{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "3d127ad1",
   "metadata": {},
   "outputs": [],
   "source": [
    "import pymc as pm\n",
    "import numpy as np\n",
    "import arviz as az\n",
    "\n",
    "%load_ext lab_black"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "122c2e96",
   "metadata": {},
   "source": [
    "# Rasch*\n",
    "\n",
    "Adapted from [Unit 10: rasch.odc](https://raw.githubusercontent.com/areding/6420-pymc/main/original_examples/Codes4Unit10/rasch.odc).\n",
    "\n",
    "Data can be found [here](https://raw.githubusercontent.com/areding/6420-pymc/main/data/rasch.txt).\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7dbd4c14",
   "metadata": {},
   "source": [
    "* True/False Questions  \n",
    "* 1 if answered correctly, 0 otherwise\n",
    "* n students\n",
    "* k questions\n",
    "* Assess (relative) ability of students\n",
    "* Assess (relative) difficulty of questions\n",
    "* Originally motivated by testing/education, applicable in different contexts\n",
    "\n",
    "\n",
    "## Notes: \n",
    "\n",
    "- Model works well and matches BUGS results with simple broadcasting. Just need to figure out a better way to display the results."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "ea318d60",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(162, 33)"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y = np.loadtxt(\"../data/rasch.txt\")\n",
    "n, k = y.shape\n",
    "n, k"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "49bca796",
   "metadata": {
    "tags": [
     "hide-output"
    ]
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Auto-assigning NUTS sampler...\n",
      "Initializing NUTS using jitter+adapt_diag...\n",
      "Multiprocess sampling (4 chains in 4 jobs)\n",
      "NUTS: [tau_alpha, tau_delta, mu_delta, delta, alpha]\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "\n",
       "<style>\n",
       "    /* 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",
       "        background-size: auto;\n",
       "    }\n",
       "    progress:not([value]), progress:not([value])::-webkit-progress-bar {\n",
       "        background: repeating-linear-gradient(45deg, #7e7e7e, #7e7e7e 10px, #5c5c5c 10px, #5c5c5c 20px);\n",
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       "        background: #F44336;\n",
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       "</style>\n"
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      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
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     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "\n",
       "    <div>\n",
       "      <progress value='16000' class='' max='16000' style='width:300px; height:20px; vertical-align: middle;'></progress>\n",
       "      100.00% [16000/16000 00:14&lt;00:00 Sampling 4 chains, 0 divergences]\n",
       "    </div>\n",
       "    "
      ],
      "text/plain": [
       "<IPython.core.display.HTML object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Sampling 4 chains for 1_000 tune and 3_000 draw iterations (4_000 + 12_000 draws total) took 15 seconds.\n"
     ]
    }
   ],
   "source": [
    "with pm.Model() as m:\n",
    "    tau_alpha = pm.Gamma(\"tau_alpha\", 0.01, 0.01)\n",
    "    var_alpha = pm.Deterministic(\"var_alpha\", 1 / tau_alpha)\n",
    "    tau_delta = pm.Gamma(\"tau_delta\", 0.01, 0.01)\n",
    "    # there's a typo for mu in BUGS version\n",
    "    mu_delta = pm.Normal(\"mu_delta\", 0, tau=0.001)\n",
    "\n",
    "    # the 1s in the shapes are for broadcasting\n",
    "    delta = pm.Normal(\"delta\", mu_delta, tau=tau_delta, shape=(1, 33))\n",
    "    alpha = pm.Normal(\"alpha\", 0, tau=tau_alpha, shape=(162, 1))\n",
    "\n",
    "    p = alpha - delta\n",
    "\n",
    "    pm.Bernoulli(\"likelihood\", logit_p=p, observed=y)\n",
    "\n",
    "    trace = pm.sample(3000)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "3119daff",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
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       "\n",
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       "</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>delta[0, 0]</th>\n",
       "      <td>0.951</td>\n",
       "      <td>0.231</td>\n",
       "      <td>0.509</td>\n",
       "      <td>1.375</td>\n",
       "      <td>0.005</td>\n",
       "      <td>0.003</td>\n",
       "      <td>2540.0</td>\n",
       "      <td>6221.0</td>\n",
       "      <td>1.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>delta[0, 1]</th>\n",
       "      <td>0.574</td>\n",
       "      <td>0.222</td>\n",
       "      <td>0.171</td>\n",
       "      <td>1.010</td>\n",
       "      <td>0.005</td>\n",
       "      <td>0.003</td>\n",
       "      <td>2302.0</td>\n",
       "      <td>4440.0</td>\n",
       "      <td>1.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>delta[0, 2]</th>\n",
       "      <td>0.255</td>\n",
       "      <td>0.219</td>\n",
       "      <td>-0.163</td>\n",
       "      <td>0.659</td>\n",
       "      <td>0.004</td>\n",
       "      <td>0.003</td>\n",
       "      <td>2399.0</td>\n",
       "      <td>5453.0</td>\n",
       "      <td>1.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>delta[0, 3]</th>\n",
       "      <td>1.451</td>\n",
       "      <td>0.245</td>\n",
       "      <td>0.982</td>\n",
       "      <td>1.901</td>\n",
       "      <td>0.005</td>\n",
       "      <td>0.003</td>\n",
       "      <td>2841.0</td>\n",
       "      <td>6140.0</td>\n",
       "      <td>1.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>delta[0, 4]</th>\n",
       "      <td>-1.018</td>\n",
       "      <td>0.225</td>\n",
       "      <td>-1.428</td>\n",
       "      <td>-0.584</td>\n",
       "      <td>0.005</td>\n",
       "      <td>0.003</td>\n",
       "      <td>2242.0</td>\n",
       "      <td>4786.0</td>\n",
       "      <td>1.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>delta[0, 5]</th>\n",
       "      <td>0.684</td>\n",
       "      <td>0.229</td>\n",
       "      <td>0.246</td>\n",
       "      <td>1.103</td>\n",
       "      <td>0.005</td>\n",
       "      <td>0.003</td>\n",
       "      <td>2355.0</td>\n",
       "      <td>5108.0</td>\n",
       "      <td>1.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>delta[0, 6]</th>\n",
       "      <td>0.953</td>\n",
       "      <td>0.231</td>\n",
       "      <td>0.511</td>\n",
       "      <td>1.378</td>\n",
       "      <td>0.005</td>\n",
       "      <td>0.003</td>\n",
       "      <td>2375.0</td>\n",
       "      <td>4748.0</td>\n",
       "      <td>1.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>delta[0, 7]</th>\n",
       "      <td>0.759</td>\n",
       "      <td>0.227</td>\n",
       "      <td>0.338</td>\n",
       "      <td>1.189</td>\n",
       "      <td>0.004</td>\n",
       "      <td>0.003</td>\n",
       "      <td>2814.0</td>\n",
       "      <td>5476.0</td>\n",
       "      <td>1.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>delta[0, 8]</th>\n",
       "      <td>0.649</td>\n",
       "      <td>0.227</td>\n",
       "      <td>0.219</td>\n",
       "      <td>1.070</td>\n",
       "      <td>0.005</td>\n",
       "      <td>0.003</td>\n",
       "      <td>2536.0</td>\n",
       "      <td>5835.0</td>\n",
       "      <td>1.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>delta[0, 9]</th>\n",
       "      <td>1.682</td>\n",
       "      <td>0.251</td>\n",
       "      <td>1.217</td>\n",
       "      <td>2.158</td>\n",
       "      <td>0.005</td>\n",
       "      <td>0.003</td>\n",
       "      <td>2722.0</td>\n",
       "      <td>5779.0</td>\n",
       "      <td>1.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>delta[0, 10]</th>\n",
       "      <td>1.152</td>\n",
       "      <td>0.235</td>\n",
       "      <td>0.713</td>\n",
       "      <td>1.595</td>\n",
       "      <td>0.004</td>\n",
       "      <td>0.003</td>\n",
       "      <td>2770.0</td>\n",
       "      <td>5242.0</td>\n",
       "      <td>1.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>delta[0, 11]</th>\n",
       "      <td>-1.201</td>\n",
       "      <td>0.227</td>\n",
       "      <td>-1.627</td>\n",
       "      <td>-0.779</td>\n",
       "      <td>0.005</td>\n",
       "      <td>0.003</td>\n",
       "      <td>2415.0</td>\n",
       "      <td>5362.0</td>\n",
       "      <td>1.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>delta[0, 12]</th>\n",
       "      <td>-0.187</td>\n",
       "      <td>0.223</td>\n",
       "      <td>-0.620</td>\n",
       "      <td>0.225</td>\n",
       "      <td>0.005</td>\n",
       "      <td>0.003</td>\n",
       "      <td>2194.0</td>\n",
       "      <td>5197.0</td>\n",
       "      <td>1.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>delta[0, 13]</th>\n",
       "      <td>-0.088</td>\n",
       "      <td>0.215</td>\n",
       "      <td>-0.500</td>\n",
       "      <td>0.301</td>\n",
       "      <td>0.005</td>\n",
       "      <td>0.003</td>\n",
       "      <td>1996.0</td>\n",
       "      <td>4825.0</td>\n",
       "      <td>1.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>delta[0, 14]</th>\n",
       "      <td>0.873</td>\n",
       "      <td>0.229</td>\n",
       "      <td>0.445</td>\n",
       "      <td>1.305</td>\n",
       "      <td>0.005</td>\n",
       "      <td>0.003</td>\n",
       "      <td>2496.0</td>\n",
       "      <td>5520.0</td>\n",
       "      <td>1.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>delta[0, 15]</th>\n",
       "      <td>2.271</td>\n",
       "      <td>0.274</td>\n",
       "      <td>1.743</td>\n",
       "      <td>2.776</td>\n",
       "      <td>0.005</td>\n",
       "      <td>0.003</td>\n",
       "      <td>3313.0</td>\n",
       "      <td>6834.0</td>\n",
       "      <td>1.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>delta[0, 16]</th>\n",
       "      <td>1.543</td>\n",
       "      <td>0.243</td>\n",
       "      <td>1.084</td>\n",
       "      <td>2.002</td>\n",
       "      <td>0.004</td>\n",
       "      <td>0.003</td>\n",
       "      <td>2953.0</td>\n",
       "      <td>6441.0</td>\n",
       "      <td>1.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>delta[0, 17]</th>\n",
       "      <td>-0.291</td>\n",
       "      <td>0.220</td>\n",
       "      <td>-0.724</td>\n",
       "      <td>0.091</td>\n",
       "      <td>0.005</td>\n",
       "      <td>0.003</td>\n",
       "      <td>2148.0</td>\n",
       "      <td>5395.0</td>\n",
       "      <td>1.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>delta[0, 18]</th>\n",
       "      <td>0.049</td>\n",
       "      <td>0.218</td>\n",
       "      <td>-0.360</td>\n",
       "      <td>0.461</td>\n",
       "      <td>0.004</td>\n",
       "      <td>0.003</td>\n",
       "      <td>2413.0</td>\n",
       "      <td>5464.0</td>\n",
       "      <td>1.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>delta[0, 19]</th>\n",
       "      <td>-1.315</td>\n",
       "      <td>0.233</td>\n",
       "      <td>-1.742</td>\n",
       "      <td>-0.873</td>\n",
       "      <td>0.005</td>\n",
       "      <td>0.003</td>\n",
       "      <td>2466.0</td>\n",
       "      <td>5697.0</td>\n",
       "      <td>1.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>delta[0, 20]</th>\n",
       "      <td>2.805</td>\n",
       "      <td>0.300</td>\n",
       "      <td>2.209</td>\n",
       "      <td>3.342</td>\n",
       "      <td>0.005</td>\n",
       "      <td>0.004</td>\n",
       "      <td>3432.0</td>\n",
       "      <td>6759.0</td>\n",
       "      <td>1.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>delta[0, 21]</th>\n",
       "      <td>-0.458</td>\n",
       "      <td>0.217</td>\n",
       "      <td>-0.854</td>\n",
       "      <td>-0.043</td>\n",
       "      <td>0.005</td>\n",
       "      <td>0.003</td>\n",
       "      <td>2296.0</td>\n",
       "      <td>4904.0</td>\n",
       "      <td>1.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>delta[0, 22]</th>\n",
       "      <td>0.573</td>\n",
       "      <td>0.224</td>\n",
       "      <td>0.134</td>\n",
       "      <td>0.984</td>\n",
       "      <td>0.004</td>\n",
       "      <td>0.003</td>\n",
       "      <td>2548.0</td>\n",
       "      <td>5782.0</td>\n",
       "      <td>1.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>delta[0, 23]</th>\n",
       "      <td>0.359</td>\n",
       "      <td>0.218</td>\n",
       "      <td>-0.053</td>\n",
       "      <td>0.769</td>\n",
       "      <td>0.004</td>\n",
       "      <td>0.003</td>\n",
       "      <td>2473.0</td>\n",
       "      <td>5795.0</td>\n",
       "      <td>1.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>delta[0, 24]</th>\n",
       "      <td>-1.767</td>\n",
       "      <td>0.243</td>\n",
       "      <td>-2.206</td>\n",
       "      <td>-1.291</td>\n",
       "      <td>0.005</td>\n",
       "      <td>0.003</td>\n",
       "      <td>2708.0</td>\n",
       "      <td>6129.0</td>\n",
       "      <td>1.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>delta[0, 25]</th>\n",
       "      <td>-1.474</td>\n",
       "      <td>0.232</td>\n",
       "      <td>-1.906</td>\n",
       "      <td>-1.038</td>\n",
       "      <td>0.005</td>\n",
       "      <td>0.003</td>\n",
       "      <td>2537.0</td>\n",
       "      <td>6302.0</td>\n",
       "      <td>1.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>delta[0, 26]</th>\n",
       "      <td>0.872</td>\n",
       "      <td>0.226</td>\n",
       "      <td>0.447</td>\n",
       "      <td>1.293</td>\n",
       "      <td>0.004</td>\n",
       "      <td>0.003</td>\n",
       "      <td>2635.0</td>\n",
       "      <td>5833.0</td>\n",
       "      <td>1.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>delta[0, 27]</th>\n",
       "      <td>0.084</td>\n",
       "      <td>0.219</td>\n",
       "      <td>-0.329</td>\n",
       "      <td>0.494</td>\n",
       "      <td>0.005</td>\n",
       "      <td>0.003</td>\n",
       "      <td>2220.0</td>\n",
       "      <td>5476.0</td>\n",
       "      <td>1.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>delta[0, 28]</th>\n",
       "      <td>1.319</td>\n",
       "      <td>0.239</td>\n",
       "      <td>0.881</td>\n",
       "      <td>1.774</td>\n",
       "      <td>0.005</td>\n",
       "      <td>0.003</td>\n",
       "      <td>2536.0</td>\n",
       "      <td>6135.0</td>\n",
       "      <td>1.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>delta[0, 29]</th>\n",
       "      <td>-0.394</td>\n",
       "      <td>0.219</td>\n",
       "      <td>-0.804</td>\n",
       "      <td>0.013</td>\n",
       "      <td>0.005</td>\n",
       "      <td>0.003</td>\n",
       "      <td>2209.0</td>\n",
       "      <td>4431.0</td>\n",
       "      <td>1.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>delta[0, 30]</th>\n",
       "      <td>0.398</td>\n",
       "      <td>0.224</td>\n",
       "      <td>-0.029</td>\n",
       "      <td>0.810</td>\n",
       "      <td>0.005</td>\n",
       "      <td>0.003</td>\n",
       "      <td>2232.0</td>\n",
       "      <td>5299.0</td>\n",
       "      <td>1.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>delta[0, 31]</th>\n",
       "      <td>-0.874</td>\n",
       "      <td>0.222</td>\n",
       "      <td>-1.279</td>\n",
       "      <td>-0.455</td>\n",
       "      <td>0.005</td>\n",
       "      <td>0.003</td>\n",
       "      <td>2324.0</td>\n",
       "      <td>5660.0</td>\n",
       "      <td>1.0</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>delta[0, 32]</th>\n",
       "      <td>0.723</td>\n",
       "      <td>0.227</td>\n",
       "      <td>0.308</td>\n",
       "      <td>1.159</td>\n",
       "      <td>0.005</td>\n",
       "      <td>0.003</td>\n",
       "      <td>2448.0</td>\n",
       "      <td>5629.0</td>\n",
       "      <td>1.0</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "               mean     sd  hdi_3%  hdi_97%  mcse_mean  mcse_sd  ess_bulk  \\\n",
       "delta[0, 0]   0.951  0.231   0.509    1.375      0.005    0.003    2540.0   \n",
       "delta[0, 1]   0.574  0.222   0.171    1.010      0.005    0.003    2302.0   \n",
       "delta[0, 2]   0.255  0.219  -0.163    0.659      0.004    0.003    2399.0   \n",
       "delta[0, 3]   1.451  0.245   0.982    1.901      0.005    0.003    2841.0   \n",
       "delta[0, 4]  -1.018  0.225  -1.428   -0.584      0.005    0.003    2242.0   \n",
       "delta[0, 5]   0.684  0.229   0.246    1.103      0.005    0.003    2355.0   \n",
       "delta[0, 6]   0.953  0.231   0.511    1.378      0.005    0.003    2375.0   \n",
       "delta[0, 7]   0.759  0.227   0.338    1.189      0.004    0.003    2814.0   \n",
       "delta[0, 8]   0.649  0.227   0.219    1.070      0.005    0.003    2536.0   \n",
       "delta[0, 9]   1.682  0.251   1.217    2.158      0.005    0.003    2722.0   \n",
       "delta[0, 10]  1.152  0.235   0.713    1.595      0.004    0.003    2770.0   \n",
       "delta[0, 11] -1.201  0.227  -1.627   -0.779      0.005    0.003    2415.0   \n",
       "delta[0, 12] -0.187  0.223  -0.620    0.225      0.005    0.003    2194.0   \n",
       "delta[0, 13] -0.088  0.215  -0.500    0.301      0.005    0.003    1996.0   \n",
       "delta[0, 14]  0.873  0.229   0.445    1.305      0.005    0.003    2496.0   \n",
       "delta[0, 15]  2.271  0.274   1.743    2.776      0.005    0.003    3313.0   \n",
       "delta[0, 16]  1.543  0.243   1.084    2.002      0.004    0.003    2953.0   \n",
       "delta[0, 17] -0.291  0.220  -0.724    0.091      0.005    0.003    2148.0   \n",
       "delta[0, 18]  0.049  0.218  -0.360    0.461      0.004    0.003    2413.0   \n",
       "delta[0, 19] -1.315  0.233  -1.742   -0.873      0.005    0.003    2466.0   \n",
       "delta[0, 20]  2.805  0.300   2.209    3.342      0.005    0.004    3432.0   \n",
       "delta[0, 21] -0.458  0.217  -0.854   -0.043      0.005    0.003    2296.0   \n",
       "delta[0, 22]  0.573  0.224   0.134    0.984      0.004    0.003    2548.0   \n",
       "delta[0, 23]  0.359  0.218  -0.053    0.769      0.004    0.003    2473.0   \n",
       "delta[0, 24] -1.767  0.243  -2.206   -1.291      0.005    0.003    2708.0   \n",
       "delta[0, 25] -1.474  0.232  -1.906   -1.038      0.005    0.003    2537.0   \n",
       "delta[0, 26]  0.872  0.226   0.447    1.293      0.004    0.003    2635.0   \n",
       "delta[0, 27]  0.084  0.219  -0.329    0.494      0.005    0.003    2220.0   \n",
       "delta[0, 28]  1.319  0.239   0.881    1.774      0.005    0.003    2536.0   \n",
       "delta[0, 29] -0.394  0.219  -0.804    0.013      0.005    0.003    2209.0   \n",
       "delta[0, 30]  0.398  0.224  -0.029    0.810      0.005    0.003    2232.0   \n",
       "delta[0, 31] -0.874  0.222  -1.279   -0.455      0.005    0.003    2324.0   \n",
       "delta[0, 32]  0.723  0.227   0.308    1.159      0.005    0.003    2448.0   \n",
       "\n",
       "              ess_tail  r_hat  \n",
       "delta[0, 0]     6221.0    1.0  \n",
       "delta[0, 1]     4440.0    1.0  \n",
       "delta[0, 2]     5453.0    1.0  \n",
       "delta[0, 3]     6140.0    1.0  \n",
       "delta[0, 4]     4786.0    1.0  \n",
       "delta[0, 5]     5108.0    1.0  \n",
       "delta[0, 6]     4748.0    1.0  \n",
       "delta[0, 7]     5476.0    1.0  \n",
       "delta[0, 8]     5835.0    1.0  \n",
       "delta[0, 9]     5779.0    1.0  \n",
       "delta[0, 10]    5242.0    1.0  \n",
       "delta[0, 11]    5362.0    1.0  \n",
       "delta[0, 12]    5197.0    1.0  \n",
       "delta[0, 13]    4825.0    1.0  \n",
       "delta[0, 14]    5520.0    1.0  \n",
       "delta[0, 15]    6834.0    1.0  \n",
       "delta[0, 16]    6441.0    1.0  \n",
       "delta[0, 17]    5395.0    1.0  \n",
       "delta[0, 18]    5464.0    1.0  \n",
       "delta[0, 19]    5697.0    1.0  \n",
       "delta[0, 20]    6759.0    1.0  \n",
       "delta[0, 21]    4904.0    1.0  \n",
       "delta[0, 22]    5782.0    1.0  \n",
       "delta[0, 23]    5795.0    1.0  \n",
       "delta[0, 24]    6129.0    1.0  \n",
       "delta[0, 25]    6302.0    1.0  \n",
       "delta[0, 26]    5833.0    1.0  \n",
       "delta[0, 27]    5476.0    1.0  \n",
       "delta[0, 28]    6135.0    1.0  \n",
       "delta[0, 29]    4431.0    1.0  \n",
       "delta[0, 30]    5299.0    1.0  \n",
       "delta[0, 31]    5660.0    1.0  \n",
       "delta[0, 32]    5629.0    1.0  "
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "az.summary(trace, var_names=[\"delta\"])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "cacc39d4",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Last updated: Wed Mar 22 2023\n",
      "\n",
      "Python implementation: CPython\n",
      "Python version       : 3.11.0\n",
      "IPython version      : 8.9.0\n",
      "\n",
      "pytensor: 2.10.1\n",
      "\n",
      "arviz: 0.15.1\n",
      "numpy: 1.24.2\n",
      "pymc : 5.1.2\n",
      "\n"
     ]
    }
   ],
   "source": [
    "%load_ext watermark\n",
    "%watermark -n -u -v -iv -p pytensor"
   ]
  }
 ],
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  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "language": "python",
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    "name": "ipython",
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   "file_extension": ".py",
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   "nbconvert_exporter": "python",
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