY7Pbh0ZpEcmd9fZPeMsTgYAqIvDYWr7kTz9fPC41iUf188Hs5WRX3uTT/NQf3WJC1PXuPCqv8PUKTZMQf6sDlSt3o0/klRaUqwv3/23lv4wW5lH0xQWEam+Z47Wlbfdr+YtWjVaWADWyz2epU/felk//7RA2ZnpCgmPUO+Bw3TxTXeqfceuVsf7TRwEA2A1ZzX+SNRjgLtZt2yhVi2coz1bNyo785iKiwoVHBqmdoldNHTcHzTxoqvk5x/glOui5gEAHMws1FvLkvT5uhSV20352gzdPb6T/jgqUb4+9VoA3GkyMjIsuV7Ak8XExFgdoVGYpqkNh7L17eY0/bDtqI7mlZx0mjZRQeoaF67uLcPUtWW4urUMV/voYNlshgWJAVhl77F8vTR/t+ZtPyZJ8vex6eaRHXT7mI4K9ve1OF39UCsB3slT67mGME1Tx/JKtTMtTzvS8rTjSJ52puXpQFahautoMQwpvlmIusaFqUtcmLq3DFeftpFe+0WXBjX+AIA74iAYAKtRSwFoCtQ8AOCdyiocWronQ1+sP6x5O47WfDA6qnOMHpncTR1jwyzNZxgcjAecrR4f7bsth8PUjrQ8rTlwXGsPZGnL4Vyl5Z7cCCRJYQG+6t4qXL1aR6hf+ygNSWim6BD/Jk4MoCmkHC/Sqwv3ataGw3KYks2QLujbRneP76S20cFWx2sQaiXAO3lDPXe6isoqtOdYgXYfzdOuo/naXfUnq7Cs1tO3jAhUn7aRNX/OaBupQD/PXxmIxh8AXoODYACsRi0FoClQ8wCA9ygpt2vdwWz9sD1N329JU3ZRec2+sV1jdevIBA1OaGZhwv/hYBbgfN56oCinqEy7juZrZ1qedqXla1fVQaDSCsdJp+3eMlxnJjbTsE7NdWZiMwX4ev5BH8CTbUvN1dvLkvTdljTZHZW/Ayf2aKH7J3RRpxbWNjmfLmolwDt5az3nDBn5pdp99H+14LbUXO05li/Hr+5SPx9DvVpHaGB8tAbER2tA+yhFeWBzOI0/ALwGB8EAWI1aCkBToOYBAM9lmqb2ZxRq6Z4MLd2bodVJWSop/9+B7piwAJ3bu5UuG9hWXeJc6+AXB7MA5+NA0f9U2B3al1GgrYdzteVwrtYcyNKeYwUnnCY0wFdjusbq7B5xGt0lRiEB7jUGCPBWDoepn/Zm6O2lSVq5P6tm+/COzXXfhM7q2y7KwnTOQ60EeCfqOecqLK3Q1tRcbUrJ0aZDOdpwKFvp+aUnna5TbKgGxEdrUIcoDWgfrTZRQW7/e5jGHwBeg4NgAKxGLQWgKVDzAIBnyS0u18p9mVq6N0NL92QqNaf4hP2xYQEa1TlG5/VppaEJzeTrY7Mo6W9z9w9RAVfEgaLflpFfqpX7M7Vqf5YW707Xsbz/HfQJ9LNpTJdY/aFXS43tGksTEOCCDmcX6fN1h/XF+sM19Y+PzdDk3i1184gE9WwdYXFC56JWArwT9VzjMk1TKceL9fPB4zV/9mcUnnS6uPBADewQrYHxURoYH63OLcLkY3Ov38s0/gDwGhwEA2A1aikATYGaBwDcW2mFXeuTs7VyX5ZW7M/UlsO5NaMsJMnfx6ZBHaI1snNzjewcoy4twtziQFFGRobVEQCPExMTY3UEt+FwmNp0OEfzth3V3G1Hdeh4Uc2+QD+bxnb9XxNQsD9NQIBVUo4X6ccdxzRv+1GtPXhc1UcwwwJ9demAtrpheAe1jgyyNmQjoVYCvBP1XNPLKijV+uTsqkagbG1LzVXFr+aDhQX6qn/7KPVrF6VOsaFKjA1V+2bBLj02lsYfAF6Dg2AArEYtBaApUPMAgHuxO0xtP5KrFfuytHJ/pn4+ePyE8V2SlBATopGdYjSqc4wGJ0RzUBoAToNpmtp+JE/fb03TnK1pSs46uQlofLcWGt6xuWLDAy1MCni+tNxirU/O1vrkbK1JOq4daXkn7D8zsZkuG9hWE3vEKdDPdQ+2AgDcV3GZXZtScmpWBNqQnK3CMvtJp7MZleO1W4QHKjYsQM1CAhTk76NAPx8F+tkUGeSnqcM6WHALKtH4A8BrcBAMgNWopQA0BWoeAHBtWQWl2nw4R5tScrU5JUebUnKUW1x+wmliwgI0LLGZzuzYXMM6NvfYb7YDgNV+2QT0/Za0E1YCkqQuLcI0rGNz9W0XqT5tI9UmKsgtVlkDXIFpmioorVBOUbmOF5Ypq7BUKceLdTCrUIeyirQzLU9HcktOOI/NkAbGR2tCjzhN6N5CbaODLUoPAPBWFXaHdh3N188Hj2vr4VztzyxUUnqB8ksrfvN8LcIDtOZv45so5cn4ehAAAAAAAAC8UsrxIi3ceUy+Pjb5+Rjytdnk62PIz8cmfx+bAvxsCvD1UYDvr/7ta1OAX+W/fW2GHKZUVuFQWYVDpRV25ZVUKCO/VJkFpTqWV6IDmYVKyihUUmaBjuWVnpQjLMBXgxOaaVjHZhrWsbk6xYZyYBkAmoBhGOrZOkI9W0fogYldtC01T3O3pWnp3gxtP5Kn3cfytftYvrSi8vShAb6Kbx6s+GYhim8Wouah/ooODVCzEH+FBvjK39dW+cen8rXC18cm0zRlSnKYpkyz8m+HWdkU8ev/n/C3TDkc1ftN2R2Vf1fYTdnNyn0VDkfVPsnucFT+bZqyOxyqsJuqcJiqsDtUbjdV4aj6u+rfZfaq09gdKneY6hYXpmuGxlv5cMANpeeX6PL/rFa5wyG73VR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",
      "text/plain": [
       "<Figure size 2944x552 with 4 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "az.plot_posterior(trace, var_names=\"~disasters\");"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>mean</th>\n",
       "      <th>sd</th>\n",
       "      <th>hdi_3%</th>\n",
       "      <th>hdi_97%</th>\n",
       "      <th>mcse_mean</th>\n",
       "      <th>mcse_sd</th>\n",
       "      <th>ess_bulk</th>\n",
       "      <th>ess_tail</th>\n",
       "      <th>r_hat</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>disasters_missing[0]</th>\n",
       "      <td>2.272</td>\n",
       "      <td>1.906</td>\n",
       "      <td>0.000</td>\n",
       "      <td>6.000</td>\n",
       "      <td>0.079</td>\n",
       "      <td>0.060</td>\n",
       "      <td>618.0</td>\n",
       "      <td>685.0</td>\n",
       "      <td>1.01</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>disasters_missing[1]</th>\n",
       "      <td>0.990</td>\n",
       "      <td>1.045</td>\n",
       "      <td>0.000</td>\n",
       "      <td>3.000</td>\n",
       "      <td>0.040</td>\n",
       "      <td>0.030</td>\n",
       "      <td>712.0</td>\n",
       "      <td>652.0</td>\n",
       "      <td>1.01</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>early_rate</th>\n",
       "      <td>3.085</td>\n",
       "      <td>0.283</td>\n",
       "      <td>2.538</td>\n",
       "      <td>3.592</td>\n",
       "      <td>0.005</td>\n",
       "      <td>0.003</td>\n",
       "      <td>3770.0</td>\n",
       "      <td>3235.0</td>\n",
       "      <td>1.00</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>late_rate</th>\n",
       "      <td>0.929</td>\n",
       "      <td>0.118</td>\n",
       "      <td>0.711</td>\n",
       "      <td>1.147</td>\n",
       "      <td>0.002</td>\n",
       "      <td>0.001</td>\n",
       "      <td>3214.0</td>\n",
       "      <td>2258.0</td>\n",
       "      <td>1.00</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "                       mean     sd  hdi_3%  hdi_97%  mcse_mean  mcse_sd  \\\n",
       "disasters_missing[0]  2.272  1.906   0.000    6.000      0.079    0.060   \n",
       "disasters_missing[1]  0.990  1.045   0.000    3.000      0.040    0.030   \n",
       "early_rate            3.085  0.283   2.538    3.592      0.005    0.003   \n",
       "late_rate             0.929  0.118   0.711    1.147      0.002    0.001   \n",
       "\n",
       "                      ess_bulk  ess_tail  r_hat  \n",
       "disasters_missing[0]     618.0     685.0   1.01  \n",
       "disasters_missing[1]     712.0     652.0   1.01  \n",
       "early_rate              3770.0    3235.0   1.00  \n",
       "late_rate               3214.0    2258.0   1.00  "
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "az.summary(trace, var_names=\"~disasters\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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