13-Chapter 13. Models with Memory.ipynb

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   "source": [
    "using Turing\n",
    "using DataFrames\n",
    "using CSV\n",
    "using Random\n",
    "using Distributions\n",
    "using StatisticalRethinking\n",
    "using StatisticalRethinking: link\n",
    "using StatisticalRethinkingPlots\n",
    "using ParetoSmooth\n",
    "using StatsPlots\n",
    "using Plots.PlotMeasures\n",
    "using StatsBase\n",
    "using FreqTables\n",
    "using Logging\n",
    "\n",
    "default(label=false);\n",
    "Logging.disable_logging(Logging.Warn);"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e9c522b7",
   "metadata": {},
   "source": [
    "# 13.1 Example: multilevel tadpoles"
   ]
  },
  {
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   "id": "39bfd9fe",
   "metadata": {},
   "source": [
    "Code 13.1"
   ]
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  {
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   "execution_count": 2,
   "id": "35c8aef1",
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       "<div class=\"data-frame\"><p>5 rows × 7 columns</p><table class=\"data-frame\"><thead><tr><th></th><th>variable</th><th>mean</th><th>min</th><th>median</th><th>max</th><th>nmissing</th><th>eltype</th></tr><tr><th></th><th title=\"Symbol\">Symbol</th><th title=\"Union{Nothing, Float64}\">Union…</th><th title=\"Any\">Any</th><th title=\"Union{Nothing, Float64}\">Union…</th><th title=\"Any\">Any</th><th title=\"Int64\">Int64</th><th title=\"DataType\">DataType</th></tr></thead><tbody><tr><th>1</th><td>density</td><td>23.3333</td><td>10</td><td>25.0</td><td>35</td><td>0</td><td>Int64</td></tr><tr><th>2</th><td>pred</td><td></td><td>no</td><td></td><td>pred</td><td>0</td><td>String7</td></tr><tr><th>3</th><td>size</td><td></td><td>big</td><td></td><td>small</td><td>0</td><td>String7</td></tr><tr><th>4</th><td>surv</td><td>16.3125</td><td>4</td><td>12.5</td><td>35</td><td>0</td><td>Int64</td></tr><tr><th>5</th><td>propsurv</td><td>0.721607</td><td>0.114286</td><td>0.885714</td><td>1.0</td><td>0</td><td>Float64</td></tr></tbody></table></div>"
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       "\\begin{tabular}{r|ccccccc}\n",
       "\t& variable & mean & min & median & max & nmissing & eltype\\\\\n",
       "\t\\hline\n",
       "\t& Symbol & Union… & Any & Union… & Any & Int64 & DataType\\\\\n",
       "\t\\hline\n",
       "\t1 & density & 23.3333 & 10 & 25.0 & 35 & 0 & Int64 \\\\\n",
       "\t2 & pred &  & no &  & pred & 0 & String7 \\\\\n",
       "\t3 & size &  & big &  & small & 0 & String7 \\\\\n",
       "\t4 & surv & 16.3125 & 4 & 12.5 & 35 & 0 & Int64 \\\\\n",
       "\t5 & propsurv & 0.721607 & 0.114286 & 0.885714 & 1.0 & 0 & Float64 \\\\\n",
       "\\end{tabular}\n"
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       "\u001b[1m5×7 DataFrame\u001b[0m\n",
       "\u001b[1m Row \u001b[0m│\u001b[1m variable \u001b[0m\u001b[1m mean     \u001b[0m\u001b[1m min      \u001b[0m\u001b[1m median   \u001b[0m\u001b[1m max   \u001b[0m\u001b[1m nmissing \u001b[0m\u001b[1m eltype   \u001b[0m\n",
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       "─────┼───────────────────────────────────────────────────────────────────\n",
       "   1 │ density   23.3333   10        25.0      35            0  Int64\n",
       "   2 │ pred     \u001b[90m          \u001b[0m no       \u001b[90m          \u001b[0m pred          0  String7\n",
       "   3 │ size     \u001b[90m          \u001b[0m big      \u001b[90m          \u001b[0m small         0  String7\n",
       "   4 │ surv      16.3125   4         12.5      35            0  Int64\n",
       "   5 │ propsurv  0.721607  0.114286  0.885714  1.0           0  Float64"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "d = DataFrame(CSV.File(\"data/reedfrogs.csv\"))\n",
    "describe(d)"
   ]
  },
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   "id": "b8cc1130",
   "metadata": {},
   "source": [
    "Code 13.2"
   ]
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  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "bb7e6294",
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   "outputs": [],
   "source": [
    "d.tank = 1:nrow(d)\n",
    "\n",
    "@model function m13_1(S, N, tank)\n",
    "    tank_size = length(levels(d.tank))\n",
    "    a ~ filldist(Normal(0, 1.5), tank_size)\n",
    "    p = logistic.(a)\n",
    "    @. S ~ Binomial(N, p)\n",
    "end\n",
    "\n",
    "Random.seed!(1)\n",
    "m13_1_ch = sample(m13_1(d.surv, d.density, d.tank), NUTS(200, 0.65, init_ϵ=0.5), 1000)\n",
    "m13_1_df = DataFrame(m13_1_ch);"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "11e0b2f8",
   "metadata": {},
   "source": [
    "Code 13.3"
   ]
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  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "d6d144be",
   "metadata": {},
   "outputs": [],
   "source": [
    "@model function m13_2(S, N, tank)\n",
    "    tank_size = length(levels(d.tank))\n",
    "    σ ~ Exponential()\n",
    "    ā ~ Normal(0, 1.5)\n",
    "    a ~ filldist(Normal(ā, σ), tank_size)\n",
    "    p = logistic.(a)\n",
    "    @. S ~ Binomial(N, p)\n",
    "end\n",
    "\n",
    "Random.seed!(1)\n",
    "m13_2_ch = sample(m13_2(d.surv, d.density, d.tank), NUTS(200, 0.65, init_ϵ=0.2), 1000)\n",
    "m13_2_df = DataFrame(m13_2_ch);"
   ]
  },
  {
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   "id": "3a843440",
   "metadata": {},
   "source": [
    "Code 13.4"
   ]
  },
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   "cell_type": "code",
   "execution_count": 5,
   "id": "8df69df6",
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       "<div class=\"data-frame\"><p>2 rows × 8 columns</p><table class=\"data-frame\"><thead><tr><th></th><th>models</th><th>WAIC</th><th>lppd</th><th>SE</th><th>dWAIC</th><th>dSE</th><th>pWAIC</th><th>weight</th></tr><tr><th></th><th title=\"String\">String</th><th title=\"Float64\">Float64</th><th title=\"Float64\">Float64</th><th title=\"Float64\">Float64</th><th title=\"Float64\">Float64</th><th title=\"Float64\">Float64</th><th title=\"Float64\">Float64</th><th title=\"Float64\">Float64</th></tr></thead><tbody><tr><th>1</th><td>m13.2</td><td>201.2</td><td>158.41</td><td>7.47</td><td>0.0</td><td>0.0</td><td>21.4</td><td>1.0</td></tr><tr><th>2</th><td>m13.1</td><td>214.7</td><td>163.28</td><td>4.68</td><td>13.5</td><td>3.99</td><td>25.71</td><td>0.0</td></tr></tbody></table></div>"
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      "text/latex": [
       "\\begin{tabular}{r|cccccccc}\n",
       "\t& models & WAIC & lppd & SE & dWAIC & dSE & pWAIC & weight\\\\\n",
       "\t\\hline\n",
       "\t& String & Float64 & Float64 & Float64 & Float64 & Float64 & Float64 & Float64\\\\\n",
       "\t\\hline\n",
       "\t1 & m13.2 & 201.2 & 158.41 & 7.47 & 0.0 & 0.0 & 21.4 & 1.0 \\\\\n",
       "\t2 & m13.1 & 214.7 & 163.28 & 4.68 & 13.5 & 3.99 & 25.71 & 0.0 \\\\\n",
       "\\end{tabular}\n"
      ],
      "text/plain": [
       "\u001b[1m2×8 DataFrame\u001b[0m\n",
       "\u001b[1m Row \u001b[0m│\u001b[1m models \u001b[0m\u001b[1m WAIC    \u001b[0m\u001b[1m lppd    \u001b[0m\u001b[1m SE      \u001b[0m\u001b[1m dWAIC   \u001b[0m\u001b[1m dSE     \u001b[0m\u001b[1m pWAIC   \u001b[0m\u001b[1m weight  \u001b[0m\n",
       "\u001b[1m     \u001b[0m│\u001b[90m String \u001b[0m\u001b[90m Float64 \u001b[0m\u001b[90m Float64 \u001b[0m\u001b[90m Float64 \u001b[0m\u001b[90m Float64 \u001b[0m\u001b[90m Float64 \u001b[0m\u001b[90m Float64 \u001b[0m\u001b[90m Float64 \u001b[0m\n",
       "─────┼───────────────────────────────────────────────────────────────────────\n",
       "   1 │ m13.2     201.2   158.41     7.47      0.0     0.0     21.4       1.0\n",
       "   2 │ m13.1     214.7   163.28     4.68     13.5     3.99    25.71      0.0"
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     "execution_count": 5,
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   ],
   "source": [
    "link_fun = (r, dr) -> begin\n",
    "    a = get(r, \"a[$(dr.tank)]\", 0)\n",
    "    p = logistic(a)\n",
    "    binomlogpdf(dr.density, p, dr.surv)\n",
    "end\n",
    "\n",
    "m1_ll = link(m13_1_df, link_fun, eachrow(d))\n",
    "m1_ll = hcat(m1_ll...);\n",
    "\n",
    "m2_ll = link(m13_2_df, link_fun, eachrow(d))\n",
    "m2_ll = hcat(m2_ll...);\n",
    "\n",
    "compare([m1_ll, m2_ll], :waic, mnames=[\"m13.1\", \"m13.2\"])"
   ]
  },
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   "source": [
    "Code 13.5"
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      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Random.seed!()\n",
    "post = sample(resetrange(m13_2_ch), 10000)\n",
    "post_df = DataFrame(post)\n",
    "\n",
    "propsurv_est = [\n",
    "    logistic(mean(post_df[:,\"a[$i]\"]))\n",
    "    for i ∈ 1:nrow(d)\n",
    "]\n",
    "\n",
    "scatter(propsurv_est, mc=:white, xlab=\"tank\", ylab=\"proportion survival\", ylim=(-0.05, 1.05))\n",
    "scatter!(d.propsurv, mc=:blue, ms=3)\n",
    "hline!([mean(logistic.(post_df.ā))], ls=:dash, c=:black)\n",
    "vline!([16.5, 32.5], c=:black)\n",
    "annotate!([\n",
    "        (8, 0, (\"small tanks\", 10)),\n",
    "        (16+8, 0, (\"medium tanks\", 10)),\n",
    "        (32+8, 0, (\"large tanks\", 10))\n",
    "    ])"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e42dd7a9",
   "metadata": {},
   "source": [
    "Code 13.6"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "4b7c8b33",
   "metadata": {},
   "outputs": [
    {
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       "  1784.07,1389.31 1791.13,1389.32 1798.19,1389.32 1805.25,1389.32 1812.3,1389.31 1819.36,1389.29 1826.42,1389.25 1833.47,1389.19 1840.53,1389.1 1847.59,1388.95 \n",
       "  1854.64,1388.71 1861.7,1388.37 1868.76,1387.86 1875.81,1387.14 1882.87,1386.16 1889.93,1384.84 1896.98,1383.11 1904.04,1380.92 1911.1,1378.2 1918.15,1374.9 \n",
       "  1925.21,1371 1932.27,1366.49 1939.32,1361.4 1946.38,1355.76 1953.44,1349.65 1960.49,1343.15 1967.55,1336.38 1974.61,1329.43 1981.66,1322.43 1988.72,1315.47 \n",
       "  1995.78,1308.65 2002.83,1302.04 2009.89,1295.7 2016.95,1289.67 2024,1283.97 2031.06,1278.63 2038.12,1273.63 2045.17,1268.96 2052.23,1264.61 2059.29,1260.53 \n",
       "  2066.34,1256.7 2073.4,1253.07 2080.46,1249.6 2087.52,1246.27 2094.57,1243.03 2101.63,1239.88 2108.69,1236.8 2115.74,1233.81 2122.8,1230.91 2129.86,1228.12 \n",
       "  2136.91,1225.44 2143.97,1222.88 2151.03,1220.43 2158.08,1218.11 2165.14,1215.89 2172.2,1213.76 2179.25,1211.71 2186.31,1209.72 2193.37,1207.78 2200.42,1205.85 \n",
       "  2207.48,1203.9 2214.54,1201.89 2221.59,1199.78 2228.65,1197.52 2235.71,1195.07 2242.76,1192.39 2249.82,1189.47 2256.88,1186.3 2263.93,1182.91 2270.99,1179.33 \n",
       "  2278.05,1175.62 2285.1,1171.82 2292.16,1167.99 2299.22,1164.17 2306.27,1160.4 2313.33,1156.7 2320.39,1153.08 2327.44,1149.53 2334.5,1146.05 2341.56,1142.62 \n",
       "  2348.61,1139.24 2355.67,1135.89 2362.73,1132.55 2369.79,1129.22 2376.84,1125.89 2383.9,1122.55 2390.96,1119.21 2398.01,1115.86 2405.07,1112.49 2412.13,1109.1 \n",
       "  2419.18,1105.67 2426.24,1102.17 2433.3,1098.57 2440.35,1094.84 2447.41,1090.95 2454.47,1086.88 2461.52,1082.61 2468.58,1078.17 2475.64,1073.57 2482.69,1068.88 \n",
       "  2489.75,1064.13 2496.81,1059.37 2503.86,1054.64 2510.92,1049.95 2517.98,1045.29 2525.03,1040.64 2532.09,1035.93 2539.15,1031.1 2546.2,1026.07 2553.26,1020.77 \n",
       "  2560.32,1015.14 2567.37,1009.13 2574.43,1002.7 2581.49,995.838 2588.54,988.559 2595.6,980.883 2602.66,972.844 2609.71,964.485 2616.77,955.847 2623.83,946.971 \n",
       "  2630.88,937.892 2637.94,928.637 2645,919.234 2652.06,909.711 2659.11,900.103 2666.17,890.457 2673.23,880.833 2680.28,871.3 2687.34,861.929 2694.4,852.783 \n",
       "  2701.45,843.905 2708.51,835.306 2715.57,826.957 2722.62,818.786 2729.68,810.675 2736.74,802.468 2743.79,793.983 2750.85,785.023 2757.91,775.396 2764.96,764.926 \n",
       "  2772.02,753.468 2779.08,740.916 2786.13,727.208 2793.19,712.323 2800.25,696.276 2807.3,679.105 2814.36,660.868 2821.42,641.622 2828.47,621.421 2835.53,600.304 \n",
       "  2842.59,578.287 2849.64,555.358 2856.7,531.473 2863.76,506.556 2870.81,480.501 2877.87,453.186 2884.93,424.489 2891.98,394.315 2899.04,362.637 2906.1,329.541 \n",
       "  2913.15,295.278 2920.21,260.327 2927.27,225.446 2934.33,191.726 2941.38,160.611 2948.44,133.891 2955.5,113.645 2962.55,102.132 2969.61,101.623 2976.67,114.184 \n",
       "  2983.72,141.435 2990.78,184.317 2997.84,242.889 3004.89,316.218 3011.95,402.363 3019.01,498.491 3026.06,601.097 3033.12,706.309 3040.18,810.243 3047.23,909.344 \n",
       "  3054.29,1000.67 3061.35,1082.09 3068.4,1152.36 3075.46,1211.09 3082.52,1258.66 3089.57,1296 3096.63,1324.4 3103.69,1345.36 3110.74,1360.35 3117.8,1370.74 \n",
       "  3124.86,1377.73 3131.91,1382.3 3138.97,1385.18 3146.03,1386.95 3153.08,1388 3160.14,1388.61 3167.2,1388.95 3174.25,1389.14 3181.31,1389.24 3188.37,1389.28 \n",
       "  \n",
       "  \"/>\n",
       "</svg>\n"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p1 = plot(xlim=(-3, 4), xlab=\"log-odds survive\", ylab=\"Density\")\n",
    "\n",
    "for r ∈ first(eachrow(post_df), 100)\n",
    "    plot!(Normal(r.ā, r.σ), c=:black, alpha=0.2)\n",
    "end\n",
    "\n",
    "sim_tanks = @. rand(Normal(post_df.ā[1:8000], post_df.σ[1:8000]));\n",
    "p2 = plot(xlab=\"probability survive\", ylab=\"Density\", xlim=(-0.1, 1.1))\n",
    "density!(logistic.(sim_tanks), lw=2)\n",
    "\n",
    "plot(p1, p2, size=(800, 400), margin=2mm)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "267517dc",
   "metadata": {},
   "source": [
    "# 13.2 Varying effects and the underfitting/overfitting trade-off"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a496f7dd",
   "metadata": {},
   "source": [
    "Code 13.7"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "eae64343",
   "metadata": {},
   "outputs": [],
   "source": [
    "ā = 1.5\n",
    "σ = 1.5\n",
    "nponds = 60\n",
    "Ni = repeat([5, 10, 25, 35], inner=15);"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b34889e0",
   "metadata": {},
   "source": [
    "Code 13.8"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "6819c5e7",
   "metadata": {},
   "outputs": [],
   "source": [
    "Random.seed!(5005)\n",
    "a_pond = rand(Normal(ā, σ), nponds);"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8f362cfb",
   "metadata": {},
   "source": [
    "Code 13.9"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "92e33938",
   "metadata": {},
   "outputs": [],
   "source": [
    "dsim = DataFrame(pond=1:nponds, Ni=Ni, true_a=a_pond);"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c1e0373d",
   "metadata": {},
   "source": [
    "Code 13.10\n",
    "\n",
    "Doesn't make much sense in Julia, but anyways"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "20a013a0",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(UnitRange{Int64}, Vector{Int64})"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "typeof(1:3), typeof([1,2,3])"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "acc7358e",
   "metadata": {},
   "source": [
    "Code 13.11"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "95b53dd0",
   "metadata": {},
   "outputs": [],
   "source": [
    "Random.seed!(1)\n",
    "dsim.Si = @. rand(Binomial(dsim.Ni, logistic(dsim.true_a)));"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0da27a88",
   "metadata": {},
   "source": [
    "Code 13.12"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "73899251",
   "metadata": {},
   "outputs": [],
   "source": [
    "dsim.p_nopool = dsim.Si ./ dsim.Ni;"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "19b4eb80",
   "metadata": {},
   "source": [
    "Code 13.13"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "0f0f33db",
   "metadata": {},
   "outputs": [],
   "source": [
    "@model function m13_3(Si, Ni, pond)\n",
    "    σ ~ Exponential()\n",
    "    ā ~ Normal(0, 1.5)\n",
    "    a_pond ~ filldist(Normal(ā, σ), nponds)\n",
    "    p = logistic.(a_pond)\n",
    "    @. Si ~ Binomial(Ni, p)\n",
    "end\n",
    "\n",
    "Random.seed!(1)\n",
    "m13_3_ch = sample(m13_3(dsim.Si, dsim.Ni, dsim.pond), NUTS(), 1000)\n",
    "m13_3_df = DataFrame(m13_3_ch);"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "54317a8b",
   "metadata": {},
   "source": [
    "Code 13.14"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "7d8b444c",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "┌────────────┬────────────────────────────────────────────────────────────┐\n",
      "│\u001b[1m      param \u001b[0m│\u001b[1m    mean \u001b[0m\u001b[1m    std \u001b[0m\u001b[1m    5.5% \u001b[0m\u001b[1m     50% \u001b[0m\u001b[1m   94.5% \u001b[0m\u001b[1m      histogram \u001b[0m│\n",
      "├────────────┼────────────────────────────────────────────────────────────┤\n",
      "│ a_pond[10] │  2.7871  1.3037   0.8877    2.668   5.0628       ▁▂▆█▇▃▁▁▁ │\n",
      "│ a_pond[11] │  2.7634  1.2303    0.991   2.6744   4.9192       ▁▂▇█▇▃▂▁▁ │\n",
      "│ a_pond[12] │ -0.0117  0.8542  -1.2945  -0.0151   1.2766        ▁▁▃██▃▁▁ │\n",
      "│ a_pond[13] │  1.5263   0.988  -0.0357   1.5169   3.0946  ▁▁▂▄▇▇█▆▅▂▁▁▁▁ │\n",
      "│ a_pond[14] │  2.7699   1.234   1.0144   2.7041   4.8799       ▁▂▇█▇▃▁▁▁ │\n",
      "│ a_pond[15] │  -0.011  0.7978  -1.2401  -0.0285   1.2438    ▁▁▁▃▅██▅▃▁▁▁ │\n",
      "│ a_pond[16] │  2.1949  0.9062   0.9032   2.1323   3.6539   ▁▁▃▆██▆▄▂▁▁▁▁ │\n",
      "│ a_pond[17] │  2.1436  0.8338   0.9518   2.0839   3.6436     ▁▂▅▇█▅▃▂▁▁▁ │\n",
      "│ a_pond[18] │  2.1927  0.9286   0.7956   2.1145   3.7073  ▁▁▃▆██▆▅▂▁▁▁▁▁ │\n",
      "│ a_pond[19] │   2.175  0.8629   0.8435   2.1215   3.5511    ▁▁▃▅██▆▄▂▁▁▁ │\n",
      "│  a_pond[1] │ -1.6247  1.0774  -3.5333  -1.5273  -0.0803        ▁▁▂▅█▆▁▁ │\n",
      "│ a_pond[20] │  0.5822  0.6324   -0.382   0.5682   1.6015       ▁▁▅██▅▂▁▁ │\n",
      "│ a_pond[21] │   3.128  1.1414   1.4743   3.0138   5.1205        ▁▄█▇▄▂▁▁ │\n",
      "│ a_pond[22] │  2.1806  0.9028   0.8011   2.1086   3.7658  ▁▁▃▅██▆▄▂▁▁▁▁▁ │\n",
      "│ a_pond[23] │   1.508  0.7347    0.451   1.4471   2.8261     ▁▁▂▆█▇▄▃▁▁▁ │\n",
      "│ a_pond[24] │ -2.2068  0.9207  -3.8283  -2.0923  -0.8888  ▁▁▁▁▁▂▄▆██▆▃▁▁ │\n",
      "│ a_pond[25] │ -1.5408  0.7909  -2.9362  -1.4729  -0.3867     ▁▁▁▂▃▅██▇▃▁ │\n",
      "│ a_pond[26] │  0.5944  0.6518  -0.4036   0.5621   1.6623       ▁▁▅██▅▂▁▁ │\n",
      "│ a_pond[27] │  3.1811  1.1907   1.4076   3.0804   5.1803       ▁▄██▅▂▁▁▁ │\n",
      "│ a_pond[28] │   1.468  0.7848   0.3416   1.4274   2.7582   ▁▁▃▇██▄▃▁▁▁▁▁ │\n",
      "│ a_pond[29] │  3.1396  1.0943   1.5897   3.0105   4.9926        ▁▃█▇▄▁▁▁ │\n",
      "│  a_pond[2] │   2.756  1.1946   0.9952   2.6579   4.6886        ▂▆█▆▃▁▁▁ │\n",
      "│ a_pond[30] │  2.1475  0.9018   0.8159   2.0692   3.6994   ▁▁▃▇██▇▄▂▁▁▁▁ │\n",
      "│ a_pond[31] │   0.834  0.4247    0.187   0.8166   1.5212        ▁▁▄█▅▁▁▁ │\n",
      "│ a_pond[32] │  -1.697  0.5355  -2.5934  -1.6737  -0.8806        ▁▁▂▅██▂▁ │\n",
      "│ a_pond[33] │   0.826  0.4281   0.1572   0.8139   1.5343         ▁▁▄█▅▂▁ │\n",
      "│ a_pond[34] │  0.3391  0.4084  -0.2986   0.3346    0.992         ▁▃█▆▁▁▁ │\n",
      "│ a_pond[35] │  0.8264   0.403   0.1995   0.8195   1.4725   ▁▁▂▄▇██▆▅▃▁▁▁ │\n",
      "│ a_pond[36] │  2.0323  0.5928   1.1086    1.992   3.0308        ▁▁▄█▇▄▂▁ │\n",
      "│ a_pond[37] │  1.4378  0.4895   0.7066   1.3936   2.2765         ▁▄█▇▂▁▁ │\n",
      "│ a_pond[38] │  2.9338  0.8079   1.7247   2.8616   4.2329     ▁▁▃▇█▇▅▂▁▁▁ │\n",
      "│ a_pond[39] │  2.4147  0.6783   1.4143   2.3627   3.4993     ▁▂▆█▆▄▂▁▁▁▁ │\n",
      "│  a_pond[3] │   2.736  1.2323   0.8996   2.6837   4.8082        ▁▂▇█▇▄▁▁ │\n",
      "│ a_pond[40] │  1.4392  0.5257   0.6512     1.41   2.2673        ▁▁▄█▇▃▁▁ │\n",
      "│ a_pond[41] │  1.6916  0.5244   0.9124   1.6683   2.5374        ▁▂██▆▂▁▁ │\n",
      "│ a_pond[42] │  3.7426  1.0807    2.149   3.6661   5.4972        ▁▆█▇▃▁▁▁ │\n",
      "│ a_pond[43] │  3.7672  1.0446    2.268   3.6938   5.5001   ▁▁▃▇██▇▅▃▂▁▁▁ │\n",
      "│ a_pond[44] │   2.974  0.8327   1.7429   2.8818   4.4708    ▁▁▃▆█▆▄▃▂▁▁▁ │\n",
      "│ a_pond[45] │ -0.3246  0.4108  -0.9746  -0.3269   0.3301          ▁▁▅█▄▁ │\n",
      "│ a_pond[46] │  1.2722  0.4274   0.6247   1.2772   1.9499  ▁▁▁▂▅▆███▅▃▂▁▁ │\n",
      "│ a_pond[47] │  2.3605  0.5572    1.544   2.3244   3.2767       ▁▂▅█▆▃▁▁▁ │\n",
      "│ a_pond[48] │  0.4596  0.3525  -0.0991   0.4657   1.0399    ▁▁▁▃▅▇█▇▄▂▁▁ │\n",
      "│ a_pond[49] │  0.4414  0.3381  -0.0893   0.4476   0.9788   ▁▁▁▃▆██▇▄▂▁▁▁ │\n",
      "│  a_pond[4] │ -0.0099  0.8433  -1.3751   0.0001   1.3435  ▁▁▁▁▁▃▅██▆▂▁▁▁ │\n",
      "│ a_pond[50] │ -0.1176  0.3398  -0.6709  -0.1086   0.4311    ▁▁▃▅▇██▄▃▁▁▁ │\n",
      "│ a_pond[51] │ -0.4502  0.3454  -1.0103  -0.4489   0.0896    ▁▁▁▂▄▆██▆▃▁▁ │\n",
      "│ a_pond[52] │  0.1215  0.3239  -0.3864   0.1219   0.6283     ▁▁▂▄██▇▅▂▁▁ │\n",
      "│ a_pond[53] │  1.4367  0.4463   0.7709   1.4048   2.2121          ▁▃█▆▂▁ │\n",
      "│ a_pond[54] │  3.2507    0.84   2.0457   3.1751   4.6379   ▁▂▅█▇▆▄▂▁▁▁▁▁ │\n",
      "│ a_pond[55] │ -1.8379  0.4738  -2.6312  -1.8214  -1.1006          ▁▂▆█▄▁ │\n",
      "│ a_pond[56] │  1.2708  0.4044    0.662   1.2434   1.9225  ▁▁▂▃▆█▇▆▄▃▁▁▁▁ │\n",
      "│ a_pond[57] │  1.4354  0.4271   0.7706   1.4289   2.1378         ▁▃█▇▂▁▁ │\n",
      "│ a_pond[58] │  1.8178  0.4508   1.1293   1.7849    2.573  ▁▂▃▅███▆▄▄▂▁▁▁ │\n",
      "│ a_pond[59] │ -1.2641  0.3962  -1.9016  -1.2475  -0.6651         ▁▁▄█▄▁▁ │\n",
      "│  a_pond[5] │  2.6734  1.2277   0.8274   2.6071   4.7021       ▁▂▇█▆▃▁▁▁ │\n",
      "│ a_pond[60] │  3.9789  1.0396   2.4838   3.8705   5.8658        ▁▄█▆▃▁▁▁ │\n",
      "│  a_pond[6] │  0.7089  0.8728  -0.6575   0.7054   2.1006   ▁▁▁▃▅▇██▄▂▁▁▁ │\n",
      "│  a_pond[7] │  1.5054  0.9959  -0.0121   1.4492   3.2375   ▁▁▂▅███▇▄▂▁▁▁ │\n",
      "│  a_pond[8] │  2.7786   1.293   0.8393   2.6775    4.926      ▁▂▆█▆▃▁▁▁▁ │\n",
      "│  a_pond[9] │  0.7212  0.8901  -0.6507   0.7049   2.1479    ▁▁▂▆███▄▂▁▁▁ │\n",
      "│          ā │   1.338  0.2493   0.9468   1.3273   1.7511        ▁▂▆█▇▃▁▁ │\n",
      "│          σ │   1.719  0.2275   1.3806   1.6944   2.1208        ▁▂▆█▅▃▁▁ │\n",
      "└────────────┴────────────────────────────────────────────────────────────┘\n"
     ]
    }
   ],
   "source": [
    "precis(m13_3_df)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7c07863b",
   "metadata": {},
   "source": [
    "Code 13.15"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "38c449f1",
   "metadata": {},
   "outputs": [],
   "source": [
    "dsim.p_partpool = [\n",
    "    mean(logistic.(m13_3_df[:,\"a_pond[$i]\"]))\n",
    "    for i ∈ 1:nponds\n",
    "];"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "444b3e51",
   "metadata": {},
   "source": [
    "Code 13.16"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "2947abef",
   "metadata": {},
   "outputs": [],
   "source": [
    "dsim.p_true = logistic.(dsim.true_a);"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2f11cf15",
   "metadata": {},
   "source": [
    "Code 13.17"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "fc974cf6",
   "metadata": {},
   "outputs": [],
   "source": [
    "nopool_error = @. abs(dsim.p_nopool - dsim.p_true)\n",
    "partpool_error = @. abs(dsim.p_partpool - dsim.p_true);"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bf88b62f",
   "metadata": {},
   "source": [
    "Code 13.18"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "f317220e",
   "metadata": {},
   "outputs": [
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       "</svg>\n"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "scatter(nopool_error, xlab=\"pond\", ylab=\"absolute error\")\n",
    "scatter!(partpool_error, mc=:white)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "088ecb30",
   "metadata": {},
   "source": [
    "Code 13.19"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "ddfef911",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(\u001b[1m4×2 DataFrame\u001b[0m\n",
       "\u001b[1m Row \u001b[0m│\u001b[1m Ni    \u001b[0m\u001b[1m nopool_error_mean \u001b[0m\n",
       "\u001b[1m     \u001b[0m│\u001b[90m Int64 \u001b[0m\u001b[90m Float64           \u001b[0m\n",
       "─────┼──────────────────────────\n",
       "   1 │     5          0.147649\n",
       "   2 │    10          0.0649205\n",
       "   3 │    25          0.0689178\n",
       "   4 │    35          0.0477076, \u001b[1m4×2 DataFrame\u001b[0m\n",
       "\u001b[1m Row \u001b[0m│\u001b[1m Ni    \u001b[0m\u001b[1m partpool_error_mean \u001b[0m\n",
       "\u001b[1m     \u001b[0m│\u001b[90m Int64 \u001b[0m\u001b[90m Float64             \u001b[0m\n",
       "─────┼────────────────────────────\n",
       "   1 │     5            0.138281\n",
       "   2 │    10            0.0502345\n",
       "   3 │    25            0.0621743\n",
       "   4 │    35            0.0450389)"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dsim.nopool_error = nopool_error\n",
    "dsim.partpool_error = partpool_error\n",
    "\n",
    "gb = groupby(dsim, :Ni)\n",
    "nopool_avg = combine(gb, :nopool_error => mean)\n",
    "partpool_avg = combine(gb, :partpool_error => mean);\n",
    "nopool_avg, partpool_avg"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "da9e5a7b",
   "metadata": {},
   "source": [
    "Code 13.20"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "8d3668f1",
   "metadata": {},
   "outputs": [
    {
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      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ā = 1.5\n",
    "σ = 1.5\n",
    "nponds = 60\n",
    "Ni = repeat([5, 10, 25, 35], inner=15)\n",
    "a_pond = rand(Normal(ā, σ), nponds)\n",
    "\n",
    "dsim = DataFrame(pond=1:nponds, Ni=Ni, true_a=a_pond)\n",
    "dsim.Si = @. rand(Binomial(dsim.Ni, logistic(dsim.true_a)))\n",
    "dsim.p_nopool = dsim.Si ./ dsim.Ni\n",
    "\n",
    "m13_3_ch = sample(m13_3(dsim.Si, dsim.Ni, dsim.pond), NUTS(), 1000)\n",
    "m13_3_df = DataFrame(m13_3_ch)\n",
    "\n",
    "dsim.p_partpool = [\n",
    "    mean(logistic.(m13_3_df[:,\"a_pond[$i]\"]))\n",
    "    for i ∈ 1:nponds\n",
    "]\n",
    "dsim.p_true = logistic.(dsim.true_a)\n",
    "nopool_error = @. abs(dsim.p_nopool - dsim.p_true)\n",
    "partpool_error = @. abs(dsim.p_partpool - dsim.p_true)\n",
    "\n",
    "scatter(nopool_error, xlab=\"pond\", ylab=\"absolute error\")\n",
    "scatter!(partpool_error, mc=:white)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3620d906",
   "metadata": {},
   "source": [
    "# 13.3 More than one type of cluster"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7c414ef4",
   "metadata": {},
   "source": [
    "Code 13.21"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "id": "83ec0616",
   "metadata": {},
   "outputs": [],
   "source": [
    "d = DataFrame(CSV.File(\"data/chimpanzees.csv\", delim=';'));\n",
    "d.treatment = 1 .+ d.prosoc_left .+ 2*d.condition;\n",
    "\n",
    "Random.seed!(13)\n",
    "\n",
    "@model function m13_4(pulled_left, actor, block_id, treatment)\n",
    "    σ_a ~ Exponential()\n",
    "    σ_g ~ Exponential()\n",
    "    ā ~ Normal(0, 1.5)\n",
    "    actors_count = length(levels(actor))\n",
    "    blocks_count = length(levels(block_id))\n",
    "    treats_count = length(levels(treatment))\n",
    "    a ~ filldist(Normal(ā, σ_a), actors_count)\n",
    "    g ~ filldist(Normal(0, σ_g), blocks_count)\n",
    "    b ~ filldist(Normal(0, 0.5), treats_count)\n",
    "    \n",
    "    p = @. logistic(a[actor] + g[block_id] + b[treatment])\n",
    "    @. pulled_left ~ Binomial(1, p)\n",
    "end\n",
    "\n",
    "m13_4_ch = sample(m13_4(d.pulled_left, d.actor, d.block, d.treatment), NUTS(), 4000)\n",
    "m13_4_df = DataFrame(m13_4_ch);"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0d133853",
   "metadata": {},
   "source": [
    "Code 13.22"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "id": "34b369cd",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "┌───────┬────────────────────────────────────────────────────────────┐\n",
      "│\u001b[1m param \u001b[0m│\u001b[1m    mean \u001b[0m\u001b[1m    std \u001b[0m\u001b[1m    5.5% \u001b[0m\u001b[1m     50% \u001b[0m\u001b[1m   94.5% \u001b[0m\u001b[1m      histogram \u001b[0m│\n",
      "├───────┼────────────────────────────────────────────────────────────┤\n",
      "│  a[1] │ -0.3696  0.3736  -0.9527  -0.3614   0.2245         ▁▁▆█▃▁▁ │\n",
      "│  a[2] │  4.6715  1.2414   3.0805   4.4924   6.9113     ▁▁▇█▅▃▁▁▁▁▁ │\n",
      "│  a[3] │ -0.6737  0.3789  -1.2842  -0.6824  -0.0673          ▁▃█▅▁▁ │\n",
      "│  a[4] │ -0.6797  0.3725  -1.2943  -0.6711  -0.1029          ▁▃█▅▁▁ │\n",
      "│  a[5] │ -0.3664  0.3667  -0.9526  -0.3526   0.1959         ▁▁▆█▃▁▁ │\n",
      "│  a[6] │  0.5701  0.3695  -0.0321   0.5718   1.1599          ▁▂▇█▃▁ │\n",
      "│  a[7] │  2.1083  0.4609   1.3834      2.1    2.891         ▁▂▇█▄▁▁ │\n",
      "│  b[1] │  -0.119  0.3015  -0.5939  -0.1217   0.3687   ▁▁▁▂▄▇█▆▃▂▁▁▁ │\n",
      "│  b[2] │  0.4116  0.3054  -0.0736   0.4147   0.8785   ▁▁▁▃▆██▆▃▁▁▁▁ │\n",
      "│  b[3] │  -0.474  0.2976  -0.9366  -0.4818   0.0123    ▁▁▁▁▄▇█▇▄▂▁▁ │\n",
      "│  b[4] │  0.2868  0.2959  -0.1621    0.276   0.7579  ▁▁▁▂▄▇█▆▄▁▁▁▁▁ │\n",
      "│  g[1] │ -0.1717  0.2207  -0.5788  -0.1181    0.078    ▁▁▁▁▁▂▄█▄▁▁▁ │\n",
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      "│   σ_g │  0.2236  0.1785   0.0422   0.1829   0.5409    █▅▂▁▁▁▁▁▁▁▁▁ │\n",
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      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "precis(m13_4_df)\n",
    "coeftab_plot(m13_4_df, size=(800,600))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "562aeb17",
   "metadata": {},
   "source": [
    "Code 13.23"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "id": "f83a3fc8",
   "metadata": {},
   "outputs": [],
   "source": [
    "Random.seed!(14)\n",
    "\n",
    "@model function m13_5(pulled_left, actor, treatment)\n",
    "    σ_a ~ Exponential()\n",
    "    ā ~ Normal(0, 1.5)\n",
    "    actors_count = length(levels(actor))\n",
    "    treats_count = length(levels(treatment))\n",
    "    a ~ filldist(Normal(ā, σ_a), actors_count)\n",
    "    b ~ filldist(Normal(0, 0.5), treats_count)\n",
    "    \n",
    "    p = @. logistic(a[actor] + b[treatment])\n",
    "    @. pulled_left ~ Binomial(1, p)\n",
    "end\n",
    "\n",
    "m13_5_ch = sample(m13_5(d.pulled_left, d.actor, d.treatment), NUTS(), 4000)\n",
    "m13_5_df = DataFrame(m13_5_ch);"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6c624a4d",
   "metadata": {},
   "source": [
    "Code 13.24"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "id": "520d5352",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div class=\"data-frame\"><p>2 rows × 8 columns</p><table class=\"data-frame\"><thead><tr><th></th><th>models</th><th>WAIC</th><th>lppd</th><th>SE</th><th>dWAIC</th><th>dSE</th><th>pWAIC</th><th>weight</th></tr><tr><th></th><th title=\"String\">String</th><th title=\"Float64\">Float64</th><th title=\"Float64\">Float64</th><th title=\"Float64\">Float64</th><th title=\"Float64\">Float64</th><th title=\"Float64\">Float64</th><th title=\"Float64\">Float64</th><th title=\"Float64\">Float64</th></tr></thead><tbody><tr><th>1</th><td>m5</td><td>531.1</td><td>514.03</td><td>19.2</td><td>0.0</td><td>0.0</td><td>8.51</td><td>0.63</td></tr><tr><th>2</th><td>m4</td><td>532.2</td><td>510.79</td><td>19.43</td><td>1.1</td><td>1.72</td><td>10.72</td><td>0.37</td></tr></tbody></table></div>"
      ],
      "text/latex": [
       "\\begin{tabular}{r|cccccccc}\n",
       "\t& models & WAIC & lppd & SE & dWAIC & dSE & pWAIC & weight\\\\\n",
       "\t\\hline\n",
       "\t& String & Float64 & Float64 & Float64 & Float64 & Float64 & Float64 & Float64\\\\\n",
       "\t\\hline\n",
       "\t1 & m5 & 531.1 & 514.03 & 19.2 & 0.0 & 0.0 & 8.51 & 0.63 \\\\\n",
       "\t2 & m4 & 532.2 & 510.79 & 19.43 & 1.1 & 1.72 & 10.72 & 0.37 \\\\\n",
       "\\end{tabular}\n"
      ],
      "text/plain": [
       "\u001b[1m2×8 DataFrame\u001b[0m\n",
       "\u001b[1m Row \u001b[0m│\u001b[1m models \u001b[0m\u001b[1m WAIC    \u001b[0m\u001b[1m lppd    \u001b[0m\u001b[1m SE      \u001b[0m\u001b[1m dWAIC   \u001b[0m\u001b[1m dSE     \u001b[0m\u001b[1m pWAIC   \u001b[0m\u001b[1m weight  \u001b[0m\n",
       "\u001b[1m     \u001b[0m│\u001b[90m String \u001b[0m\u001b[90m Float64 \u001b[0m\u001b[90m Float64 \u001b[0m\u001b[90m Float64 \u001b[0m\u001b[90m Float64 \u001b[0m\u001b[90m Float64 \u001b[0m\u001b[90m Float64 \u001b[0m\u001b[90m Float64 \u001b[0m\n",
       "─────┼───────────────────────────────────────────────────────────────────────\n",
       "   1 │ m5        531.1   514.03    19.2       0.0     0.0      8.51     0.63\n",
       "   2 │ m4        532.2   510.79    19.43      1.1     1.72    10.72     0.37"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "l_fun = (r, dr) -> begin\n",
    "    a = get(r, \"a[$(dr.actor)]\", 0)\n",
    "    g = get(r, \"g[$(dr.block)]\", 0)\n",
    "    b = get(r, \"b[$(dr.treatment)]\", 0)\n",
    "    p = logistic(a + g + b)\n",
    "    binomlogpdf(1, p, dr.pulled_left)\n",
    "end\n",
    "\n",
    "m13_4_ll = link(m13_4_df, l_fun, eachrow(d))\n",
    "m13_4_ll = hcat(m13_4_ll...)\n",
    "\n",
    "l_fun = (r, dr) -> begin\n",
    "    a = get(r, \"a[$(dr.actor)]\", 0)\n",
    "    b = get(r, \"b[$(dr.treatment)]\", 0)\n",
    "    p = logistic(a + b)\n",
    "    binomlogpdf(1, p, dr.pulled_left)\n",
    "end\n",
    "\n",
    "m13_5_ll = link(m13_5_df, l_fun, eachrow(d))\n",
    "m13_5_ll = hcat(m13_5_ll...);\n",
    "\n",
    "compare([m13_4_ll, m13_5_ll], :waic, mnames=[\"m4\", \"m5\"])"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8e9557a7",
   "metadata": {},
   "source": [
    "Code 13.25"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "id": "1c4d2f1f",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "┌───────┬──────────────────────────────────────────────────────────┐\n",
      "│\u001b[1m param \u001b[0m│\u001b[1m   mean \u001b[0m\u001b[1m    std \u001b[0m\u001b[1m    5.5% \u001b[0m\u001b[1m     50% \u001b[0m\u001b[1m  94.5% \u001b[0m\u001b[1m      histogram \u001b[0m│\n",
      "├───────┼──────────────────────────────────────────────────────────┤\n",
      "│  b[1] │ -0.119  0.3015  -0.5939  -0.1217  0.3687   ▁▁▁▂▄▇█▆▃▂▁▁▁ │\n",
      "│  b[2] │ 0.4116  0.3054  -0.0736   0.4147  0.8785   ▁▁▁▃▆██▆▃▁▁▁▁ │\n",
      "│  b[3] │ -0.474  0.2976  -0.9366  -0.4818  0.0123    ▁▁▁▁▄▇█▇▄▂▁▁ │\n",
      "│  b[4] │ 0.2868  0.2959  -0.1621    0.276  0.7579  ▁▁▁▂▄▇█▆▄▁▁▁▁▁ │\n",
      "└───────┴──────────────────────────────────────────────────────────┘\n",
      "┌───────┬───────────────────────────────────────────────────────┐\n",
      "│\u001b[1m param \u001b[0m│\u001b[1m    mean \u001b[0m\u001b[1m    std \u001b[0m\u001b[1m    5.5% \u001b[0m\u001b[1m     50% \u001b[0m\u001b[1m  94.5% \u001b[0m\u001b[1m  histogram \u001b[0m│\n",
      "├───────┼───────────────────────────────────────────────────────┤\n",
      "│  b[1] │ -0.1116  0.3607  -0.6868  -0.1029  0.4391   ▁▁▂█▅▁▁▁▁ │\n",
      "│  b[2] │  0.3877  0.3609  -0.1153   0.3614  0.9866   ▁▁▂█▅▁▁▁▁ │\n",
      "│  b[3] │ -0.4464  0.3655  -1.0599   -0.422  0.0814    ▁▁▁▆█▂▁▁ │\n",
      "│  b[4] │  0.2806    0.36  -0.2171    0.246  0.8956    ▁▁▃█▃▁▁▁ │\n",
      "│   σ_b │  0.5973  0.3767   0.2072   0.5049  1.2311  █▇▂▁▁▁▁▁▁▁ │\n",
      "└───────┴───────────────────────────────────────────────────────┘\n"
     ]
    }
   ],
   "source": [
    "Random.seed!(15)\n",
    "\n",
    "@model function m13_6(pulled_left, actor, block_id, treatment)\n",
    "    σ_a ~ Exponential()\n",
    "    σ_g ~ Exponential()\n",
    "    σ_b ~ Exponential()\n",
    "    ā ~ Normal(0, 1.5)\n",
    "    actors_count = length(levels(actor))\n",
    "    blocks_count = length(levels(block_id))\n",
    "    treats_count = length(levels(treatment))\n",
    "    a ~ filldist(Normal(ā, σ_a), actors_count)\n",
    "    g ~ filldist(Normal(0, σ_g), blocks_count)\n",
    "    b ~ filldist(Normal(0, σ_b), treats_count)\n",
    "    \n",
    "    p = @. logistic(a[actor] + g[block_id] + b[treatment])\n",
    "    @. pulled_left ~ Binomial(1, p)\n",
    "end\n",
    "\n",
    "m13_6_ch = sample(m13_6(d.pulled_left, d.actor, d.block, d.treatment), NUTS(), 4000)\n",
    "m13_6_df = DataFrame(m13_6_ch);\n",
    "\n",
    "precis(m13_4_df[:,r\"b\"])\n",
    "precis(m13_6_df[:,r\"b\"])"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "130be50a",
   "metadata": {},
   "source": [
    "# 13.4 Divirgent transitions and non-centered priors"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1b7d8d66",
   "metadata": {},
   "source": [
    "Code 13.26"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "id": "c05a2fa3",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Chains MCMC chain (1000×14×1 Array{Float64, 3}):\n",
       "\n",
       "Iterations        = 501:1:1500\n",
       "Number of chains  = 1\n",
       "Samples per chain = 1000\n",
       "Wall duration     = 3.42 seconds\n",
       "Compute duration  = 3.42 seconds\n",
       "parameters        = v, x\n",
       "internals         = lp, n_steps, is_accept, acceptance_rate, log_density, hamiltonian_energy, hamiltonian_energy_error, max_hamiltonian_energy_error, tree_depth, numerical_error, step_size, nom_step_size\n",
       "\n",
       "Summary Statistics\n",
       " \u001b[1m parameters \u001b[0m \u001b[1m    mean \u001b[0m \u001b[1m      std \u001b[0m \u001b[1m naive_se \u001b[0m \u001b[1m    mcse \u001b[0m \u001b[1m      ess \u001b[0m \u001b[1m    rhat \u001b[0m \u001b[1m \u001b[0m ⋯\n",
       " \u001b[90m     Symbol \u001b[0m \u001b[90m Float64 \u001b[0m \u001b[90m  Float64 \u001b[0m \u001b[90m  Float64 \u001b[0m \u001b[90m Float64 \u001b[0m \u001b[90m  Float64 \u001b[0m \u001b[90m Float64 \u001b[0m \u001b[90m \u001b[0m ⋯\n",
       "\n",
       "           v    4.1270     1.3028     0.0412    0.2004    10.3585    1.0110    ⋯\n",
       "           x   30.6167   318.1477    10.0607   27.9820   141.9414    1.0026    ⋯\n",
       "\u001b[36m                                                                1 column omitted\u001b[0m\n",
       "\n",
       "Quantiles\n",
       " \u001b[1m parameters \u001b[0m \u001b[1m      2.5% \u001b[0m \u001b[1m    25.0% \u001b[0m \u001b[1m   50.0% \u001b[0m \u001b[1m   75.0% \u001b[0m \u001b[1m    97.5% \u001b[0m\n",
       " \u001b[90m     Symbol \u001b[0m \u001b[90m   Float64 \u001b[0m \u001b[90m  Float64 \u001b[0m \u001b[90m Float64 \u001b[0m \u001b[90m Float64 \u001b[0m \u001b[90m  Float64 \u001b[0m\n",
       "\n",
       "           v      2.7433     3.0582    3.7470    4.9860     7.2314\n",
       "           x   -475.4665   -14.7917   -2.4200   21.4529   766.0837\n"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "@model function m13_7(N)\n",
    "    v ~ Normal(0, 3)\n",
    "    x ~ Normal(0, exp(v))\n",
    "end\n",
    "\n",
    "Random.seed!(5)\n",
    "m13_7_ch = sample(m13_7(1), NUTS(), 1000)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "dccb3b89",
   "metadata": {},
   "source": [
    "Code 13.27"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "id": "01ee230f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Chains MCMC chain (1000×14×1 Array{Float64, 3}):\n",
       "\n",
       "Iterations        = 501:1:1500\n",
       "Number of chains  = 1\n",
       "Samples per chain = 1000\n",
       "Wall duration     = 2.92 seconds\n",
       "Compute duration  = 2.92 seconds\n",
       "parameters        = v, z\n",
       "internals         = lp, n_steps, is_accept, acceptance_rate, log_density, hamiltonian_energy, hamiltonian_energy_error, max_hamiltonian_energy_error, tree_depth, numerical_error, step_size, nom_step_size\n",
       "\n",
       "Summary Statistics\n",
       " \u001b[1m parameters \u001b[0m \u001b[1m    mean \u001b[0m \u001b[1m     std \u001b[0m \u001b[1m naive_se \u001b[0m \u001b[1m    mcse \u001b[0m \u001b[1m      ess \u001b[0m \u001b[1m    rhat \u001b[0m \u001b[1m e\u001b[0m ⋯\n",
       " \u001b[90m     Symbol \u001b[0m \u001b[90m Float64 \u001b[0m \u001b[90m Float64 \u001b[0m \u001b[90m  Float64 \u001b[0m \u001b[90m Float64 \u001b[0m \u001b[90m  Float64 \u001b[0m \u001b[90m Float64 \u001b[0m \u001b[90m  \u001b[0m ⋯\n",
       "\n",
       "           v   -0.0217    2.9165     0.0922    0.1000   975.4488    1.0034     ⋯\n",
       "           z   -0.0017    1.0110     0.0320    0.0324   928.7026    0.9995     ⋯\n",
       "\u001b[36m                                                                1 column omitted\u001b[0m\n",
       "\n",
       "Quantiles\n",
       " \u001b[1m parameters \u001b[0m \u001b[1m    2.5% \u001b[0m \u001b[1m   25.0% \u001b[0m \u001b[1m   50.0% \u001b[0m \u001b[1m   75.0% \u001b[0m \u001b[1m   97.5% \u001b[0m\n",
       " \u001b[90m     Symbol \u001b[0m \u001b[90m Float64 \u001b[0m \u001b[90m Float64 \u001b[0m \u001b[90m Float64 \u001b[0m \u001b[90m Float64 \u001b[0m \u001b[90m Float64 \u001b[0m\n",
       "\n",
       "           v   -5.8505   -1.9989    0.0482    1.8963    5.6774\n",
       "           z   -1.9249   -0.6809   -0.0099    0.6728    1.9642\n"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "@model function m13_7nc(N)\n",
    "    v ~ Normal(0, 3)\n",
    "    z ~ Normal()\n",
    "    x = z * exp(v)\n",
    "end\n",
    "\n",
    "Random.seed!(5)\n",
    "m13_7nc_ch = sample(m13_7nc(1), NUTS(), 1000)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1fc4c1c2",
   "metadata": {},
   "source": [
    "Code 13.28\n",
    "\n",
    "There is no way to get amount of divergent samples, but they could be estimated by comparing `ess` values from the chain"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "id": "276ee9f0",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Chains MCMC chain (4000×32×1 Array{Float64, 3}):\n",
       "\n",
       "Iterations        = 1001:1:5000\n",
       "Number of chains  = 1\n",
       "Samples per chain = 4000\n",
       "Wall duration     = 22.54 seconds\n",
       "Compute duration  = 22.54 seconds\n",
       "parameters        = σ_a, σ_g, ā, a[1], a[2], a[3], a[4], a[5], a[6], a[7], g[1], g[2], g[3], g[4], g[5], g[6], b[1], b[2], b[3], b[4]\n",
       "internals         = lp, n_steps, is_accept, acceptance_rate, log_density, hamiltonian_energy, hamiltonian_energy_error, max_hamiltonian_energy_error, tree_depth, numerical_error, step_size, nom_step_size\n",
       "\n",
       "Summary Statistics\n",
       " \u001b[1m parameters \u001b[0m \u001b[1m    mean \u001b[0m \u001b[1m     std \u001b[0m \u001b[1m naive_se \u001b[0m \u001b[1m    mcse \u001b[0m \u001b[1m       ess \u001b[0m \u001b[1m    rhat \u001b[0m \u001b[1m \u001b[0m ⋯\n",
       " \u001b[90m     Symbol \u001b[0m \u001b[90m Float64 \u001b[0m \u001b[90m Float64 \u001b[0m \u001b[90m  Float64 \u001b[0m \u001b[90m Float64 \u001b[0m \u001b[90m   Float64 \u001b[0m \u001b[90m Float64 \u001b[0m \u001b[90m \u001b[0m ⋯\n",
       "\n",
       "         σ_a    2.0123    0.6448     0.0102    0.0154   1928.9317    0.9998    ⋯\n",
       "         σ_g    0.2254    0.1768     0.0028    0.0072    655.0168    0.9998    ⋯\n",
       "           ā    0.5905    0.7271     0.0115    0.0172   1668.0941    1.0004    ⋯\n",
       "        a[1]   -0.3596    0.3647     0.0058    0.0095    982.0526    1.0025    ⋯\n",
       "        a[2]    4.6656    1.2885     0.0204    0.0296   1481.5114    1.0000    ⋯\n",
       "        a[3]   -0.6614    0.3712     0.0059    0.0092   1108.9323    1.0016    ⋯\n",
       "        a[4]   -0.6603    0.3602     0.0057    0.0094   1006.5836    1.0025    ⋯\n",
       "        a[5]   -0.3495    0.3711     0.0059    0.0088   1061.5711    1.0021    ⋯\n",
       "        a[6]    0.5874    0.3676     0.0058    0.0089   1079.3178    1.0016    ⋯\n",
       "        a[7]    2.1131    0.4571     0.0072    0.0109   1248.0935    1.0010    ⋯\n",
       "        g[1]   -0.1777    0.2190     0.0035    0.0069   1138.3718    0.9998    ⋯\n",
       "        g[2]    0.0429    0.1864     0.0029    0.0042   2022.7685    0.9998    ⋯\n",
       "        g[3]    0.0554    0.1834     0.0029    0.0040   2201.4475    0.9999    ⋯\n",
       "        g[4]    0.0107    0.1906     0.0030    0.0037   2323.5314    0.9998    ⋯\n",
       "        g[5]   -0.0337    0.1789     0.0028    0.0036   2211.6014    0.9998    ⋯\n",
       "        g[6]    0.1166    0.1983     0.0031    0.0052   1464.2777    0.9998    ⋯\n",
       "        b[1]   -0.1316    0.2961     0.0047    0.0074   1053.2554    1.0014    ⋯\n",
       "        b[2]    0.3919    0.3039     0.0048    0.0074    999.4859    1.0040    ⋯\n",
       "        b[3]   -0.4786    0.3022     0.0048    0.0078   1044.5529    1.0006    ⋯\n",
       "        b[4]    0.2745    0.3002     0.0047    0.0079   1003.2132    1.0016    ⋯\n",
       "\u001b[36m                                                                1 column omitted\u001b[0m\n",
       "\n",
       "Quantiles\n",
       " \u001b[1m parameters \u001b[0m \u001b[1m    2.5% \u001b[0m \u001b[1m   25.0% \u001b[0m \u001b[1m   50.0% \u001b[0m \u001b[1m   75.0% \u001b[0m \u001b[1m   97.5% \u001b[0m\n",
       " \u001b[90m     Symbol \u001b[0m \u001b[90m Float64 \u001b[0m \u001b[90m Float64 \u001b[0m \u001b[90m Float64 \u001b[0m \u001b[90m Float64 \u001b[0m \u001b[90m Float64 \u001b[0m\n",
       "\n",
       "         σ_a    1.0938    1.5534    1.9003    2.3560    3.5495\n",
       "         σ_g    0.0257    0.1006    0.1860    0.2985    0.6869\n",
       "           ā   -0.9008    0.1319    0.5915    1.0443    1.9871\n",
       "        a[1]   -1.0718   -0.6056   -0.3566   -0.1153    0.3601\n",
       "        a[2]    2.7792    3.7892    4.4519    5.3011    7.8573\n",
       "        a[3]   -1.3976   -0.8984   -0.6568   -0.4202    0.0797\n",
       "        a[4]   -1.3599   -0.9004   -0.6646   -0.4125    0.0514\n",
       "        a[5]   -1.0684   -0.5981   -0.3589   -0.1002    0.3986\n",
       "        a[6]   -0.1205    0.3389    0.5795    0.8366    1.3073\n",
       "        a[7]    1.2231    1.7957    2.1121    2.4170    3.0189\n",
       "        g[1]   -0.7116   -0.3000   -0.1309   -0.0192    0.1410\n",
       "        g[2]   -0.3066   -0.0561    0.0255    0.1351    0.4609\n",
       "        g[3]   -0.3032   -0.0439    0.0323    0.1527    0.4739\n",
       "        g[4]   -0.3961   -0.0746    0.0062    0.0985    0.4263\n",
       "        g[5]   -0.4315   -0.1180   -0.0176    0.0572    0.3120\n",
       "        g[6]   -0.2165   -0.0089    0.0744    0.2209    0.5842\n",
       "        b[1]   -0.7123   -0.3282   -0.1300    0.0674    0.4568\n",
       "        b[2]   -0.2180    0.1924    0.3873    0.5893    0.9994\n",
       "        b[3]   -1.0814   -0.6817   -0.4807   -0.2661    0.0956\n",
       "        b[4]   -0.3056    0.0748    0.2815    0.4757    0.8592\n"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Random.seed!(13)\n",
    "m13_4b_ch = sample(m13_4(d.pulled_left, d.actor, d.block, d.treatment), NUTS(0.95, init_ϵ=0.1), 4000)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "id": "2c6f498e",
   "metadata": {},
   "outputs": [
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      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "t = ess_rhat(m13_4_ch)\n",
    "ess_4 = t[:,:ess]\n",
    "t = ess_rhat(m13_4b_ch)\n",
    "ess_4b = t[:,:ess]\n",
    "\n",
    "plot(ess_4, lw=2, label=\"ESS m13_4\")\n",
    "plot!(ess_4b, lw=2, label=\"ESS m13_4b\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5c701199",
   "metadata": {},
   "source": [
    "Code 13.29"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "id": "04e5a040",
   "metadata": {},
   "outputs": [],
   "source": [
    "Random.seed!(13)\n",
    "\n",
    "@model function m13_4nc(pulled_left, actor, block_id, treatment)\n",
    "    σ_a ~ Exponential()\n",
    "    σ_g ~ Exponential()\n",
    "    ā ~ Normal(0, 1.5)\n",
    "    actors_count = length(levels(actor))\n",
    "    blocks_count = length(levels(block_id))\n",
    "    treats_count = length(levels(treatment))\n",
    "    b ~ filldist(Normal(0, 0.5), treats_count)\n",
    "    z ~ filldist(Normal(), actors_count)\n",
    "    x ~ filldist(Normal(), blocks_count)\n",
    "    a = @. ā + σ_a*z\n",
    "    g = σ_g*x\n",
    "    \n",
    "    p = @. logistic(a[actor] + g[block_id] + b[treatment])\n",
    "    @. pulled_left ~ Binomial(1, p)\n",
    "end\n",
    "\n",
    "m13_4nc_ch = sample(m13_4nc(d.pulled_left, d.actor, d.block, d.treatment), NUTS(), 4000);"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b2d2f789",
   "metadata": {},
   "source": [
    "Code 13.30"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "id": "cf051185",
   "metadata": {},
   "outputs": [
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       "  1114.87,916.963 1195.73,833.028 1277.07,748.595 1367.01,655.243 1452.58,566.42 1541.55,474.066 1616.45,396.323 1707.89,301.408 1793.85,212.185 1931.56,69.2423 \n",
       "  1952.76,47.2441 \n",
       "  \"/>\n",
       "</svg>\n"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "t = ess_rhat(m13_4_ch)\n",
    "ess_4 = t[:,:ess]\n",
    "t = ess_rhat(m13_4nc_ch)\n",
    "ess_4nc = t[:,:ess]\n",
    "\n",
    "lims = extrema(vcat(ess_4, ess_4nc)) .+ (-100, 100)\n",
    "plot(xlim=lims, ylims=lims, xlab=\"n_eff (centered)\", ylab=\"n_eff (non-centered)\", size=(500,500))\n",
    "scatter!(ess_4, ess_4nc)\n",
    "plot!(identity, c=:gray, s=:dash)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "72e53a33",
   "metadata": {},
   "source": [
    "# 13.5 Multilevel posterior predictions"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b48545ff",
   "metadata": {},
   "source": [
    "Code 13.31"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "id": "18d076e9",
   "metadata": {},
   "outputs": [],
   "source": [
    "chimp = 2\n",
    "d_pred = DataFrame(\n",
    "    actor = fill(chimp, 4),\n",
    "    treatment = 1:4,\n",
    "    block = fill(1, 4)\n",
    ")\n",
    "\n",
    "l_fun = (r, dr) -> begin\n",
    "    a = get(r, \"a[$(dr.actor)]\", 0)\n",
    "    g = get(r, \"g[$(dr.block)]\", 0)\n",
    "    b = get(r, \"b[$(dr.treatment)]\", 0)\n",
    "    logistic(a + g + b)\n",
    "end\n",
    "\n",
    "p = link(m13_4_df, l_fun, eachrow(d_pred))\n",
    "p = hcat(p...)\n",
    "p_μ = mean.(eachcol(p))\n",
    "p_ci = PI.(eachcol(p));"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3f3fe05f",
   "metadata": {},
   "source": [
    "Code 13.32"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "id": "ce41f848",
   "metadata": {},
   "outputs": [
    {
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       "<div class=\"data-frame\"><p>20 rows × 7 columns</p><table class=\"data-frame\"><thead><tr><th></th><th>variable</th><th>mean</th><th>min</th><th>median</th><th>max</th><th>nmissing</th><th>eltype</th></tr><tr><th></th><th title=\"Symbol\">Symbol</th><th title=\"Float64\">Float64</th><th title=\"Float64\">Float64</th><th title=\"Float64\">Float64</th><th title=\"Float64\">Float64</th><th title=\"Int64\">Int64</th><th title=\"DataType\">DataType</th></tr></thead><tbody><tr><th>1</th><td>a[1]</td><td>-0.362603</td><td>-1.74662</td><td>-0.343984</td><td>1.16849</td><td>0</td><td>Float64</td></tr><tr><th>2</th><td>a[2]</td><td>4.6697</td><td>1.99411</td><td>4.47706</td><td>11.9458</td><td>0</td><td>Float64</td></tr><tr><th>3</th><td>a[3]</td><td>-0.658423</td><td>-1.95394</td><td>-0.672465</td><td>0.759445</td><td>0</td><td>Float64</td></tr><tr><th>4</th><td>a[4]</td><td>-0.669022</td><td>-1.94008</td><td>-0.667599</td><td>0.702358</td><td>0</td><td>Float64</td></tr><tr><th>5</th><td>a[5]</td><td>-0.36085</td><td>-1.61727</td><td>-0.360718</td><td>0.808689</td><td>0</td><td>Float64</td></tr><tr><th>6</th><td>a[6]</td><td>0.582945</td><td>-0.568726</td><td>0.584092</td><td>1.77474</td><td>0</td><td>Float64</td></tr><tr><th>7</th><td>a[7]</td><td>2.12467</td><td>0.73358</td><td>2.12324</td><td>3.41372</td><td>0</td><td>Float64</td></tr><tr><th>8</th><td>b[1]</td><td>-0.125846</td><td>-1.19189</td><td>-0.123159</td><td>0.838821</td><td>0</td><td>Float64</td></tr><tr><th>9</th><td>b[2]</td><td>0.406341</td><td>-0.594921</td><td>0.409354</td><td>1.60454</td><td>0</td><td>Float64</td></tr><tr><th>10</th><td>b[3]</td><td>-0.483899</td><td>-1.71984</td><td>-0.487185</td><td>0.435965</td><td>0</td><td>Float64</td></tr><tr><th>11</th><td>b[4]</td><td>0.283934</td><td>-0.838933</td><td>0.273036</td><td>1.73403</td><td>0</td><td>Float64</td></tr><tr><th>12</th><td>g[1]</td><td>-0.166633</td><td>-1.37376</td><td>-0.117014</td><td>0.451782</td><td>0</td><td>Float64</td></tr><tr><th>13</th><td>g[2]</td><td>0.0394322</td><td>-0.80255</td><td>0.0205989</td><td>0.77087</td><td>0</td><td>Float64</td></tr><tr><th>14</th><td>g[3]</td><td>0.047988</td><td>-0.71706</td><td>0.033677</td><td>0.823652</td><td>0</td><td>Float64</td></tr><tr><th>15</th><td>g[4]</td><td>0.00900076</td><td>-1.07557</td><td>0.00703865</td><td>1.28219</td><td>0</td><td>Float64</td></tr><tr><th>16</th><td>g[5]</td><td>-0.0321617</td><td>-0.828938</td><td>-0.0175807</td><td>0.729183</td><td>0</td><td>Float64</td></tr><tr><th>17</th><td>g[6]</td><td>0.108842</td><td>-0.519147</td><td>0.082075</td><td>1.17474</td><td>0</td><td>Float64</td></tr><tr><th>18</th><td>ā</td><td>0.58869</td><td>-2.63128</td><td>0.607964</td><td>2.62595</td><td>0</td><td>Float64</td></tr><tr><th>19</th><td>σ_a</td><td>2.01588</td><td>0.792844</td><td>1.91205</td><td>5.93771</td><td>0</td><td>Float64</td></tr><tr><th>20</th><td>σ_g</td><td>0.21504</td><td>0.0285691</td><td>0.178079</td><td>2.2755</td><td>0</td><td>Float64</td></tr></tbody></table></div>"
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       "\\begin{tabular}{r|ccccccc}\n",
       "\t& variable & mean & min & median & max & nmissing & eltype\\\\\n",
       "\t\\hline\n",
       "\t& Symbol & Float64 & Float64 & Float64 & Float64 & Int64 & DataType\\\\\n",
       "\t\\hline\n",
       "\t1 & a[1] & -0.362603 & -1.74662 & -0.343984 & 1.16849 & 0 & Float64 \\\\\n",
       "\t2 & a[2] & 4.6697 & 1.99411 & 4.47706 & 11.9458 & 0 & Float64 \\\\\n",
       "\t3 & a[3] & -0.658423 & -1.95394 & -0.672465 & 0.759445 & 0 & Float64 \\\\\n",
       "\t4 & a[4] & -0.669022 & -1.94008 & -0.667599 & 0.702358 & 0 & Float64 \\\\\n",
       "\t5 & a[5] & -0.36085 & -1.61727 & -0.360718 & 0.808689 & 0 & Float64 \\\\\n",
       "\t6 & a[6] & 0.582945 & -0.568726 & 0.584092 & 1.77474 & 0 & Float64 \\\\\n",
       "\t7 & a[7] & 2.12467 & 0.73358 & 2.12324 & 3.41372 & 0 & Float64 \\\\\n",
       "\t8 & b[1] & -0.125846 & -1.19189 & -0.123159 & 0.838821 & 0 & Float64 \\\\\n",
       "\t9 & b[2] & 0.406341 & -0.594921 & 0.409354 & 1.60454 & 0 & Float64 \\\\\n",
       "\t10 & b[3] & -0.483899 & -1.71984 & -0.487185 & 0.435965 & 0 & Float64 \\\\\n",
       "\t11 & b[4] & 0.283934 & -0.838933 & 0.273036 & 1.73403 & 0 & Float64 \\\\\n",
       "\t12 & g[1] & -0.166633 & -1.37376 & -0.117014 & 0.451782 & 0 & Float64 \\\\\n",
       "\t13 & g[2] & 0.0394322 & -0.80255 & 0.0205989 & 0.77087 & 0 & Float64 \\\\\n",
       "\t14 & g[3] & 0.047988 & -0.71706 & 0.033677 & 0.823652 & 0 & Float64 \\\\\n",
       "\t15 & g[4] & 0.00900076 & -1.07557 & 0.00703865 & 1.28219 & 0 & Float64 \\\\\n",
       "\t16 & g[5] & -0.0321617 & -0.828938 & -0.0175807 & 0.729183 & 0 & Float64 \\\\\n",
       "\t17 & g[6] & 0.108842 & -0.519147 & 0.082075 & 1.17474 & 0 & Float64 \\\\\n",
       "\t18 & ā & 0.58869 & -2.63128 & 0.607964 & 2.62595 & 0 & Float64 \\\\\n",
       "\t19 & σ\\_a & 2.01588 & 0.792844 & 1.91205 & 5.93771 & 0 & Float64 \\\\\n",
       "\t20 & σ\\_g & 0.21504 & 0.0285691 & 0.178079 & 2.2755 & 0 & Float64 \\\\\n",
       "\\end{tabular}\n"
      ],
      "text/plain": [
       "\u001b[1m20×7 DataFrame\u001b[0m\n",
       "\u001b[1m Row \u001b[0m│\u001b[1m variable \u001b[0m\u001b[1m mean        \u001b[0m\u001b[1m min        \u001b[0m\u001b[1m median      \u001b[0m\u001b[1m max       \u001b[0m\u001b[1m nmissing \u001b[0m\u001b[1m el\u001b[0m ⋯\n",
       "\u001b[1m     \u001b[0m│\u001b[90m Symbol   \u001b[0m\u001b[90m Float64     \u001b[0m\u001b[90m Float64    \u001b[0m\u001b[90m Float64     \u001b[0m\u001b[90m Float64   \u001b[0m\u001b[90m Int64    \u001b[0m\u001b[90m Da\u001b[0m ⋯\n",
       "─────┼──────────────────────────────────────────────────────────────────────────\n",
       "   1 │ a[1]      -0.362603    -1.74662    -0.343984     1.16849          0  Fl ⋯\n",
       "   2 │ a[2]       4.6697       1.99411     4.47706     11.9458           0  Fl\n",
       "   3 │ a[3]      -0.658423    -1.95394    -0.672465     0.759445         0  Fl\n",
       "   4 │ a[4]      -0.669022    -1.94008    -0.667599     0.702358         0  Fl\n",
       "   5 │ a[5]      -0.36085     -1.61727    -0.360718     0.808689         0  Fl ⋯\n",
       "   6 │ a[6]       0.582945    -0.568726    0.584092     1.77474          0  Fl\n",
       "   7 │ a[7]       2.12467      0.73358     2.12324      3.41372          0  Fl\n",
       "   8 │ b[1]      -0.125846    -1.19189    -0.123159     0.838821         0  Fl\n",
       "   9 │ b[2]       0.406341    -0.594921    0.409354     1.60454          0  Fl ⋯\n",
       "  10 │ b[3]      -0.483899    -1.71984    -0.487185     0.435965         0  Fl\n",
       "  11 │ b[4]       0.283934    -0.838933    0.273036     1.73403          0  Fl\n",
       "  12 │ g[1]      -0.166633    -1.37376    -0.117014     0.451782         0  Fl\n",
       "  13 │ g[2]       0.0394322   -0.80255     0.0205989    0.77087          0  Fl ⋯\n",
       "  14 │ g[3]       0.047988    -0.71706     0.033677     0.823652         0  Fl\n",
       "  15 │ g[4]       0.00900076  -1.07557     0.00703865   1.28219          0  Fl\n",
       "  16 │ g[5]      -0.0321617   -0.828938   -0.0175807    0.729183         0  Fl\n",
       "  17 │ g[6]       0.108842    -0.519147    0.082075     1.17474          0  Fl ⋯\n",
       "  18 │ ā          0.58869     -2.63128     0.607964     2.62595          0  Fl\n",
       "  19 │ σ_a        2.01588      0.792844    1.91205      5.93771          0  Fl\n",
       "  20 │ σ_g        0.21504      0.0285691   0.178079     2.2755           0  Fl\n",
       "\u001b[36m                                                                1 column omitted\u001b[0m"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "post = sample(resetrange(m13_4_ch), 2000)\n",
    "post = DataFrame(post)\n",
    "describe(post)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a9a1881a",
   "metadata": {},
   "source": [
    "Code 13.33"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "id": "c48dcd3b",
   "metadata": {},
   "outputs": [
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       "  1993.59,1411.48 2003.86,1416.61 2014.13,1420.95 2024.4,1424.63 2034.68,1427.75 2044.95,1430.42 2055.22,1432.7 2065.49,1434.66 2075.76,1436.34 2086.03,1437.78 \n",
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       "  \n",
       "  \"/>\n",
       "</svg>\n"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "density(post.\"a[5]\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "44324743",
   "metadata": {},
   "source": [
    "Code 13.34"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "id": "b3b8d97e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "#15 (generic function with 1 method)"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p_link = (treatment, actor, block_id) -> begin\n",
    "    logodds = \n",
    "        getproperty(post, \"a[$actor]\") + \n",
    "        getproperty(post, \"b[$block_id]\") + \n",
    "        getproperty(post, \"g[$treatment]\")\n",
    "    logistic.(logodds)\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "02074fbf",
   "metadata": {},
   "source": [
    "Code 13.35"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "id": "ba5562a9",
   "metadata": {},
   "outputs": [],
   "source": [
    "p_raw = p_link.(1:4, 2, 1)\n",
    "p_raw = hcat(p_raw...)\n",
    "p_μ = mean.(eachcol(p_raw))\n",
    "p_ci = PI.(eachcol(p_raw));"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "effaa4f1",
   "metadata": {},
   "source": [
    "Code 13.36"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "id": "31a22fa8",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "#17 (generic function with 1 method)"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p_link_abar = treatment -> begin\n",
    "    logodds = \n",
    "        post.ā + getproperty(post, \"b[$treatment]\")\n",
    "    logistic.(logodds)\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "15b20b5d",
   "metadata": {},
   "source": [
    "Code 13.37"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "id": "3a33ce93",
   "metadata": {},
   "outputs": [
    {
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     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p_raw = p_link_abar.(1:4)\n",
    "p_raw = hcat(p_raw...)\n",
    "p_μ = mean.(eachcol(p_raw))\n",
    "p_ci = PI.(eachcol(p_raw))\n",
    "p_ci = vcat(p_ci'...)\n",
    "\n",
    "plot(xlab=\"treatment\", ylab=\"proportion pulled left\", title=\"average actor\", ylim=(0, 1))\n",
    "plot!([\"R/N\", \"L/N\", \"R/P\", \"L/P\"], [p_μ p_μ], fillrange=p_ci, fillalpha=0.2, c=:black, lw=1.5)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1c03657f",
   "metadata": {},
   "source": [
    "Code 13.38"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "id": "cd44ea84",
   "metadata": {},
   "outputs": [
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     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Random.seed!(1)\n",
    "a_sim = rand.(Normal.(post.ā, post.σ_a))\n",
    "\n",
    "p_link_asim = treatment -> begin\n",
    "    logodds = \n",
    "        a_sim + getproperty(post, \"b[$treatment]\")\n",
    "    logistic.(logodds)\n",
    "end\n",
    "\n",
    "p_raw_asim = p_link_asim.(1:4)\n",
    "p_raw_asim = hcat(p_raw_asim...)\n",
    "p_μ = mean.(eachcol(p_raw_asim))\n",
    "p_ci = PI.(eachcol(p_raw_asim))\n",
    "p_ci = vcat(p_ci'...)\n",
    "\n",
    "plot(xlab=\"treatment\", ylab=\"proportion pulled left\", title=\"marginal of actor\", ylim=(0, 1))\n",
    "plot!([\"R/N\", \"L/N\", \"R/P\", \"L/P\"], [p_μ p_μ], fillrange=p_ci, fillalpha=0.2, c=:black, lw=1.5)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "85f52a8f",
   "metadata": {},
   "source": [
    "Code 13.39"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "id": "d54a6e00",
   "metadata": {},
   "outputs": [
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       "  280.231,148.462 950.951,138.553 1621.67,157.332 2292.39,140.548 \n",
       "  \"/>\n",
       "</svg>\n"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p = plot(xlab=\"treatment\", ylab=\"proportion pulled left\", title=\"simulated actors\", ylim=(0, 1))\n",
    "\n",
    "for r in first(eachrow(p_raw_asim), 100)\n",
    "    plot!([\"R/N\", \"L/N\", \"R/P\", \"L/P\"], r, c=:black, alpha=0.2)\n",
    "end\n",
    "p"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "bce74423",
   "metadata": {},
   "outputs": [],
   "source": []
  }
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