Hierarchical modeling.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<thead><tr><th scope=col>chips</th><th scope=col>location</th></tr></thead>\n",
       "<tbody>\n",
       "\t<tr><td>12</td><td>1 </td></tr>\n",
       "\t<tr><td>12</td><td>1 </td></tr>\n",
       "\t<tr><td> 6</td><td>1 </td></tr>\n",
       "\t<tr><td>13</td><td>1 </td></tr>\n",
       "\t<tr><td>12</td><td>1 </td></tr>\n",
       "\t<tr><td>12</td><td>1 </td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{r|ll}\n",
       " chips & location\\\\\n",
       "\\hline\n",
       "\t 12 & 1 \\\\\n",
       "\t 12 & 1 \\\\\n",
       "\t  6 & 1 \\\\\n",
       "\t 13 & 1 \\\\\n",
       "\t 12 & 1 \\\\\n",
       "\t 12 & 1 \\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "| chips | location |\n",
       "|---|---|\n",
       "| 12 | 1  |\n",
       "| 12 | 1  |\n",
       "|  6 | 1  |\n",
       "| 13 | 1  |\n",
       "| 12 | 1  |\n",
       "| 12 | 1  |\n",
       "\n"
      ],
      "text/plain": [
       "  chips location\n",
       "1 12    1       \n",
       "2 12    1       \n",
       "3  6    1       \n",
       "4 13    1       \n",
       "5 12    1       \n",
       "6 12    1       "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "dat = read.table(\"cookies.txt\", header = TRUE)\n",
    "head(dat)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>1</li>\n",
       "\t<li>2</li>\n",
       "\t<li>3</li>\n",
       "\t<li>4</li>\n",
       "\t<li>5</li>\n",
       "</ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item 1\n",
       "\\item 2\n",
       "\\item 3\n",
       "\\item 4\n",
       "\\item 5\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 1\n",
       "2. 2\n",
       "3. 3\n",
       "4. 4\n",
       "5. 5\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "[1] 1 2 3 4 5"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "unique(dat$location)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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rQTL099LzeFG2qCIrQTw76VYZErohHa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\nGpICoSEpEBqSAqEhKRAakuIf05OwVI8buFEAAAAASUVORK5CYII=",
      "text/plain": [
       "Plot with title \"Histogram of dat$chips\""
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "options(repr.plot.width=6, repr.plot.height=6)\n",
    "\n",
    "hist(dat$chips)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "boxplot(chips ~ location, data = dat)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Prior Predictive Check\n",
    "\n",
    "\n",
    "Before implementing the model, we need to select prior distributions for α and β, the hyperparameters governing the gamma distribution for the λ parameters. First, think about what the λ’s represent. For location j, λj is the expected number of chocolate chips per cookie. Hence, α and β control the distribution of these means between locations. The mean of this gamma distribution will represent the overall mean of number of chips for all cookies. The variance of this gamma distribution controls the variability between locations. If this is high, the mean number of chips will vary widely from location to location. If it is small, the mean number of chips will be nearly the same from location to location.\n",
    "\n",
    "To see the effects of different priors on the distribution of λ’s, we can simulate. Suppose we try independent exponential priors for α and β."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<img src=\"hierarchy.PNG\"/>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "set.seed(112)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "n_sim = 1e3\n",
    "\n",
    "alpha_prior = rexp(n_sim, rate = 1.0/2.0)\n",
    "beta_prior = rexp(n_sim, rate = 5.0)\n",
    "\n",
    "mu_prior = (alpha_prior ^ 2) / (beta_prior ^ 2)\n",
    "sig_prior = sqrt( alpha_prior/ (beta_prior ^ 2) )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "    Min.  1st Qu.   Median     Mean  3rd Qu.     Max. \n",
       "       0       11       93    47997      713 37405797 "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "summary(mu_prior)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "    Min.  1st Qu.   Median     Mean  3rd Qu.     Max. \n",
       "   0.169    3.702    8.194   29.394   19.331 4660.062 "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "summary(sig_prior)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "    Min.  1st Qu.   Median     Mean  3rd Qu.     Max. \n",
       "   0.000    1.073    6.838   40.477   25.373 5211.686 "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# After simulating from the priors for α and β, we can use those samples to simulate further down the hierarchy.\n",
    "\n",
    "lambda_prior = rgamma(n = n_sim, \n",
    "                      shape = alpha_prior, \n",
    "                      rate = beta_prior)\n",
    "\n",
    "summary(lambda_prior)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. \n",
       "  1.767   4.737  11.836  16.499  17.928  46.226 "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Or for a prior predictive reconstruction of the original data set:\n",
    "\n",
    "lambda_prior = rgamma(n = 5, \n",
    "                      shape = alpha_prior, \n",
    "                      rate = beta_prior)\n",
    "\n",
    "summary(lambda_prior)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "\t<li>0</li>\n",
       "\t<li>2</li>\n",
       "\t<li>2</li>\n",
       "\t<li>0</li>\n",
       "\t<li>1</li>\n",
       "\t<li>0</li>\n",
       "\t<li>2</li>\n",
       "\t<li>1</li>\n",
       "\t<li>1</li>\n",
       "\t<li>0</li>\n",
       "\t<li>1</li>\n",
       "\t<li>0</li>\n",
       "\t<li>0</li>\n",
       "\t<li>4</li>\n",
       "\t<li>0</li>\n",
       "\t<li>1</li>\n",
       "\t<li>2</li>\n",
       "\t<li>1</li>\n",
       "\t<li>4</li>\n",
       "\t<li>2</li>\n",
       "\t<li>1</li>\n",
       "\t<li>0</li>\n",
       "\t<li>2</li>\n",
       "\t<li>0</li>\n",
       "\t<li>5</li>\n",
       "\t<li>3</li>\n",
       "\t<li>43</li>\n",
       "\t<li>42</li>\n",
       "\t<li>47</li>\n",
       "\t<li>39</li>\n",
       "\t<li>52</li>\n",
       "\t<li>50</li>\n",
       "\t<li>53</li>\n",
       "\t<li>50</li>\n",
       "\t<li>49</li>\n",
       "\t<li>54</li>\n",
       "\t<li>44</li>\n",
       "\t<li>42</li>\n",
       "\t<li>40</li>\n",
       "\t<li>53</li>\n",
       "\t<li>51</li>\n",
       "\t<li>31</li>\n",
       "\t<li>47</li>\n",
       "\t<li>44</li>\n",
       "\t<li>61</li>\n",
       "\t<li>53</li>\n",
       "\t<li>56</li>\n",
       "\t<li>40</li>\n",
       "\t<li>41</li>\n",
       "\t<li>49</li>\n",
       "\t<li>48</li>\n",
       "\t<li>54</li>\n",
       "\t<li>49</li>\n",
       "\t<li>52</li>\n",
       "\t<li>48</li>\n",
       "\t<li>43</li>\n",
       "</ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item 7\n",
       "\\item 4\n",
       "\\item 4\n",
       "\\item 3\n",
       "\\item 9\n",
       "\\item 7\n",
       "\\item 1\n",
       "\\item 2\n",
       "\\item 4\n",
       "\\item 3\n",
       "\\item 5\n",
       "\\item 4\n",
       "\\item 3\n",
       "\\item 5\n",
       "\\item 5\n",
       "\\item 3\n",
       "\\item 7\n",
       "\\item 6\n",
       "\\item 4\n",
       "\\item 5\n",
       "\\item 2\n",
       "\\item 3\n",
       "\\item 6\n",
       "\\item 7\n",
       "\\item 8\n",
       "\\item 6\n",
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       "\\item 3\n",
       "\\item 3\n",
       "\\item 3\n",
       "\\item 20\n",
       "\\item 14\n",
       "\\item 16\n",
       "\\item 15\n",
       "\\item 18\n",
       "\\item 26\n",
       "\\item 14\n",
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       "\\item 16\n",
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       "\\item 28\n",
       "\\item 18\n",
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       "\\item 22\n",
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       "\\item 14\n",
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       "\\item 17\n",
       "\\item 25\n",
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       "\\item 21\n",
       "\\item 18\n",
       "\\item 13\n",
       "\\item 11\n",
       "\\item 20\n",
       "\\item 12\n",
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       "\\item 13\n",
       "\\item 15\n",
       "\\item 11\n",
       "\\item 14\n",
       "\\item 9\n",
       "\\item 9\n",
       "\\item 11\n",
       "\\item 9\n",
       "\\item 11\n",
       "\\item 15\n",
       "\\item 7\n",
       "\\item 9\n",
       "\\item 8\n",
       "\\item 9\n",
       "\\item 12\n",
       "\\item 16\n",
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       "\\item 22\n",
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       "\\item 17\n",
       "\\item 10\n",
       "\\item 6\n",
       "\\item 9\n",
       "\\item 17\n",
       "\\item 11\n",
       "\\item 10\n",
       "\\item 3\n",
       "\\item 2\n",
       "\\item 1\n",
       "\\item 4\n",
       "\\item 0\n",
       "\\item 2\n",
       "\\item 2\n",
       "\\item 0\n",
       "\\item 1\n",
       "\\item 0\n",
       "\\item 2\n",
       "\\item 1\n",
       "\\item 1\n",
       "\\item 0\n",
       "\\item 1\n",
       "\\item 0\n",
       "\\item 0\n",
       "\\item 4\n",
       "\\item 0\n",
       "\\item 1\n",
       "\\item 2\n",
       "\\item 1\n",
       "\\item 4\n",
       "\\item 2\n",
       "\\item 1\n",
       "\\item 0\n",
       "\\item 2\n",
       "\\item 0\n",
       "\\item 5\n",
       "\\item 3\n",
       "\\item 43\n",
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       "\\item 47\n",
       "\\item 39\n",
       "\\item 52\n",
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       "\\item 53\n",
       "\\item 50\n",
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       "\\item 54\n",
       "\\item 44\n",
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       "\\item 40\n",
       "\\item 53\n",
       "\\item 51\n",
       "\\item 31\n",
       "\\item 47\n",
       "\\item 44\n",
       "\\item 61\n",
       "\\item 53\n",
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       "\\item 40\n",
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       "\\item 49\n",
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       "\\item 54\n",
       "\\item 49\n",
       "\\item 52\n",
       "\\item 48\n",
       "\\item 43\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 7\n",
       "2. 4\n",
       "3. 4\n",
       "4. 3\n",
       "5. 9\n",
       "6. 7\n",
       "7. 1\n",
       "8. 2\n",
       "9. 4\n",
       "10. 3\n",
       "11. 5\n",
       "12. 4\n",
       "13. 3\n",
       "14. 5\n",
       "15. 5\n",
       "16. 3\n",
       "17. 7\n",
       "18. 6\n",
       "19. 4\n",
       "20. 5\n",
       "21. 2\n",
       "22. 3\n",
       "23. 6\n",
       "24. 7\n",
       "25. 8\n",
       "26. 6\n",
       "27. 8\n",
       "28. 3\n",
       "29. 3\n",
       "30. 3\n",
       "31. 20\n",
       "32. 14\n",
       "33. 16\n",
       "34. 15\n",
       "35. 18\n",
       "36. 26\n",
       "37. 14\n",
       "38. 20\n",
       "39. 16\n",
       "40. 18\n",
       "41. 17\n",
       "42. 19\n",
       "43. 28\n",
       "44. 18\n",
       "45. 11\n",
       "46. 20\n",
       "47. 10\n",
       "48. 17\n",
       "49. 22\n",
       "50. 17\n",
       "51. 14\n",
       "52. 22\n",
       "53. 17\n",
       "54. 25\n",
       "55. 23\n",
       "56. 21\n",
       "57. 18\n",
       "58. 13\n",
       "59. 11\n",
       "60. 20\n",
       "61. 12\n",
       "62. 13\n",
       "63. 13\n",
       "64. 15\n",
       "65. 11\n",
       "66. 14\n",
       "67. 9\n",
       "68. 9\n",
       "69. 11\n",
       "70. 9\n",
       "71. 11\n",
       "72. 15\n",
       "73. 7\n",
       "74. 9\n",
       "75. 8\n",
       "76. 9\n",
       "77. 12\n",
       "78. 16\n",
       "79. 12\n",
       "80. 22\n",
       "81. 10\n",
       "82. 11\n",
       "83. 14\n",
       "84. 17\n",
       "85. 10\n",
       "86. 6\n",
       "87. 9\n",
       "88. 17\n",
       "89. 11\n",
       "90. 10\n",
       "91. 3\n",
       "92. 2\n",
       "93. 1\n",
       "94. 4\n",
       "95. 0\n",
       "96. 2\n",
       "97. 2\n",
       "98. 0\n",
       "99. 1\n",
       "100. 0\n",
       "101. 2\n",
       "102. 1\n",
       "103. 1\n",
       "104. 0\n",
       "105. 1\n",
       "106. 0\n",
       "107. 0\n",
       "108. 4\n",
       "109. 0\n",
       "110. 1\n",
       "111. 2\n",
       "112. 1\n",
       "113. 4\n",
       "114. 2\n",
       "115. 1\n",
       "116. 0\n",
       "117. 2\n",
       "118. 0\n",
       "119. 5\n",
       "120. 3\n",
       "121. 43\n",
       "122. 42\n",
       "123. 47\n",
       "124. 39\n",
       "125. 52\n",
       "126. 50\n",
       "127. 53\n",
       "128. 50\n",
       "129. 49\n",
       "130. 54\n",
       "131. 44\n",
       "132. 42\n",
       "133. 40\n",
       "134. 53\n",
       "135. 51\n",
       "136. 31\n",
       "137. 47\n",
       "138. 44\n",
       "139. 61\n",
       "140. 53\n",
       "141. 56\n",
       "142. 40\n",
       "143. 41\n",
       "144. 49\n",
       "145. 48\n",
       "146. 54\n",
       "147. 49\n",
       "148. 52\n",
       "149. 48\n",
       "150. 43\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "  [1]  7  4  4  3  9  7  1  2  4  3  5  4  3  5  5  3  7  6  4  5  2  3  6  7  8\n",
       " [26]  6  8  3  3  3 20 14 16 15 18 26 14 20 16 18 17 19 28 18 11 20 10 17 22 17\n",
       " [51] 14 22 17 25 23 21 18 13 11 20 12 13 13 15 11 14  9  9 11  9 11 15  7  9  8\n",
       " [76]  9 12 16 12 22 10 11 14 17 10  6  9 17 11 10  3  2  1  4  0  2  2  0  1  0\n",
       "[101]  2  1  1  0  1  0  0  4  0  1  2  1  4  2  1  0  2  0  5  3 43 42 47 39 52\n",
       "[126] 50 53 50 49 54 44 42 40 53 51 31 47 44 61 53 56 40 41 49 48 54 49 52 48 43"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "y_prior = rpois(n = 150, lambda = rep(lambda_prior, each=30))\n",
    "\n",
    "y_prior"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Because these priors have high variance and are somewhat noninformative, they produce unrealistic predictive distributions. Still, enough data would overwhelm the prior, resulting in useful posterior distributions. Alternatively, we could tweak and simulate from these prior distributions until they adequately represent our prior beliefs. Yet another approach would be to re-parameterize the gamma prior, which we’ll demonstrate as we fit the model."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# JAGS Model"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "Warning message:\n",
      "\"package 'coda' was built under R version 3.4.4\"Warning message:\n",
      "\"package 'rjags' was built under R version 3.4.4\"Linked to JAGS 4.3.0\n",
      "Loaded modules: basemod,bugs\n"
     ]
    }
   ],
   "source": [
    "library(\"coda\")\n",
    "library(\"rjags\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<thead><tr><th scope=col>chips</th><th scope=col>location</th></tr></thead>\n",
       "<tbody>\n",
       "\t<tr><td>12</td><td>1 </td></tr>\n",
       "\t<tr><td>12</td><td>1 </td></tr>\n",
       "\t<tr><td> 6</td><td>1 </td></tr>\n",
       "\t<tr><td>13</td><td>1 </td></tr>\n",
       "\t<tr><td>12</td><td>1 </td></tr>\n",
       "\t<tr><td>12</td><td>1 </td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{r|ll}\n",
       " chips & location\\\\\n",
       "\\hline\n",
       "\t 12 & 1 \\\\\n",
       "\t 12 & 1 \\\\\n",
       "\t  6 & 1 \\\\\n",
       "\t 13 & 1 \\\\\n",
       "\t 12 & 1 \\\\\n",
       "\t 12 & 1 \\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "| chips | location |\n",
       "|---|---|\n",
       "| 12 | 1  |\n",
       "| 12 | 1  |\n",
       "|  6 | 1  |\n",
       "| 13 | 1  |\n",
       "| 12 | 1  |\n",
       "| 12 | 1  |\n",
       "\n"
      ],
      "text/plain": [
       "  chips location\n",
       "1 12    1       \n",
       "2 12    1       \n",
       "3  6    1       \n",
       "4 13    1       \n",
       "5 12    1       \n",
       "6 12    1       "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "head(dat)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<img src=\"hierarchy.PNG\"/>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "mod_str = \" model {\n",
    "    # Model's Likelihood Hierarchy\n",
    "    for(i in 1:length(chips)){\n",
    "        chips[i] ~ dpois(lam[location[i]])\n",
    "    }\n",
    "\n",
    "    for(j in 1:max(location)){\n",
    "        lam[j] ~ dgamma(alpha, beta)\n",
    "    }\n",
    "    \n",
    "    # Priors (in form of mu and sig)\n",
    "\n",
    "    alpha = mu^2 / sig^2\n",
    "    beta = mu / sig^2\n",
    "\n",
    "    mu ~ dgamma(2.0, 1.0/5.0)\n",
    "    sig ~ dexp(1.0)\n",
    "} \""
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "set.seed(113)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "data_jags = as.list(dat)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "params = c(\"lam\", \"mu\", \"sig\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Compiling model graph\n",
      "   Resolving undeclared variables\n",
      "   Allocating nodes\n",
      "Graph information:\n",
      "   Observed stochastic nodes: 150\n",
      "   Unobserved stochastic nodes: 7\n",
      "   Total graph size: 315\n",
      "\n",
      "Initializing model\n",
      "\n"
     ]
    }
   ],
   "source": [
    "mod = jags.model(textConnection(mod_str), \n",
    "                 data = data_jags, \n",
    "                 n.chains = 3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [],
   "source": [
    "update(mod, 1e3)\n",
    "\n",
    "mod_sim = coda.samples(model = mod, \n",
    "                       variable.names = params, \n",
    "                       n.iter = 1e4)\n",
    "\n",
    "mod_csim = as.mcmc(do.call(rbind, mod_sim))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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B6BeaPrCNyUETgreI7AJa95i8YrH61c\nRFOX0BqmFoGpxECu74YgcNkdFH/XWi7ciwtcZuwsEYUmtr/GCFyiX9pOUX/LBgy9AQQuVt9K\nBJ7a/hIClzjHD1CxwEWjvbTAlP15tHHpFxZVmFejwCXO8QMUDQ+Bd6DAPnot7JnGhRF4oS6z\n1dqVigVOp7dstNcTOH7Zjx9yDGOUCRQ3KHOON1J4vCz9Q46CwTRimZCmBAYAnAYEBsAwEBgA\nw0BgAAwDgQEwDAQGwDAQGADDQGAADAOBATAMBAbAMBAYAMNAYAAMA4EBMAwEBsAwEBgAw0Bg\nAAwDgQEwDAQGwDAQGADDQGAADAOBATAMBAbAMK8h8OJRvsYpqJRswuvJVWun5V34l4xp/Hx5\n2tcsUjebbzar73K1D6aGOuwPzR3oaxx+xdDsbOrb/mHkex3BaNH8vPDjmfSzTWK30v09mRrq\nsD8QuFZozqSN/77yEQJni6iryMnUUIf90X91NP5tSJ+mf0Q0rYjXbWfW9pXIBab+dfCDAg9T\nme1Cn1FsD5QG0XzXkwIPaoVL6GPJelyS/pN6S/sr5jtnUJrujE+t2iBwFHCUShoInDeAyV1D\n4PoYCDxaOn4Ax1BM4OzvVEaps7cZtoeJ/Q4FHl4WQOBj6S6UU6fbPZ1IKi6hj6I3oJW6hN4u\n8L1LaMp3A4EPp39dPBiMx0ltXuW8nE8u8OSqbv06gWcS+9wI3Pc3uV5DI6mhDvvTT1h8mE/3\nq5yX81El5lY9JPA4sfN99iaBqV+mikZSQx32R7PadJ9z6Gcbw0+h81fgABZMSMPd3EOvcDcs\nUs/PQWbToszy0Y6zbbIW0wwvs6uQp4Y6AAAeBAIDYBgIDIBhIDAAhoHAABgGAgNgGAgMgGEg\nMACGgcAAGAYCA2AYCAyAYSAwAIaBwAAYBgIDYBgIDIBhIDAAhoHAABgGAgNgGAgMgGEgMACG\ngcAAGAYCA2AYCAyAYSAwAIaBwAAYBgIDYBgIDIBhIDAAhoHAABgGAgNgGAgMgGEgMACGgcAA\nGAYCA2AYCAyAYSAwAIaBwAAYBgIDYBgIDIBhIDAAhoHAABgGAgNgGAgMgGEgMACGgcAAGAYC\nA2AYCAyAYa4iMCW2bvn1i+i3hFiM3vx7J/r1pS/AQUhSf33eL9c0/37Pr79qmi3UcQ2PC3xr\nt3mXEIvR/0n4TyuZvQgxrR93yy1m8LJptlDHtTx2wtNWi5ltmvfQhv7S28M7Ao8g5/p/rVlr\ny95Zd7E0W6jjWvSEt93nrT3/f9s+9/270YsnfvJGb1+x8Pe7rO5GbX6IWxG1pX+HTX5/y6rf\nlAqZyOxF0HP9l4fPmMFW53cdlD9udPuQcjJ23rj8TVN0+TRbqONaUobCu52/JI+NXBO9xyfa\nkf/cONk/g8ymrfStV3qlu/hHvxojmb0Ieq5/gpgpg3pZ/R580+vrKPAHhbfLn3rJff00W6jj\nWlKG/jQ/bRf81XzHa6J/8uRf++Rdyn6EJ++a+W7ztBXJVh/NV+/N1S9uHiYyexHiue5lsHX3\nJ1gYnv10KeIPsvjq90076uun2UId15IE/uHHrz+/elm5ZVdY7Ysf7dYHb47yrThSFuP7TTp2\nE5m9CLnAKYOS4rCovab+8xnXc9k3tvyXbHX9NFuo41ryNznN102vmrLMZZ9SU3qr08tsf6v0\nR8p83/TCzERmL0J+CZ0y2GWvla1V+jNLU3i3/E6f+dZXTrOFOq6lJ/Ab/fn3M8pcYqZr7m/V\nz2yb2D8TkcCu6Ln+lCvn3kL17e97TCMv+AlfBel11guk2UId19ITmOg7fPkg74C+whuj9CQw\n8+YobTWR2Rv9r78jcAByrj9v4U1tymA+sKZ3s3Hxe/at8fXTbKGOa+kJzB9Phg5YPrr8Gz/D\n1O+RvuPHk/3Mpq3Gmf3oLsFNZPYi5D/kSBnsBH7PPoUO75FbX78o+ClcP80W6riWnsBtam8f\n3yG1X290+xuWhCexd01fEA6uy+JW48zejGX2IugXPfKeNmYwu4TOvgcO43T4teRb/AireYE0\nW6jj+QxSaSKzL8vPje7+cnoai2m2UMfz6T69Hr4AtdF9VfjIpubSbKGO52Mxs6/KjX593y81\nicU0W6gjAGAGCAyAYSAwAIaBwAAYBgIDYBgIDIBhIDAAhoHAABgGAgNgGAgMgGEgMACGgcAA\nGAYCA2AYCAyAYSAwAIaBwAAYBgIDYJh1AlM30xcAoB5WOUmjJ+BShL4ZHbRJIDDgtFJ8AkwB\ngQEENgwEBhDYME9/iEXgTIo1gjmBzz7AF2dV7p5OPjiLQmef5j/EQn7PBAI/iz95/3dO7/5n\n30R+tw9cRigtcF76iHOVB99RpSz08HBm9zp33O756qzZzbrV+1fgfCbb4FVEtj4C55+ebRI4\nbegbur/hQGA/vWpmDwO2CLyi7LECH91BP8+dCho4gkVOEvhew5ywwg02JBdCDwWerS0Ndknd\nVp3As9Xy/F/6GGcosMueD/awFPH+ulShPFi/lhiBl1glp2WHSwm88UOOkSmZmOHPaoF9XCoq\nZjsb7HZSYN8GmBJ4VL05gWlCYNetG+0wrJb1fnCI04O6m1rtemtCXJcOYXTiXlrg9V6aVbiQ\nwJNfM8xcYvm+ANoEs1fe9dshN/m4Lu2jjesoKNhu68LT0I6JtXZhGcvsuuIu1UNXO7YoCUxZ\nrfLOwmmdg8B6mN7HOpMbmOe8Horq3uthuqNpyw2U867bnx97OBY4xksCc8+yl8ALl8u1tvyt\nTtpUeE+BZwJ4bq26TSekTxarwGlUceE1C8jO6rBHwVrqBCbnewKzo05auguiy27EsCSwE4Gl\nM+DFjQTwSWCRty2ZBNYBXwUmkU+McuKhU4G991xb3oJLeXG+J7Dj+GGFix2GnCCW3WvvwYfk\nvHQd7ZE7353evQWm0ZPSeyjNI0OqxWH4eIFJBQ72NUFACq2ubfHeOedEJIoCyzrPfjoZ4ZwK\n7MImFGQPq3wUWHbIHolU/J/nlexzJ3DQVwQOOrDAsgHpcBxKsMBhB+2LEL3dZVhEvtFd+7Dj\ndqMgYLthWBAC8fU8scDhdaiGCBxieVZbzocPnQd3Dy5dXju203GFpQdxvANigWN35Z12d2kE\nJgjMPG6iOYWPF7iVgnVp22MSOOiVBHZBTefYNB6oeAkFudrFXgei4Ei7ljdyTg1nixpeEAKF\nMqKCU4EpLAhLvBjAIx85jeXDPkIh7lxka9kVj5GtwMFIdppkk+CjCMwV4VC640aKhyt9L367\nRg6s4Vp7qRcXCyekXUd85I0cWhNllxMR+g/n+ZLDyWFx5XTU1ssDCBx4biA1Ngwf/yEWC+xF\nYE/cLLnVi4zciqWdOxErPnix2nsRmJeLQSpQI9uQKi09Ajd0iaArJFr2XywnIzrJPimtdbK5\nFNBassBOtuSuh7So1turxCKa7JRi7xSci0v0BXmtH3cvsU/iDkUiNHqgxFcMsWINyQnzWj+v\n5m9P3930LoSrrL0XENCSwqUEntxy8oMPadA8rPGoR+pUbOg+SptcS206LqK0JK6grjhlprrY\n7HOBo/TaL3iVxneLYqfh47VAHk8vFPq70GfaB3nfLekXo3yL7jB4S9+vnKocuwAfj8THDsV1\nNe3q7txgEN7xQ6wKvwcuVZu6jmqJPQWeDpDLGMeNKSYXx1GxZ1MfGgg3Fc4nP6Z26Ce26NZl\nIo7X9jb1UbupMK7X66Tq+Nx5DZl1AoP9Dmqquy2bvq35PZGi2hlR+HiBp5v+BmaEH62/V27b\n/rZGW1F+Q+Bo6prCrypw8VHTxDBcj8CrDZkquLBxKY+nwhXc7dwwvXZxNkyXTd/W/J7ELrYZ\nUPj498DPtnRwj0LpW53m3fdwn/1Mq93hekZgUIqy6dua3xPY+Ya3qhWGwPXw2LXInTkaXkDg\n3StQs8EQ+HqUTd9CYqv4GumQ/Z99kPNA4OtRNn1b83ssh5l1ekc1w/kfYoHSFErf6jTvvof5\nXR+67yodPmkE3uuj5/JxZ39oUp6NO5otXjZ9W/N7GCf4VKHCFV5Cz/4Ea2M7LrPB4ve+a78U\nfpBVv9sYUzZ9W/N7DKepdP6b/j51CNz9Jmn8C8e8RHreK5b/inrwe8j+Zj4vPcvop4yjRd2S\nQbXiryNX/BhzeqH+1ToS3ekzpimbvq35PYCzHapJ4hPfA/cUi/fyeL1jJ9zS0/3yv2v0lO44\nyH/A71J7j/cuzbV53Yvc/zMuwCt7Pzge3DoQf9/sslsM4u5TsN4tC4Od9PuPkZ7aX8hO+W6j\nwRno7lfy2aMs1SKF0rcilyX3sKUmx+5xkkqqUfJ+4JW3E3KTa/guIL4n3cldsSIVC9yoBCwr\neb0PPolBncA9W73e5Z6PtHJ/LkUreGINuT+viXchZQO3l9uMyXWdg9cJN7RgiCT143Byw3Bf\nYDe41cG73r1V3jfRYpK7seJzqVm8OZlLNvHOK+n25BZM1/S152pJGenVtqdvXX6H4Y7/GqkW\na5QaqlNQYJopPRQ43vzqZUIbvXGW56aR2aZCi2xPDk+YQ3z/elLYseKOz5velSszWfCNsWJo\no3cKs4uhPfNEADp9Tugfwi20YVfiVSMq6swYPL2NUyFIxmO5t55vX04CaxfDH4Q6uReZtDPS\nKTykUg357sZm7/XoRVtS81RZvpU3zAEQbmWOteUb/V0TexA+/CZdw8cpgXiRTARAxwlcdA9r\nKnG+LiNOd/hEgXm+mTB7TRBK5pjk2XB4jgrZjqeG8jqxTBxcPOnUVTqblDrOUwrwdHA84Uwc\n2fWn/U6DNY1OccPPnJjL82+owDyZDjUyCWXoBtjuRmb04a6ApZSJOziK3pjPh8AT+MiEHzIv\nV8OzbfHgz/NlcJ15XG90gisdW2XWLZ7rSufb0ekuXTqBPC7LhFja58m8fHwQMiGmHzWnawhc\nwQ9G5jm1cscL3LBecSpUnuFJm2oncNxOBI4bqoG5wPkElzJ5jU8q6Wx4HfF5nC1PZqmKNUyz\n2Gk14sRY0rnElb7RWeXEUVnh4lRdUf1uZVwWJ5HsT7cpRxqslDm2tFJzk0aPFkeBu5NxQYGr\nljdxVi1LCUw0FvjOeyRdHqdTHe1stF2cBa43bXPfiOy/ZiTw1DbTU7nTYNrWrqCP3UtXQGrk\n+pWmmei9+TKcz8XILhZm8Dok5wt0itsk8MShlOCcD7Fq+KnmBs74bWm5T6G54lsSfC/wuYnr\nBB7NKD8o0KO/8CGBl6s1ErgXfTeBF9hjD1X8yvpRpu8q2WtnBUrsHeBk5kfECe4d7MKMVasE\nvhPduMAzd1RZZeEmsXL7KFBi7wCW2HiwPYGf3/tEiMJnf+It0n93oPQnPbSP/D+Sv//9Ry/C\nf/FvPHp+9h9R/0RR71QtsyllDyX4pQSuDlMjMNjMESMwOJNn04f81s39/JTOd0VR6gpTVWWG\nQdc2l4d3YC6wlchbgq0pe8XGfsVjmo5pzzMjmu0Xee33wMeO6HU19ise03RMe54Z0Wy/yOuC\n0ehnHM/FOyZKXWGqqsxMTHueGdFsv8jrf8gBgWuIAoEPCmwl8vpg6z7juGJjv+IxDYLiQyyr\nkfEh1lFhqqrM0UDgvSLja6SjwlRVmaOBwHtFttkeAAAMBAbAMBAYAMNAYAAMA4EBMAwEBsAw\nEBgAw0BgAAwDgQEwDAQGwDAFBNZfwU8+bK7Ls1EKVaZEGCoTqFCYE9ntPondTsCOt3aUDvx8\nML1VePJhWyCaCXZ8ZUqEif+AzJOBCoU5k73quO8J2Mnf0qGL/cS+IoGr6Aek/NOBqCsIgQ+K\nu2f0CwvMG11H4KaMwFlBqwIbGyP3DQ6B7wYpMZDrOxUIXIL93gLv+U+S7nbZUN17YI1SYNAr\nEYUmtr/GCFyiXzqF3erZfTywQ+x9wlY6As821W0xSl1CX1LgEuf4RHYReK/A+4WtU2DK/jza\nuPSD+yrMq1HgEuf4TIwJvO+VeWUC91rYM40LI/BCXWarVT37XULvFfi1BI5feuOHHMMYZQLF\nDcqc4zMw86OILPI+YZvydTbRAgAA00BgAAwDgQEwDAQGwDAQGADDQGAADAOBATAMBAbAMBAY\nAMNAYAAMA4EBMAwEBsAwEBgAw0BgAAwDgQEwDAQGwDAQGADDQGAADAOBATAMBAbAMBAYAMNA\nYAAM8xoCLx7la5yCusnmvZ5ctXZ23oV/2JfGz2m8ZhCr/7TG6Xzrqcme0NyBvsbh1w/NTni+\n7d9HvtcRjB0zfaUAACAASURBVBbR9Jpun9nTKifUr6YiuwKBK4fmTNr4zyzvL/B8tHOopiK7\nEv8hSur+9cL4Lx2kf/pAizXpFTiOXGDqXwc/KPAwo9ku9BnFZhEL9XcNgSsi62pJrsqot7S/\nYr5XBjvRnfipVRsEjgKOMkoDgfN2MLnrKYGzd8VPHGxRqqnIrgwEHi0dP4BDKSZw9ncqsdTZ\n2wybxcR+BwJTM9hDBdRTkz3pLpRTb9s9ncgmLqEPpruyLXgJvV3gFZfQU3s9k4qqsiP96+LB\nYDzOZvMq56UacoEnVzWZQWsEnsnvcyNwM1v4PGqqy370MxUf5vP8KuelGvSiaG7VQwKP8zvf\ndW8SuKq2UVVldkPT2XQfcOiHGsNPofNX4DgyUSZXbf8hB1HPz0GC06LM8tGOs23yHS78WOQE\nqqkIAGA7EBgAw0BgAAwDgQEwDAQGwDAQGADDQGAADAOBATAMBAbAMBAYAMNAYAAMA4EBMAwE\nBsAwEBgAw0BgAAwDgQEwDAQGwDAQGADDQGAADAOBATAMBAbAMBAYAMNAYAAMA4EBMAwEBsAw\nEBgAw0BgAAwDgQEwDAQGwDAQGADDQGAADAOBATAMBAbAMBAYAMNAYAAMA4EBMAwEBsAwEBgA\nw0BgAAwDgQEwDAQGwDAQGADDQGAADAOBATAMBAbAMBAYAMNAYAAMcxWBKbF1y69fRL8lxGL0\n5rMt+PZXX4CDkKT++rxfrmn+/Z5ff9U0W6jjGh4X+NZu8y4hFqP/T8L/sZLZixDT+nG33GIG\nL5tmC3Vcy2MnPG21mNmm+U3/a/t4LmQisxdBznWr1b+1Ze+su1iaLdRxLXrCiT5vb03zt+1z\n378bvXjiJ2/09hULf7/L6m7U5oe4FVFb+nfY5Pd3lsr/0VtjJLMXQc/1Xx4+YwZbnd91UP64\n0e1DysnYeePyN03R5dNsoY5rSRkK73b+kjyG3lSunuSJduQ/N072zyCzaSt965VecYnf7Rbf\njZHMXgQ91z9BzJRBvax+55TI9XUU+IPC2+VPveS+fpot1HEtKUN/mp+2C/5qvsOS9zaX/+TJ\nv/bJu5T9CE/eNfPd5mkrkq0+mq/szVXbGm7/GiOZvQjxXPcy2Lr7EywMz366FPEHWTx4vmlH\nff00W6jjWpLAP/z49edX75ONW3aF1b740W598OYo34oj9WJ80K/GSGYvQi5wyqCkOCxqr6n/\nfMb1XPaNLf8lW10/zRbquJb8TU7zddOrpixz2afU8jhYP9wq/elSaefTjYuQX0KnDHbZ+34L\nSn9maQrvlt/pM9/6ymm2UMe19AR+oz//fkaZS8x0zf2tepn9fftpLGX2Iui5/pQr595C+fv9\n9z2mkRe06fsXr7NeIM0W6riWnsBE3+HLB3kH9BXeGKUngZk3R2mrcWbfwxa8mY3MXgQ515+3\n8KY2ZTAfWNO72bj4PfvW+PpptlDHtfQE5o8nQwcsH13+jZ9h6vdI3/HjyX5m01bjzOoWZj6e\nvAj5DzlSBjuB37NPocN75Fa8Lwp+CtdPs4U6rqUncJva28d3SO3XG93+hiXhyf9i4fgF4eC6\nLG418eYobWEjsxdBv+iR97Qxg9kldPY9cBinw68l3+JHWM0LpNlCHc9nkEoTmX1Zfm5095fT\n01hMs4U6nk/36fXwBaiN7qvCRzY1l2YLdTwfi5l9VW706/t+qUksptlCHQEAM0BgAAwDgQEw\nDAQGwDAQGADDQGAADAOBATAMBAbAMBAYAMNAYAAMA4EBMAwEBsAwEBgAw0BgAAwDgQEwDAQG\nwDAQGADDQGAADLNO4Di9CHQHoCpWKZnkhcAAVMV6gcNfCAxAVWwQuH2AwABUxRaBJ2e6JnAm\nBZvCTO7BmdzPz7oszpfGoHwmBwi8+x7APKUEntxybR9xOSo6ZAg8uEY4uzKF2VPgUgEKxzmA\n7VXd7eBeXuCBs1ez+IICV5CdTVW4//353XC+UFUeooLzPcucq9eRuJDAS1colQq80OifZviN\nuFvaXU9gN1ti9DQDAk+ybOk1JC79IdbDAVbsYqVyzwpMw9V+qvDdIbOIwLyVh8APsEpP+w4X\nu4SeLzZYM9/YvJTuFcjbrh+W7ZGa/qgmNCgwFSHbmoY7U4H7l7qk20y6OfGbND8o5bsvz31e\n2mfVlHNBA4H9oHNZ5iUF3jC4Wh+Hj38P7N1cRG6Z5EKbdHFT6tozxTbfPjppuLmTY4F9f4Hr\nl2xNyeTbInB8kgTuNwEvX4dnNW8L+Uxgnx/MQGDXE5iWBfZ5mJlM7dg6a/1cd3OVKjyG1Rz/\nNVIQOC1KtvomFzi2YnJBA3bFjwRunXCjgU0qTOyViprJ0WTBc4FJBaZwucrbk+8unH3f28Fd\nG85Hgbk4V0kE1p6KegJTT2DSpz4WZYFjFxcE9lFg5xrZt0+npz2Vvhl0LMN8vdwI/JiMZhU+\n4VPottk5bpNtw+cnLDCxCtyGXRpkfGjP4dyGliqNlUR6r04EiaVE53xo+CowsV9O43ou5cUt\n2aBdywt8vH5nl3k3TYgSXpP0OFIPigL7UMKFqCKwp1Ch4FRYECK55HWobOiLxD3nZWwlvkT2\nEkRd5AMORXmPbck2lg8H35ZxIXoILAKHk6Pe8hN+4YYX1a8l8BNjqdFh+HiBXWitYaBrLQqt\nP7RjkaD9266QhupZ67DeR4GdjM/BQxZbR2A2qAkPwXXfCcxWqMBhmQrMKoUV7V/eAXckjqUj\nHZdJSoWgwRwZ0In15D3Ihu0LJ25yDYK5wU3PsbxzchDcNXjZq5d6hmORHsnznrwI7HjEDa9C\nh8LXCvw6GDsQmPsWF2rGfV1YR9x1uOFb/VcS+EkFTSp8hsCeLY0C8wjkQ4N10pbFcC8CB4W4\nCXtt2Y63cV4EdSKwOsCe8KgVw0ivEPYWdheeNhJFCjgVWHoO4v4huMEhpRCJPVyKHeOqy54l\nNO+TuOvhQZE3koo2HIoH+Lg7crqp9CPkYuW5W/NOTkHDf/mkyMng0lz32LeI9U6rGgR2sW8q\nmr6t+T2NAvoZVPiMD7HEniiw04EwOhe9ctqWVTfv81WyRkPxKh3zeiVdiiHiZHvw6YVXvYj/\nJwLG9bGCXErKJYGd67SMQeKuyXUCZ5XVv006vq768khdzX1e+axbo+6VWuxdd9Zkr0XTtzW/\nJ1HKPWsKn3IJnTVS3zXWnsB9/HBNGvsyN9ZtmTf10R5pXDp/GV2J42Svp8g6Bu/jCHkPP/g7\n3GJcmUGpfC/xXJRN39b8nkLJobOG41nPSQKPm3ZP4AWJZja/w7LDw1f+zg4m19NUuaX9j/da\niLLp25rfEyg8apoahI//Gmmu2RVrv/szf5lw8C6nKZS+B/N7ODtU4PRjWs+pl9BgF8qmb2t+\nj2Wv/uP8fmklEPh6lE3f1vweyZ6aGVEYAp+OL03Z9G3N73HsrZgJhSHw9Sibvq35PYpD7Kpf\nYQh8Pcqmb2t+j+Ews2ofhiHw9Sibvq35PYCDpara4dO/Rpr/Nnbq9aNs++Jny16frOHqzdfv\n5+rvgc/wqV6HTx+B7yo7+omWn/rN0nI/sNzgV22zIqRfjOTn9jT7o5a1FRj+Gq1s+rbmd+ed\nnWVSBV94T3LS3UiDn++uaK7Z7471Z8fpVf+n0vMCDH8F3RfAp18XL4ToNh9a2P0s1A9/1502\nnO6cxt3RlOHpV9Z9X7uz0i0umz4NM3E1dfwPOU6WqEaHTxCYvM9vz+naNt/y1/38P7vpJmvr\nLCwHSLfxhFsAGrkfqGvP1FeG9JXXG33Sz4pTfLlVSW4jatIu8xuX9G4fr7cdqKXhdkMvNzrJ\nbnx0SX+q7alnuHPpN9x6EL199A84exlup9B7GfJOJhy8nqf4WDZ9/SgT4V7q46QqKpFzisBh\nkomgSrhNPdzoS6yBCy9cvG9O74tXGcQ6vimf+LvOsB3fQK8jenjhpWuQ1t3EGx1UOm7ooUyj\nt/rxTYrqNQfjW/V4jgG+bZe7Btd03Qvrxvc+xprydBm8rzCDh+jPFXHxLkA+0EY2dnxzcSPV\niFaHgF1fE28AjpJzPOd12oCgMFvsXLzhkJz2JHxfYriL8aICVyRORVVpygnMt8NPH9pwGc9W\nw3fZ8tQ1TmamcHLzHU964XSclrKh8YY74NkT8VYEbvh+9oZkHescDJHJAnjGDR6U5G136Cv4\n7v5QhXinPQvM9/Y3cq8tycQ7jU76QTLphlTWyX37XFpuv+eD4zuK+UkTJyfQyXVIFOcpR1Rw\nnimLj6GRXoYPUmcwCLrrPb9SlpokcMMTDkjPxiekcXGSHbkJUuYk4BP4QPrupnch3AHvsqtS\npqnqDXEhgSkWXJNgmU6GWx8LLDPYxCniukLU6NxyXieG8tmCJgms02WlnyDpudVZrmTCLZ1V\nw8XJp6Tt8551JppGJ+zRGXB4L+kA5HJB9dc5O/K5L3x2DpLATQpEjfYZqT5cMM4FxkFlYixe\nLdOA8WwflE+RIwI3OvdQttc4mw7RwN7phDzAaQLXcLfENJXUbE+BZz/kyOZ4lOE4rRkJLOV5\n/qlsKlbf9CdUpd6Crkyvfi5b2V1nyvRXspC6XQ4iEA3M6AKOjy4XeAiNqpaV1Nn+YnXkoPNt\ndT47yurXCTydqh0/xCq7h5l97hW8DOffkHXGCJwvXZqtPVepr8LkRgOB7wbtXvcEntxLvnzG\n28UqrVq8iuHeh58wDMtb/BrpdCu2cqLHZwu8jmGLf8aAxXrMCZyzQeDjMSwwdRSKeDR0wiGc\n8CFWXdRdu+cpfHwTb5H+W4SGT8PfsFXYMPyVGHRh+DDDccsL6p8Qkv8ovRqe0DsneP8WcHVF\n6sbOCAwe4YjvgcGZPJs+5Ldu7uendL4rilJXmKoqMwy6trlsj4ww+8aBwEeFqaoyMzF3+MgZ\nYfaNA4GPClNVZWZiQuAjw0BgU2GqqsxMTAh8ZBgIbCpMVZWZiQmBjwwDgU2Fqaoyw6D4EOuE\nMBDYVJiqKnMYdR11XWEgsKkwVVXmMOo66rrCVCkwAOBAIDAAhoHAABgGAgNgGAgMgGEgMACG\ngcAAGAYCA2AYCAyAYSAwAIYpILD+Cn7yYXNdno1SqDIlwlCZQIXCnEGhCpYJk5/HZ8MU8KZQ\ndQoILDWZftgWiGaCHV+ZEmH032Z5NlChMGdQqIJlwuTn8dkw6eH06hT8eXdFAlfRD0j5pwNl\ns9FD4KeCFDGmf/KfjvNsdZ7fvAvzdOPq/uWTSwjclBE4KwiBnw5TIBT1Hp6OA4FHQUoM5Prm\nBAI/R7k3rwXiQODlKAUGvRJRaGL7a4zAJfqlYylVwdANXFbgZwMVaQCzTXVbjDLdwEUFLnGO\nD6ZkBa8qcLmR/NkYTzYunZOpCvNqFLjEOT4aCHw3TpnDKhKiQOPCCLxQl9lq1UuhCqbzWChM\nRQIXs++pCDqfIX7IMYxRJlDcoMw5PpRyP+QoEUVDPRmrpMAlpgKtuwUAABaBwAAYBgIDYBgI\nDIBhIDAAhoHAABgGAgNgGAgMgGEgMACGgcAAGAYCA2AYCAyAYSAwAIaBwAAYBgIDYBgIDIBh\nIDAAhoHAABgGAgNgGAgMgGEgMACGgcAAGOY1BF48ytc4BZWyML/5pml5F2ZopfHz5blhs0jd\nbL61zupbQRUOgOYO9DUOv2JodoLzbf8w8r2OYLRofrr48Uz62SbVzat/fg2OAALXCs2ZtPHf\nVz5C4Kxe/XVncn4NjkD/1dH4tyF9mv4R0bQiXredWdtXIheY+tfBDwo8TGW2C31GsT3EQv1d\nTwqc1QoCH0/W45JcnFFvaX/FfOcMStOd8alVGwSOAo5SSQOB8wYwuesFgZP22lNU0EoqqMIB\nDAQeLR0/gGMoJnD2dyqj1NnbDNvDxH6HAlO2tv9vUJ/N+TU4gu5COXW63dOJpNbQtb4GmQzl\nLqG3C3zvEron8HwtTuD8GhxB/7p4MBiPk9q8ynk5n1zgyVXN2JlFgWcS+9wITPlaCHw8/YTF\nh/l0v8p5OR+9Gppb9ZDA48TO99lbBL5fixM4vwZHoFltus859LON4afQ+StwAJkak6u2/5CD\nqOfnILNpUWb5aMfZNlmLWVGLE6igCgCAR4HAABgGAgNgGAgMgGEgMACGgcAAGAYCA2AYCAyA\nYSAwAIaBwAAYBgIDYBgIDIBhIDAAhoHAABgGAgNgGAgMgGEgMACGgcAAGAYCA2AYCAyAYSAw\nAIaBwAAYBgIDYBgIDIBhIDAAhoHAABgGAgNgGAgMgGEgMACGgcAAGAYCA2AYCAyAYSAwAIaB\nwAAYBgIDYBgIDIBhIDAAhoHAABgGAgNgGAgMgGEgMACGgcAAGAYCA2AYCAyAYSAwAIaBwAAY\nBgIDYJirCEyJrVt+/SL6LSEWo4fHnxt1L8ARSFJ/fd4v1zT/fs+vv2qaLdRxDY8LfGu3eZcQ\ni9HD42+yk9mLENP6cbfcYgYvm2YLdVzLYyc8bbWYWX741JyayOxFkHP9P6J/a8veWXexNFuo\n41r0hBN93t6a5m/b575/N3rxxE/e6O0rFv5+l9XdqM0PcSuitvTvsMnv7xT5djOU2Yug5/ov\nD58xg63O7zoof9zo9iHlZKi+cfmbpujyabZQx7WkDIV3O39JHpt//OQ9PtGO/OfGyf4ZZDZt\npW+90qtQ4J2+DGX2Iui5/glipgzqZfU7X+3K9XUU+IPC2+VPveS+fpot1HEtKUN/mp+2C/5q\nvsOS9zaX/+TJv/bJu5T9CE/eNfPd5mkrkq0+mq/05uqr3cRQZi9CPNe9DLbu/gQLw7OfLkX8\nQRa1V1/Nm3bU10+zhTquJQn8w49ff371Ptm4ZVdY7Ysf7dYHb47yrThSenVLhU1k9iLkAqcM\nSorDovaa+s9nXM9l39jyX7LV9dNsoY5ryd/kNF83vWrKMpd9St2lqJfZ/lbpj3T/X6YyexHy\nS+iUwS4N329B6c8sleHd8jt95ltfOc0W6riWnsBv9OffzyhziZmuub9VL7PZ11QmMnsR0ifD\n76NUyt/vv+8xjbygTd+/eJ31Amm2UMe19AQm+g5fPsg7oK/wxig9Ccy8OUpb2c/sRZBz/XkL\nb2pTBvOBNb2b7T6E6r41vn6aLdRxLT2B+ePJ0AHLR5d/42eY+j3Sd/x4sp/ZtNU4s1lsE5m9\nCFGooGTKYJeG9+xT6PAe+T18ChX8FK6fZgt1XEtP4Da1t4/vkNqvN7r9DUvCk//FwvELwkHW\n4lb2M3sR9IseeU8bM5ilIfseOIzT4deSb/EjrOYF0myhjuczSKWJzL4sPze6+8vpaSym2UId\nz6f79Hr4AtRG91XhI5uaS7OFOp6Pxcy+Kjf69X2/1CQW02yhjgCAGSAwAIaBwAAYBgIDYBgI\nDIBhIDAAhoHAABgGAgNgGAgMgGEgMACGgcAAGAYCA2AYCAyAYSAwAIaBwAAYBgIDYBgIDIBh\nIDAAhoHAABgGAgNgGAgMgGEgMACGeVpgAmdSog0gv/VyPz9PJ/jZAOAJDhB49z2AeSBwaRYO\n94wzAYGb4UXC2bUpinGB/Q4xn6wvBK6KCWUvZfGeAh/Q4z0i8L36QODa9vAoS43vKhKfMQK3\nC9yzQRW/8GqO2QPyd9bfj3BnJQQ+jhWCXsLhUgIvjLZrBHaLL2dj+cWXWaR8u8kDcs2EwBMb\nz0e4s9ItbrfHWwHhNQVebaZ5hQsJTKMnCwF0Qd5oVRVPvZfdll5fbRTYt5FoWIe1ArtUq+y/\n6Qj5jilf0u8M+J+IHu+3uXdFMrl2VJvpGC8o8LaB1fgwfJLAzmVOdM0/E7g/Ym0VmHxvs6HA\nfrJeMwL7SYHjWhdW+1jrZiRw3MhpU5kVeMFhN1VdCDzDdiEtK3yCwK4nMDd/0gbsO0lV4Gy0\nnhA411xHW2nXLHDrlpcR2FGsCUWBqU2b6wca11f6l77AJPl2qrAI7OTA2ogkPmv5VmCiqFoU\nuLNa6pHr3ztxxAc1fRp91jnxmfGHC1zl1zKPVaeyg9jAKQJ7vWIl3xPYJ4HDS8+nNUjYiA+h\nAHUjD4flUjK8tUHbNhwWyKu2kxCB286iFTg8eDaZ1LBWN58yp5fQ4f9EXB/vnMjpU4W512En\nWV6nArf1dlxBR45kQGaR+QhY4HAg3N7DGkdynFyGa8ZrHe88jauOK6KdhGtywbnKUWCXCez0\nVG5O31PU1PYfF9Gqwsd/iBVatAs2uSSwY4HDC+941OGXoZ179sNxE3dhLXEbDXYE/doNSH3j\noqFQa2r71IWdhEc2l8u4sI54Z7rTsCRUmY1kCcNGPpRLAjt9EEXalWxWOID2P+4npNOgUNn2\ngNoasXbto+eloZ8QgcMuQ1RqpF7h8JPAfEZ4c6kK140PoOGdNdrHhZLcpXnugkR032h0T68s\n8HMS2lS4lMCTW05KLW656IpnQ/gpLyMnkrDA8aW0+1iWZGyVVi7e8OVrlI/jtw9BHcd/wisR\n2MvmUeBQSErH4PrXy271NVeYa+nTQh5wXVpPoefQPoE7mGAiL1WBJThvT9y1aN+k9dCz4H13\nAhjdgx5JOFDH+gfJ9cBUdBHYu77BLyNwgWt5iwrvKfB0ANfDp/Y58CfTIyozQJq/eONTWZeV\npa6sGOJFVpfJGv2J+2ikG5BQpFXxWXDfC5n8TuvSJtStIh6vvV4UpE6gq2p8mR8p69s7I1K9\n7oz4eCwaVQ+wbPq25vccCrlnT+ETBe5Z6fvP86Y9Lufz54MCUyTnslF10p/xnqh7SdRfn3cP\nWQcU/8uRq4BczlEdabTEez8omHU9/RqnoZr/lk3f1vyeQUHvrClc6kOshV+KL47Aw0bbb93T\nQ2/fgunReWnl4oq5qtyvyGjnwyUTwe4e34oTkD99RYFLfw5e2+fqyxT7EGv1qqUWeSBLZqzT\n9nxm6vlA+p7h3Oa+h22WFC52CT1frE6BL8wj6XuECr4H3mvn1X29PcvZH2KB8pRN39b8Hse+\nkhlR+Pivkc5u3tenUPpWp3n3PUzv9oAjM+Dw8SOwv8/ZBhinbPq25vcYjlKreoVxCX09yqZv\na36P4EitKh+GIbAt1lydlE3f1vzuz+FG1ezwGQJvukS+6vX0yuN64PD90tnfgWMb9zky1atw\nvSPw7C+sFtr06ua+quBUIZpZP/7phos/cJaXkwVWVCr/NcdUgfGyKwt8okeVfrN0isBLDXLU\nNPu/QM5/SqwLfLZiOciq9p/FmqvfxG89k2c+qxRl5TqZXfdj5t4PqFxXdHiCJn5nmZ6Meohd\nBJ76RuHo74HPN+j8Gow44VPo7mfEWSuXn+PHO4+o17gpFqaeL/1mm1k1+I1ztrL79bTUgmJ0\nL3fiUdYd9PyIFuXaac+SKpSkdf3n8jj45bTvNhVhfSzpY/36RyW3Nvh4l4fWuul+351qSktn\n/9kc3v+p7E7UMgDWUo/IGbcT8i1zpPfn8a2/Tm4EjvcByW0DMtWFp04avVuvUau56VL0xavs\n2V2HfL8S33wrNy42lFxoeB9hOgy9myk8aXgTroPci+icT5f81MjNeuEe/UYd9DzeSTfAtyjz\nHQ8kEvp09wPJXY76lGsgN/6T9/EuqXjvEd9RGG975MMINSKZ/sDL3ZMN1z7uPh27v6zAdVlT\nVW2OH4FJpscId8jzXftNwzfthZkuHDdpboIytYXT5kviK0/QEW7O1wk3Grkrt0ljeIjNE2Hw\nLBwk82/wvfA8OQevkakCosDBiFAHz9MCiMB8k27wlG+Vj1aQ1pdIOh+eTYCn/tBxvJGJBZzO\n+hEmHxDRZQu+wV9ueQ4FVOAm9Sg8JY7TGXmIJydwJLN4+Dh5CEl1nRxVOAEyn08SuOETWTR9\n/SgnCFyTLolqJD7jhxzUpNvOvUwe5VlnFdh38944nXpHp7DRFw2lyfDY08ZLa/aibCNTV3Qq\npbJeX1KjU1iwyXE2LtGKBQ57IZZVZruRqbfEHB6006RXMhzKjD06nU2c64a0hNctGi7gnU5O\nSTIFGHdNFAXOZgzSLeRYuWtoGulVZJqdhmPFuYa8nkq6lMBn/9R6kdN/CC61KFBiWwCfL9JW\nJ6Z5nhQrWpZPOSmz4KSpXykP5tMccV60aBqdTDb/QjQaT9FFjkMqQNqZS1NMamoo7a4TmPXU\nlfGClXz8MzFTbDet3cQJGUwYPZhok4Y6Ul7Apb8+Pwn9LYqwZd7vgnvcJ3RBTrf4HIGzF6nN\nk46Rm3Y5I/B8aZqZYD0TeFTrqZjDmV4zryaml/SjRYNIo4rM7JumYi3OgmXva6QKbnLayolV\nLiTwUqElgSend59jenXZb02WVFuiWO7uVWCxM5ASC6/2oMgeuhkhLIk74IyDKCXwQpWLHUyp\nf1Cpau4KvDFe4aaUh9O2+t89aPBEN+P/UXwZFlyQ//6TE6RHJyeB9FTQfzQ+U9R/+d+wzJAy\nLYBG79Yak9dC18PICAwepOQl9Iyqp/V+ILClNTzE2Qf44tzPz6okplyWiXdQlLrCVFWZYdC1\nzWWfmhQ8pipD7dnNPv0h1uNFd49SV5iqKjMTc1PwKht4laFOF/jweHU19ise03RMCLxLKAh8\nhTBVVWYmJgTeJRQEvkKYqiozExMC7xIKAl8hTFWVGQbFh1h7hoLAVwhTVWVKUGUDrzIUBL5C\nmKoqU4IqG3iVoSwJDAA4EAgMgGEgMACGgcAAGAYCA2AYCAyAYSAwAIaBwAAYBgIDYBgIDIBh\nCgisv4KffNhcl2ejFKpMiTBUJlChMGV54N6HpVBFApUMlZ/0ApF2HCgL1ZCmH7YFoplgx1em\nRBhx7OlAhcLsQMGfiNcWKj/pBSKlhx0o1mNVJHAV/YCUfzpQNpk9BD4qVC93BSLlWSxNPQLL\nP1/0ZJR6BG7KCJwVrEzgoherNYYqE67+S2gNU4vAVGIg17c/EHiBUnsu+8a11Bvz5rUELjN2\nlohCE9tfYwQu0S8VpJi/BYMRlbpWfTGBZ5vqthilLqEvKXCJc1ySGgUuF+q1BKbsz6ONS7+X\nqMK84eDcHwAAAnlJREFUGgUucY5LUmy/ELhI/OdDFGhcGIEX6jJbrXOoUeB0wguGegGB43f6\n+CHHMEaZQHGDMue4FOV2XPaHHKUiabgCevQeynNaCwAAPA8EBsAwEBgAw0BgAAwDgQEwDAQG\nwDAQGADDQGAADAOBATAMBAbAMBAYAMNAYAAMA4EBMAwEBsAwEBgAw0BgAAwDgQEwDAQGwDAQ\nGADDQGAADAOBATAMBAbAMK8h8OJRvsYpqJRswuvJVWun5V34B4tp/Hx5rtcsUjebb12z+nZU\nUIUDoLkDfY3DrxianT592z+MfK8jGC2anwV+PJN+tkkl8+p3nF+DI4DAtUJzJm3895VPEvh8\nKqnGzui/Ohr/NqRP0z8imlbE67Yza/tK5AJT/zr4QYGHqcx2oc8otodYqL/rSYFnKnM+lVRj\nZ7Iel+TijHpL+yvmO2dQmu6MT63aIHAUcJRKGgicN4DJXa8RuOsEzqaCKhzAQODR0vEDOIZi\nAmd/pzJKnb3NsD1M7HcocM9kUn0ntz6a82twBN2Fcup0u6cTSa2ha30Nuivbptwl9HaB711C\n02DFTC1O4PwaHEH/ungwGI+T2rzKeTmfXODJVU0myxqBZxL73Ag8MhsCH0s/YfFhPt2vcl7O\nR6+G5lY9JPA4sfN99gMCZwEqaCbn1+AINKtN9zmHfrYx/BQ6fwUOIFNjctX2H3IQ9fwcZDYt\nyiwf7TjbhvK3ut3us1cnU0EVAACPAoEBMAwEBsAwEBgAw0BgAAwDgQEwDAQGwDAQGADDQGAA\nDAOBATAMBAbAMBAYAMNAYAAMA4EBMAwEBsAwEBgAw0BgAAwDgQEwDAQGwDAQGADD/B/5uxY3\nLo9wXgAAAABJRU5ErkJggg==",
      "text/plain": [
       "Plot with title \"Density of lam[4]\""
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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WBNEBjAMAgMYBgEBjAMAgMYBoEBDIPA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\ndPcphbbH6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kccJy2qFwopTAKbgjqfsC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\nQmCC",
      "text/plain": [
       "Plot with title \"Density of sig\""
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Convergence Dignosis\n",
    "\n",
    "options(repr.plot.width = 8, repr.model.height=10)\n",
    "\n",
    "plot(mod_sim)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Potential scale reduction factors:\n",
       "\n",
       "       Point est. Upper C.I.\n",
       "lam[1]          1          1\n",
       "lam[2]          1          1\n",
       "lam[3]          1          1\n",
       "lam[4]          1          1\n",
       "lam[5]          1          1\n",
       "mu              1          1\n",
       "sig             1          1\n",
       "\n",
       "Multivariate psrf\n",
       "\n",
       "1"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<thead><tr><th></th><th scope=col>lam[1]</th><th scope=col>lam[2]</th><th scope=col>lam[3]</th><th scope=col>lam[4]</th><th scope=col>lam[5]</th><th scope=col>mu</th><th scope=col>sig</th></tr></thead>\n",
       "<tbody>\n",
       "\t<tr><th scope=row>Lag 0</th><td> 1.0000000000</td><td> 1.000000000 </td><td> 1.000000000 </td><td> 1.0000000000</td><td> 1.000000000 </td><td> 1.000000000 </td><td>1.000000000  </td></tr>\n",
       "\t<tr><th scope=row>Lag 1</th><td> 0.0284118498</td><td> 0.112904124 </td><td> 0.020633769 </td><td> 0.0191066167</td><td> 0.056872758 </td><td> 0.368014181 </td><td>0.570981911  </td></tr>\n",
       "\t<tr><th scope=row>Lag 5</th><td> 0.0063559360</td><td> 0.012770716 </td><td>-0.009500648 </td><td> 0.0006777028</td><td>-0.001150007 </td><td> 0.026097797 </td><td>0.097693729  </td></tr>\n",
       "\t<tr><th scope=row>Lag 10</th><td>-0.0005808907</td><td>-0.002649832 </td><td> 0.011823858 </td><td>-0.0016467998</td><td> 0.002356198 </td><td> 0.003666997 </td><td>0.023958109  </td></tr>\n",
       "\t<tr><th scope=row>Lag 50</th><td> 0.0036976033</td><td> 0.004942986 </td><td> 0.001338190 </td><td> 0.0010566590</td><td>-0.011968066 </td><td>-0.009260772 </td><td>0.004680512  </td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{r|lllllll}\n",
       "  & lam{[}1{]} & lam{[}2{]} & lam{[}3{]} & lam{[}4{]} & lam{[}5{]} & mu & sig\\\\\n",
       "\\hline\n",
       "\tLag 0 &  1.0000000000 &  1.000000000  &  1.000000000  &  1.0000000000 &  1.000000000  &  1.000000000  & 1.000000000  \\\\\n",
       "\tLag 1 &  0.0284118498 &  0.112904124  &  0.020633769  &  0.0191066167 &  0.056872758  &  0.368014181  & 0.570981911  \\\\\n",
       "\tLag 5 &  0.0063559360 &  0.012770716  & -0.009500648  &  0.0006777028 & -0.001150007  &  0.026097797  & 0.097693729  \\\\\n",
       "\tLag 10 & -0.0005808907 & -0.002649832  &  0.011823858  & -0.0016467998 &  0.002356198  &  0.003666997  & 0.023958109  \\\\\n",
       "\tLag 50 &  0.0036976033 &  0.004942986  &  0.001338190  &  0.0010566590 & -0.011968066  & -0.009260772  & 0.004680512  \\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "| <!--/--> | lam[1] | lam[2] | lam[3] | lam[4] | lam[5] | mu | sig |\n",
       "|---|---|---|---|---|---|---|---|\n",
       "| Lag 0 |  1.0000000000 |  1.000000000  |  1.000000000  |  1.0000000000 |  1.000000000  |  1.000000000  | 1.000000000   |\n",
       "| Lag 1 |  0.0284118498 |  0.112904124  |  0.020633769  |  0.0191066167 |  0.056872758  |  0.368014181  | 0.570981911   |\n",
       "| Lag 5 |  0.0063559360 |  0.012770716  | -0.009500648  |  0.0006777028 | -0.001150007  |  0.026097797  | 0.097693729   |\n",
       "| Lag 10 | -0.0005808907 | -0.002649832  |  0.011823858  | -0.0016467998 |  0.002356198  |  0.003666997  | 0.023958109   |\n",
       "| Lag 50 |  0.0036976033 |  0.004942986  |  0.001338190  |  0.0010566590 | -0.011968066  | -0.009260772  | 0.004680512   |\n",
       "\n"
      ],
      "text/plain": [
       "       lam[1]        lam[2]       lam[3]       lam[4]        lam[5]      \n",
       "Lag 0   1.0000000000  1.000000000  1.000000000  1.0000000000  1.000000000\n",
       "Lag 1   0.0284118498  0.112904124  0.020633769  0.0191066167  0.056872758\n",
       "Lag 5   0.0063559360  0.012770716 -0.009500648  0.0006777028 -0.001150007\n",
       "Lag 10 -0.0005808907 -0.002649832  0.011823858 -0.0016467998  0.002356198\n",
       "Lag 50  0.0036976033  0.004942986  0.001338190  0.0010566590 -0.011968066\n",
       "       mu           sig        \n",
       "Lag 0   1.000000000 1.000000000\n",
       "Lag 1   0.368014181 0.570981911\n",
       "Lag 5   0.026097797 0.097693729\n",
       "Lag 10  0.003666997 0.023958109\n",
       "Lag 50 -0.009260772 0.004680512"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<dl class=dl-horizontal>\n",
       "\t<dt>lam[1]</dt>\n",
       "\t\t<dd>27882.7318154214</dd>\n",
       "\t<dt>lam[2]</dt>\n",
       "\t\t<dd>22280.376176607</dd>\n",
       "\t<dt>lam[3]</dt>\n",
       "\t\t<dd>28424.7414889463</dd>\n",
       "\t<dt>lam[4]</dt>\n",
       "\t\t<dd>29128.8096088254</dd>\n",
       "\t<dt>lam[5]</dt>\n",
       "\t\t<dd>23034.0820789961</dd>\n",
       "\t<dt>mu</dt>\n",
       "\t\t<dd>13238.044313942</dd>\n",
       "\t<dt>sig</dt>\n",
       "\t\t<dd>7119.86055873172</dd>\n",
       "</dl>\n"
      ],
      "text/latex": [
       "\\begin{description*}\n",
       "\\item[lam\\{{[}\\}1\\{{]}\\}] 27882.7318154214\n",
       "\\item[lam\\{{[}\\}2\\{{]}\\}] 22280.376176607\n",
       "\\item[lam\\{{[}\\}3\\{{]}\\}] 28424.7414889463\n",
       "\\item[lam\\{{[}\\}4\\{{]}\\}] 29128.8096088254\n",
       "\\item[lam\\{{[}\\}5\\{{]}\\}] 23034.0820789961\n",
       "\\item[mu] 13238.044313942\n",
       "\\item[sig] 7119.86055873172\n",
       "\\end{description*}\n"
      ],
      "text/markdown": [
       "lam[1]\n",
       ":   27882.7318154214lam[2]\n",
       ":   22280.376176607lam[3]\n",
       ":   28424.7414889463lam[4]\n",
       ":   29128.8096088254lam[5]\n",
       ":   23034.0820789961mu\n",
       ":   13238.044313942sig\n",
       ":   7119.86055873172\n",
       "\n"
      ],
      "text/plain": [
       "   lam[1]    lam[2]    lam[3]    lam[4]    lam[5]        mu       sig \n",
       "27882.732 22280.376 28424.741 29128.810 23034.082 13238.044  7119.861 "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Convergence Dignosis\n",
    "\n",
    "gelman.diag(mod_sim)\n",
    "autocorr.diag(mod_sim)\n",
    "effectiveSize(mod_sim)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Mean deviance:  783.7 \n",
       "penalty 4.805 \n",
       "Penalized deviance: 788.5 "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Compute Dic\n",
    "\n",
    "dic = dic.samples(model = mod, n.iter = 1e4)\n",
    "dic"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<dl class=dl-horizontal>\n",
       "\t<dt>lam[1]</dt>\n",
       "\t\t<dd>9.28157350066371</dd>\n",
       "\t<dt>lam[2]</dt>\n",
       "\t\t<dd>6.22679421550139</dd>\n",
       "\t<dt>lam[3]</dt>\n",
       "\t\t<dd>9.5296696327295</dd>\n",
       "\t<dt>lam[4]</dt>\n",
       "\t\t<dd>8.94468329704611</dd>\n",
       "\t<dt>lam[5]</dt>\n",
       "\t\t<dd>11.7551604486328</dd>\n",
       "\t<dt>mu</dt>\n",
       "\t\t<dd>9.09618836869898</dd>\n",
       "\t<dt>sig</dt>\n",
       "\t\t<dd>2.07494977586327</dd>\n",
       "</dl>\n"
      ],
      "text/latex": [
       "\\begin{description*}\n",
       "\\item[lam\\{{[}\\}1\\{{]}\\}] 9.28157350066371\n",
       "\\item[lam\\{{[}\\}2\\{{]}\\}] 6.22679421550139\n",
       "\\item[lam\\{{[}\\}3\\{{]}\\}] 9.5296696327295\n",
       "\\item[lam\\{{[}\\}4\\{{]}\\}] 8.94468329704611\n",
       "\\item[lam\\{{[}\\}5\\{{]}\\}] 11.7551604486328\n",
       "\\item[mu] 9.09618836869898\n",
       "\\item[sig] 2.07494977586327\n",
       "\\end{description*}\n"
      ],
      "text/markdown": [
       "lam[1]\n",
       ":   9.28157350066371lam[2]\n",
       ":   6.22679421550139lam[3]\n",
       ":   9.5296696327295lam[4]\n",
       ":   8.94468329704611lam[5]\n",
       ":   11.7551604486328mu\n",
       ":   9.09618836869898sig\n",
       ":   2.07494977586327\n",
       "\n"
      ],
      "text/plain": [
       "   lam[1]    lam[2]    lam[3]    lam[4]    lam[5]        mu       sig \n",
       " 9.281574  6.226794  9.529670  8.944683 11.755160  9.096188  2.074950 "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "colMeans(mod_csim)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Model checking\n",
    "\n",
    "After assessing convergence, we can check the fit via residuals. With a hierarchical model, there are now two levels of residuals: the observation level and the location mean level. To simplify, we’ll look at the residuals associated with the posterior means of the parameters."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [],
   "source": [
    "# First, we have observation residuals, based on the estimates of location means.\n",
    "# Observation level Residuals\n",
    "\n",
    "pm_params = colMeans(mod_csim)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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f4p39Rz9Mv/TmEOvOoQrGn40vSMpxh0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\nRK5CYII=",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "yhat = rep(pm_params[1:5], each = 30)\n",
    "resid = dat$chips - yhat\n",
    "\n",
    "options(repr.plot.width = 8, repr.plot.height=6)\n",
    "plot(resid)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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weuXsCAj6cznNcYmlXo6oUL+PgO495q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\nMAQmYAjsH/Gwl1lmF5bHAAAAAElFTkSuQmCC",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot(jitter(yhat), resid)\n",
    "\n",
    "# We can observe here that that variance has incresed as yhat increases like our hypothesis ie. Poisson."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If out hypothesis is not violated then mean will be equal to variance."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "6.44712643678161"
      ],
      "text/latex": [
       "6.44712643678161"
      ],
      "text/markdown": [
       "6.44712643678161"
      ],
      "text/plain": [
       "[1] 6.447126"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "var(resid[yhat<7])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "13.7241379310345"
      ],
      "text/latex": [
       "13.7241379310345"
      ],
      "text/markdown": [
       "13.7241379310345"
      ],
      "text/plain": [
       "[1] 13.72414"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "var(resid[yhat>11])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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8s+rMb/5M99XYTeGBf7LnRP0GbZ3eQJ\n2c2ddwSF3mfK8d+e4dB602fu6yL0xjg7dKkb0EeryG3oxem4XfJ95visa9F0eXNfF8k3Rt/2\nppjWIXEv1ur8lCX5PtN8+rmvi9wbY7/o93PPUCPxmi+/zD3J9QR8H7axO6Avx4H3jTflJhEd\n8Nv13na/ydzXRORNMZwPR849wrXOZ2IdlpHPgc9C7zPr057/w7rxgYu5r4vQG+P4bCz5oeAG\nhzMmFHmln8/hPml8/G7u6yL0xhiSt+WGdVcWsY+/ufeZwy2u99DrAjgRMAQTMAQTMAQTMAQT\nMAQTMAQTMAQTMAQTMAQTMAQTMAQTMAQTMAQTMAQTMAQTMAQTMAQTMAQTMAQTMAQTMAQTMAQT\nMAQTMAQTMAQTMAQTMAQTMAQTMAQTMAQTMAQTMAQT8KP7++fZp37KPR+4HR+dgB+a2/HRCfih\nuR0fnYAfmtvx0Z1TLWW/LN3T+Q/WXVm/BrxZlG5z/N6Xl+PXl7Kab0yuI+BH9xpwV45OBfen\nC8vzny5PF0s/DPvSHX/bdYd5R+XfCfjRvQbcH4ZNWQzDc+l2w647/en29IeHvmyPD8XHtp/K\n89yz8s8E/OheA355vbg8X9peLp4ecQ9lOZwelzfn74QR8KN7Dfjt4uveq8vFV8NpI/r4NHnG\nKbmSgB/duICHdVnPNyNXE/Cj+yrg33/LI3AoAT+6DwEvT/ushpffFy+Wx+fA/UwTUkHAj+5D\nwNvfe6HPO6SH886r5+MG9FPZzDwq/07Aj+5DwJeDv6vzxfMh4dLth0N3Pg5sIzqPgB/dx4CH\npz/OxCqrY7Wr1zOxbETHETAEEzAEEzAEEzAEEzAEEzAEEzAEEzAEEzAEEzAEEzAEEzAEEzAE\nEzAEEzAEEzAEEzAEEzAEEzAEEzAEEzAEEzAEEzAEEzAEEzAEEzAEEzAEEzAEEzAEEzAEEzAE\n+w+/ESv1ElH/5wAAAABJRU5ErkJggg==",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Location level Residuals\n",
    "\n",
    "lam_residual = pm_params[1:5] - pm_params[\"mu\"]\n",
    "plot(lam_residual)\n",
    "abline(h=0, lty=3)\n",
    "\n",
    "# We don’t see any obvious violations of our model assumptions."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Results"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "\n",
       "Iterations = 2001:12000\n",
       "Thinning interval = 1 \n",
       "Number of chains = 3 \n",
       "Sample size per chain = 10000 \n",
       "\n",
       "1. Empirical mean and standard deviation for each variable,\n",
       "   plus standard error of the mean:\n",
       "\n",
       "         Mean     SD Naive SE Time-series SE\n",
       "lam[1]  9.282 0.5369 0.003100       0.003218\n",
       "lam[2]  6.227 0.4615 0.002664       0.003092\n",
       "lam[3]  9.530 0.5463 0.003154       0.003241\n",
       "lam[4]  8.945 0.5266 0.003040       0.003085\n",
       "lam[5] 11.755 0.6175 0.003565       0.004085\n",
       "mu      9.096 0.9738 0.005622       0.008515\n",
       "sig     2.075 0.7036 0.004062       0.008422\n",
       "\n",
       "2. Quantiles for each variable:\n",
       "\n",
       "         2.5%    25%    50%    75%  97.5%\n",
       "lam[1]  8.266  8.911  9.273  9.636 10.367\n",
       "lam[2]  5.353  5.912  6.211  6.528  7.171\n",
       "lam[3]  8.494  9.154  9.517  9.890 10.640\n",
       "lam[4]  7.940  8.587  8.935  9.292  9.999\n",
       "lam[5] 10.580 11.331 11.743 12.161 13.006\n",
       "mu      7.199  8.491  9.069  9.670 11.099\n",
       "sig     1.097  1.583  1.944  2.421  3.801\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "summary(mod_sim)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Another Example"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<thead><tr><th scope=col>y</th><th scope=col>grp</th></tr></thead>\n",
       "<tbody>\n",
       "\t<tr><td> 1.2</td><td>1   </td></tr>\n",
       "\t<tr><td> 1.4</td><td>1   </td></tr>\n",
       "\t<tr><td>-0.5</td><td>1   </td></tr>\n",
       "\t<tr><td> 0.3</td><td>1   </td></tr>\n",
       "\t<tr><td> 0.9</td><td>1   </td></tr>\n",
       "\t<tr><td> 2.3</td><td>1   </td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{r|ll}\n",
       " y & grp\\\\\n",
       "\\hline\n",
       "\t  1.2 & 1   \\\\\n",
       "\t  1.4 & 1   \\\\\n",
       "\t -0.5 & 1   \\\\\n",
       "\t  0.3 & 1   \\\\\n",
       "\t  0.9 & 1   \\\\\n",
       "\t  2.3 & 1   \\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "| y | grp |\n",
       "|---|---|\n",
       "|  1.2 | 1    |\n",
       "|  1.4 | 1    |\n",
       "| -0.5 | 1    |\n",
       "|  0.3 | 1    |\n",
       "|  0.9 | 1    |\n",
       "|  2.3 | 1    |\n",
       "\n"
      ],
      "text/plain": [
       "  y    grp\n",
       "1  1.2 1  \n",
       "2  1.4 1  \n",
       "3 -0.5 1  \n",
       "4  0.3 1  \n",
       "5  0.9 1  \n",
       "6  2.3 1  "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "dat = read.csv(file=\"pctgrowth.csv\", header=TRUE)\n",
    "\n",
    "head(dat)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<ol class=list-inline>\n",
       "\t<li>1</li>\n",
       "\t<li>2</li>\n",
       "\t<li>3</li>\n",
       "\t<li>4</li>\n",
       "\t<li>5</li>\n",
       "</ol>\n"
      ],
      "text/latex": [
       "\\begin{enumerate*}\n",
       "\\item 1\n",
       "\\item 2\n",
       "\\item 3\n",
       "\\item 4\n",
       "\\item 5\n",
       "\\end{enumerate*}\n"
      ],
      "text/markdown": [
       "1. 1\n",
       "2. 2\n",
       "3. 3\n",
       "4. 4\n",
       "5. 5\n",
       "\n",
       "\n"
      ],
      "text/plain": [
       "[1] 1 2 3 4 5"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "unique(dat$grp)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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gkYigkYigkYigkYigkYigkYigkYigkY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",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "boxplot(y ~ grp, data = dat)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [],
   "source": [
    "mod_str = \" model {\n",
    "    # Model's Likelihood Hierarchy\n",
    "    for(i in 1:length(y)){\n",
    "        y[i] ~ dnorm(mu_grp[grp[i]], prec)\n",
    "    }\n",
    "\n",
    "    for(j in 1:max(grp)){\n",
    "        mu_grp[j] ~ dnorm(mu, prec_grps)\n",
    "    }\n",
    "    \n",
    "    # Priors\n",
    "    mu ~ dnorm(0, 1/1e6)\n",
    "    prec ~ dgamma(2/2.0, 2*1/2.0)\n",
    "    prec_grps ~ dgamma(1/2.0, 1*3/2.0)\n",
    "\n",
    "    #varaibles\n",
    "    sig = 1/prec\n",
    "    sig_grps =  1/prec_grps\n",
    "\n",
    "} \""
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Compiling model graph\n",
      "   Resolving undeclared variables\n",
      "   Allocating nodes\n",
      "Graph information:\n",
      "   Observed stochastic nodes: 53\n",
      "   Unobserved stochastic nodes: 8\n",
      "   Total graph size: 127\n",
      "\n",
      "Initializing model\n",
      "\n"
     ]
    }
   ],
   "source": [
    "model = jags.model(textConnection(mod_str), \n",
    "                   data = as.list(dat), \n",
    "                   n.chains = 3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [],
   "source": [
    "params = c(\"mu_grp\", \"mu\", \"sig\", \"sig_grps\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [],
   "source": [
    "mod_sim = coda.samples(model = model, \n",
    "                       variable.names = params, \n",
    "                       n.iter = 1e4)\n",
    "\n",
    "mod_csim = as.mcmc(do.call(rbind, mod_sim))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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+U5kgw/fhAaGULumkmyKpOGmeYaP4dcTGtTy88E3LFWzELbLYnGBNwpvKAQcM+2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",
      "text/plain": [
       "Plot with title \"Density of mu_grp[3]\""
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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XYA3yvJDx/gA4KmIX+nszwsyMr9wv6hATyfpJXyZA3r7/n3j9aRe5HdpuAmsa6N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\ngg==",
      "text/plain": [
       "Plot with title \"Density of sig_grps\""
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot(mod_sim)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Potential scale reduction factors:\n",
       "\n",
       "          Point est. Upper C.I.\n",
       "mu              1.00       1.00\n",
       "mu_grp[1]       1.00       1.00\n",
       "mu_grp[2]       1.00       1.00\n",
       "mu_grp[3]       1.00       1.00\n",
       "mu_grp[4]       1.00       1.00\n",
       "mu_grp[5]       1.00       1.00\n",
       "sig             1.00       1.00\n",
       "sig_grps        1.02       1.02\n",
       "\n",
       "Multivariate psrf\n",
       "\n",
       "1"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<table>\n",
       "<thead><tr><th></th><th scope=col>mu</th><th scope=col>mu_grp[1]</th><th scope=col>mu_grp[2]</th><th scope=col>mu_grp[3]</th><th scope=col>mu_grp[4]</th><th scope=col>mu_grp[5]</th><th scope=col>sig</th><th scope=col>sig_grps</th></tr></thead>\n",
       "<tbody>\n",
       "\t<tr><th scope=row>Lag 0</th><td> 1.000000000 </td><td> 1.000000000 </td><td> 1.000000000 </td><td>1.000000000  </td><td> 1.000000000 </td><td> 1.000000000 </td><td> 1.0000000000</td><td> 1.000000000 </td></tr>\n",
       "\t<tr><th scope=row>Lag 1</th><td> 0.047009202 </td><td> 0.048732973 </td><td> 0.064431936 </td><td>0.029640793  </td><td> 0.024271122 </td><td> 0.027449194 </td><td> 0.0972906518</td><td> 0.190684648 </td></tr>\n",
       "\t<tr><th scope=row>Lag 5</th><td>-0.001216101 </td><td> 0.008353765 </td><td> 0.001470163 </td><td>0.001754329  </td><td> 0.007333394 </td><td> 0.005873229 </td><td>-0.0067783378</td><td>-0.002072516 </td></tr>\n",
       "\t<tr><th scope=row>Lag 10</th><td>-0.004532311 </td><td> 0.002516124 </td><td>-0.007086858 </td><td>0.003374025  </td><td>-0.008556482 </td><td> 0.010879192 </td><td> 0.0032631002</td><td>-0.009338486 </td></tr>\n",
       "\t<tr><th scope=row>Lag 50</th><td> 0.001422383 </td><td>-0.005008029 </td><td>-0.002966232 </td><td>0.008757688  </td><td> 0.003492457 </td><td>-0.005676707 </td><td>-0.0001816587</td><td> 0.019804808 </td></tr>\n",
       "</tbody>\n",
       "</table>\n"
      ],
      "text/latex": [
       "\\begin{tabular}{r|llllllll}\n",
       "  & mu & mu\\_grp{[}1{]} & mu\\_grp{[}2{]} & mu\\_grp{[}3{]} & mu\\_grp{[}4{]} & mu\\_grp{[}5{]} & sig & sig\\_grps\\\\\n",
       "\\hline\n",
       "\tLag 0 &  1.000000000  &  1.000000000  &  1.000000000  & 1.000000000   &  1.000000000  &  1.000000000  &  1.0000000000 &  1.000000000 \\\\\n",
       "\tLag 1 &  0.047009202  &  0.048732973  &  0.064431936  & 0.029640793   &  0.024271122  &  0.027449194  &  0.0972906518 &  0.190684648 \\\\\n",
       "\tLag 5 & -0.001216101  &  0.008353765  &  0.001470163  & 0.001754329   &  0.007333394  &  0.005873229  & -0.0067783378 & -0.002072516 \\\\\n",
       "\tLag 10 & -0.004532311  &  0.002516124  & -0.007086858  & 0.003374025   & -0.008556482  &  0.010879192  &  0.0032631002 & -0.009338486 \\\\\n",
       "\tLag 50 &  0.001422383  & -0.005008029  & -0.002966232  & 0.008757688   &  0.003492457  & -0.005676707  & -0.0001816587 &  0.019804808 \\\\\n",
       "\\end{tabular}\n"
      ],
      "text/markdown": [
       "\n",
       "| <!--/--> | mu | mu_grp[1] | mu_grp[2] | mu_grp[3] | mu_grp[4] | mu_grp[5] | sig | sig_grps |\n",
       "|---|---|---|---|---|---|---|---|---|\n",
       "| Lag 0 |  1.000000000  |  1.000000000  |  1.000000000  | 1.000000000   |  1.000000000  |  1.000000000  |  1.0000000000 |  1.000000000  |\n",
       "| Lag 1 |  0.047009202  |  0.048732973  |  0.064431936  | 0.029640793   |  0.024271122  |  0.027449194  |  0.0972906518 |  0.190684648  |\n",
       "| Lag 5 | -0.001216101  |  0.008353765  |  0.001470163  | 0.001754329   |  0.007333394  |  0.005873229  | -0.0067783378 | -0.002072516  |\n",
       "| Lag 10 | -0.004532311  |  0.002516124  | -0.007086858  | 0.003374025   | -0.008556482  |  0.010879192  |  0.0032631002 | -0.009338486  |\n",
       "| Lag 50 |  0.001422383  | -0.005008029  | -0.002966232  | 0.008757688   |  0.003492457  | -0.005676707  | -0.0001816587 |  0.019804808  |\n",
       "\n"
      ],
      "text/plain": [
       "       mu           mu_grp[1]    mu_grp[2]    mu_grp[3]   mu_grp[4]   \n",
       "Lag 0   1.000000000  1.000000000  1.000000000 1.000000000  1.000000000\n",
       "Lag 1   0.047009202  0.048732973  0.064431936 0.029640793  0.024271122\n",
       "Lag 5  -0.001216101  0.008353765  0.001470163 0.001754329  0.007333394\n",
       "Lag 10 -0.004532311  0.002516124 -0.007086858 0.003374025 -0.008556482\n",
       "Lag 50  0.001422383 -0.005008029 -0.002966232 0.008757688  0.003492457\n",
       "       mu_grp[5]    sig           sig_grps    \n",
       "Lag 0   1.000000000  1.0000000000  1.000000000\n",
       "Lag 1   0.027449194  0.0972906518  0.190684648\n",
       "Lag 5   0.005873229 -0.0067783378 -0.002072516\n",
       "Lag 10  0.010879192  0.0032631002 -0.009338486\n",
       "Lag 50 -0.005676707 -0.0001816587  0.019804808"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/html": [
       "<dl class=dl-horizontal>\n",
       "\t<dt>mu</dt>\n",
       "\t\t<dd>26912.4495006898</dd>\n",
       "\t<dt>mu_grp[1]</dt>\n",
       "\t\t<dd>26599.4568266271</dd>\n",
       "\t<dt>mu_grp[2]</dt>\n",
       "\t\t<dd>26062.9732908242</dd>\n",
       "\t<dt>mu_grp[3]</dt>\n",
       "\t\t<dd>27044.7074136453</dd>\n",
       "\t<dt>mu_grp[4]</dt>\n",
       "\t\t<dd>27833.8551625249</dd>\n",
       "\t<dt>mu_grp[5]</dt>\n",
       "\t\t<dd>28355.9087315553</dd>\n",
       "\t<dt>sig</dt>\n",
       "\t\t<dd>25222.9900538354</dd>\n",
       "\t<dt>sig_grps</dt>\n",
       "\t\t<dd>19921.8943031047</dd>\n",
       "</dl>\n"
      ],
      "text/latex": [
       "\\begin{description*}\n",
       "\\item[mu] 26912.4495006898\n",
       "\\item[mu\\textbackslash{}\\_grp\\{{[}\\}1\\{{]}\\}] 26599.4568266271\n",
       "\\item[mu\\textbackslash{}\\_grp\\{{[}\\}2\\{{]}\\}] 26062.9732908242\n",
       "\\item[mu\\textbackslash{}\\_grp\\{{[}\\}3\\{{]}\\}] 27044.7074136453\n",
       "\\item[mu\\textbackslash{}\\_grp\\{{[}\\}4\\{{]}\\}] 27833.8551625249\n",
       "\\item[mu\\textbackslash{}\\_grp\\{{[}\\}5\\{{]}\\}] 28355.9087315553\n",
       "\\item[sig] 25222.9900538354\n",
       "\\item[sig\\textbackslash{}\\_grps] 19921.8943031047\n",
       "\\end{description*}\n"
      ],
      "text/markdown": [
       "mu\n",
       ":   26912.4495006898mu_grp[1]\n",
       ":   26599.4568266271mu_grp[2]\n",
       ":   26062.9732908242mu_grp[3]\n",
       ":   27044.7074136453mu_grp[4]\n",
       ":   27833.8551625249mu_grp[5]\n",
       ":   28355.9087315553sig\n",
       ":   25222.9900538354sig_grps\n",
       ":   19921.8943031047\n",
       "\n"
      ],
      "text/plain": [
       "       mu mu_grp[1] mu_grp[2] mu_grp[3] mu_grp[4] mu_grp[5]       sig  sig_grps \n",
       " 26912.45  26599.46  26062.97  27044.71  27833.86  28355.91  25222.99  19921.89 "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "gelman.diag(mod_sim)\n",
    "autocorr.diag(mod_sim)\n",
    "effectiveSize(mod_sim)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<dl class=dl-horizontal>\n",
       "\t<dt>mu_grp[1]</dt>\n",
       "\t\t<dd>0.896092225364672</dd>\n",
       "\t<dt>mu_grp[2]</dt>\n",
       "\t\t<dd>-1.48956970016204</dd>\n",
       "\t<dt>mu_grp[3]</dt>\n",
       "\t\t<dd>-1.09663515688816</dd>\n",
       "\t<dt>mu_grp[4]</dt>\n",
       "\t\t<dd>0.223977414567504</dd>\n",
       "\t<dt>mu_grp[5]</dt>\n",
       "\t\t<dd>-0.372012739436644</dd>\n",
       "</dl>\n"
      ],
      "text/latex": [
       "\\begin{description*}\n",
       "\\item[mu\\textbackslash{}\\_grp\\{{[}\\}1\\{{]}\\}] 0.896092225364672\n",
       "\\item[mu\\textbackslash{}\\_grp\\{{[}\\}2\\{{]}\\}] -1.48956970016204\n",
       "\\item[mu\\textbackslash{}\\_grp\\{{[}\\}3\\{{]}\\}] -1.09663515688816\n",
       "\\item[mu\\textbackslash{}\\_grp\\{{[}\\}4\\{{]}\\}] 0.223977414567504\n",
       "\\item[mu\\textbackslash{}\\_grp\\{{[}\\}5\\{{]}\\}] -0.372012739436644\n",
       "\\end{description*}\n"
      ],
      "text/markdown": [
       "mu_grp[1]\n",
       ":   0.896092225364672mu_grp[2]\n",
       ":   -1.48956970016204mu_grp[3]\n",
       ":   -1.09663515688816mu_grp[4]\n",
       ":   0.223977414567504mu_grp[5]\n",
       ":   -0.372012739436644\n",
       "\n"
      ],
      "text/plain": [
       " mu_grp[1]  mu_grp[2]  mu_grp[3]  mu_grp[4]  mu_grp[5] \n",
       " 0.8960922 -1.4895697 -1.0966352  0.2239774 -0.3720127 "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "colMeans(mod_csim)[2:6]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {},
   "outputs": [],
   "source": [
    "means_anova = tapply(dat$y, INDEX=dat$grp, FUN=mean)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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i4BQzABQzABQzABQzABQzABQzABQzAB\nQzABQ7A7DRjo5Ya66gcbNoEBgucePHVzH/UJ4yYwQPDcg6du7qM+YdwEBgiee/DUzX3UJ4yb\nwADBcw+eurmP+oRxExggeO7BUzf3UZ8wbgIDBM89eOrmPuoTxk1ggOC5B0/d3Ed9wrgJDBA8\n9+Cpm/uoTxg3gQGC5x48dXMf9QnjJjBA8NyDp27uoz5h3AQGCJ578NTNfdQnjJvAAMFzD566\nuY/6hMB0BAzBBAzBBAzBBAzBBAzBBAzBBAzBBAzBBAzBBAzBBAzBBAzBBAzBBAzBBAzB5g54\nM/cEbrZZlKY9zD2LWxxeSnnZzT2L272F/s7c/AFmV5+07tP9q13t/z+TabsfRpNYcNNNPbbg\nQ5P5O7N7wIB3TWrAu/JyOK8/vMw9kX/XnifdltXc87jVKvR3ZjfKSz7ra7Epy9AfxvtvUfct\ncfpNOa82JM6881r9TWwim7Ie4VlnfS1Km/t79CF3+qWZewa32cf+o78pmxGeddbXYpdcwNmh\nLOeewo3aUX6bJrAs+9DfmVXZvpSmrfysc78WoT+MHzZlO/cUbvK+Glr7F2ki6/Ka+juz+tiH\nVfmf/Llfi9Afxod9E7onaLNqRtkgG123Iyj0d6a8/9tzOtRe9Zn7tQj9YXQOTeoK9LuXyHXo\nxfm4XfLvzPtW16Lq8839WiT/MJZ1fxTTOiTuxXrpNlmSf2eqz37u1yL3h7FfLPdzz2GIxFe+\n/DL3TG4n4Puwjd0B/XEceF95VW4S0QH/fN3r7jeZ+5WI/FGcusORc0/hVt2ZWIdV5DZwJ/R3\npj3v+T+0lQ9czP1ahP4w3rfGkt8KRjicMaHIF707h/us8vG7uV+L0B/GKXld7tQ2ZRH7/pv7\nO3MY43UPfS2AMwFDMAFDMAFDMAFDMAFDMAFDMAFDMAFDMAFDMAFDMAFDMAFDMAFDMAFDMAFD\nMAFDMAFDMAFDMAFDMAFDMAFDMAFDMAFDMAFDMAFDMAFDMAFDMAFDMAFDMAFDMAFDMAFDMAE/\nur9/nn3qp9zziZ/joxPwQ/NzfHQCfmh+jo9OwA/Nz/HRdamWsl+VZt39h7Yp7Y+AN4vSbN6/\nL8vb+9e38jLfNLmNgB/dj4Cb8u5c8PK8sOr+6+q8WJan0740739smsO8U+XfCfjR/Qh4eTht\nyuJ0ei3N7rRrzv91e/6Ph2XZvr8Vv7e9Lq9zz5V/JuBH9yPgtx+Lq25p+7F4fsc9lNXp/L68\n6b4TRsCP7kfAPxd/7L36WPzhdF6Jft9MnnGW3EjAj65fwKe2tPPNkZsJ+NFdC/j33/IOHErA\nj+5TwKvzPqvT2+/FD6v3beDlTDNkAAE/uk8Bb3/vhe52SJ+6nVev7yvQ67KZear8OwE/uk8B\nfxz8fekWu0PCpdmfDk13HNhKdB4BP7rPAZ/W/zsTq7y8V/vy40wsK9FxBAzBBAzBBAzBBAzB\nBAzBBAzBBAzBBAzBBAzBBAzBBAzBBAzBBAzBBAzBBAzBBAzBBAzBBAzBBAzBBAzBBAzBBAzB\nBAzBBAzBBAzBBAzBBAzBBAzBBAzBBAzBBAzB/gOELo/cfBhDbQAAAABJRU5ErkJggg==",
      "text/plain": [
       "plot without title"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "options(repr.plot.width=8, repr.plot.height=5)\n",
    "\n",
    "plot(means_anova)\n",
    "points(jitter(colMeans(mod_csim)[2:6]), col=\"red\")\n",
    "\n",
    "# The estimates from the hierarchical model have less variability than those from the ANOVA model, \n",
    "# tending toward smaller magnitudes."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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