Modeling/Turing.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "12a4c144",
   "metadata": {},
   "outputs": [],
   "source": [
    "using Turing, StatsPlots, Random\n",
    "# using Distributions:Normal"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "1946f4eb",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2-element Vector{ChainDataFrame}:\n",
       " Summary Statistics (2 x 8)\n",
       " Quantiles (2 x 6)"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Define a simple Normal model with unknown mean and variance.\n",
    "@model function gdemo(x, y)\n",
    "  s² ~ InverseGamma(2, 3)\n",
    "  m ~ Normal(0, sqrt(s²))\n",
    "  x ~ Normal(m, sqrt(s²))\n",
    "  y ~ Normal(m, sqrt(s²))\n",
    "end\n",
    "\n",
    "#  Run sampler, collect results\n",
    "chn = sample(gdemo(1.5, 2), HMC(0.1, 5), 1000)\n",
    "\n",
    "# Summarise results\n",
    "describe(chn)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "be184938",
   "metadata": {},
   "outputs": [
    {
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997.681,1311.31 999.318,1339.96 \n  1000.95,1376.72 1002.59,1358.64 1004.23,1374.91 1005.86,1286.89 1007.5,1315.69 1009.14,1363.31 1010.77,1320.28 1012.41,1348.69 1014.05,1445.42 1015.68,1484.02 \n  1017.32,1415.34 1018.96,1436.42 1020.59,1506.26 1022.23,1507.12 1023.87,1519.38 1025.5,1521.44 1027.14,1517.83 1028.78,1554.59 1030.41,1592.07 1032.05,1605.18 \n  1033.69,1633.44 1035.32,1569.77 1036.96,1564.63 1038.6,1560.65 1040.23,1491.93 1041.87,1459.29 1043.51,1424.97 1045.14,1431.01 1046.78,1334.03 1048.41,1439.28 \n  1050.05,1366.59 1051.69,1326.32 1053.32,1275.36 1054.96,1370.95 1056.6,1327 1058.23,1382.13 1059.87,1452.58 1061.51,1437.52 1063.14,1360.67 1064.78,1414.03 \n  1066.42,1458.5 1068.05,1432.14 1069.69,1400.47 1071.33,1479.49 1072.96,1492.22 1074.6,1490.54 1076.24,1552.28 1077.87,1532.6 1079.51,1575.77 1081.15,1559.82 \n  1082.78,1584.82 1084.42,1623.15 1086.06,1622.72 1087.69,1683.88 1089.33,1716.71 1090.97,1761.95 1092.6,1814.16 1094.24,1819 1095.88,1801.81 1097.51,1759.67 \n  1099.15,1671.39 1100.79,1695.34 1102.42,1706.22 1104.06,1699.2 1105.7,1664.78 1107.33,1674.65 1108.97,1579.81 1110.61,1511.07 1112.24,1539.23 1113.88,1490.96 \n  1115.51,1445.55 1117.15,1422.73 1118.79,1400.73 1120.42,1379.36 1122.06,1338.64 1123.7,1262.18 1125.33,1377.61 1126.97,1426.61 1128.61,1480.17 1130.24,1403.89 \n  1131.88,1329.35 1133.52,1252.74 1135.15,1232.51 1136.79,1235.42 1138.43,1305.28 1140.06,1274.26 1141.7,1328.47 1143.34,1286.61 1144.97,1234.25 1146.61,1296.75 \n  1148.25,1300.43 1149.88,1346.84 1151.52,1317.01 1153.16,1318.26 1154.79,1295.09 1156.43,1296.3 1158.07,1366.63 1159.7,1350.18 1161.34,1358.28 1162.98,1354.62 \n  1164.61,1344.28 1166.25,1336.95 1167.89,1345.94 1169.52,1330.95 1171.16,1347.51 1172.8,1272.87 1174.43,1339.69 1176.07,1258.78 1177.7,1286.6 1179.34,1228.43 \n  1180.98,1212.48 1182.61,1264.4 1184.25,1304.36 1185.89,1321.75 1187.52,1357.34 1189.16,1405.58 1190.8,1336.79 1192.43,1374.98 1194.07,1396.4 1195.71,1376.43 \n  1197.34,1326.92 1198.98,1380.46 1200.62,1351.72 1202.25,1316.36 1203.89,1294.63 1205.53,1347.48 1207.16,1455.79 1208.8,1429.41 1210.44,1438 1212.07,1454.51 \n  1213.71,1398.55 1215.35,1316.45 1216.98,1294.49 1218.62,1284.58 1220.26,1295.23 1221.89,1216.78 1223.53,1230.14 1225.17,1182.13 1226.8,1210.85 1228.44,1188.33 \n  1230.08,1185.17 1231.71,1148.48 1233.35,1197.66 1234.99,1200.28 1236.62,1200.87 1238.26,1272.77 1239.89,1363.66 1241.53,1303.4 1243.17,1312.34 1244.8,1344.94 \n  1246.44,1311.87 1248.08,1287.71 1249.71,1304.04 1251.35,1377.47 1252.99,1287.91 1254.62,1316.12 1256.26,1369.23 1257.9,1343.75 1259.53,1388.42 1261.17,1323.6 \n  1262.81,1281.44 1264.44,1324.26 1266.08,1385.86 1267.72,1325.69 1269.35,1286.37 1270.99,1361.74 1272.63,1318.91 1274.26,1382.55 1275.9,1379.05 1277.54,1357.72 \n  1279.17,1316.29 1280.81,1288.76 1282.45,1362.98 1284.08,1349.46 1285.72,1312.75 1287.36,1352.55 1288.99,1453.74 1290.63,1356.56 1292.27,1386.69 1293.9,1400.97 \n  1295.54,1368.75 1297.18,1385.47 1298.81,1347.4 1300.45,1371.76 1302.08,1356.46 1303.72,1351.38 1305.36,1376.6 1306.99,1391.08 1308.63,1397.56 1310.27,1375.69 \n  1311.9,1366.03 1313.54,1430.34 1315.18,1381.5 1316.81,1400.77 1318.45,1377.01 1320.09,1344.26 1321.72,1357.06 1323.36,1380.04 1325,1468.17 1326.63,1500.59 \n  1328.27,1460.8 1329.91,1493.74 1331.54,1496.94 1333.18,1504.52 1334.82,1406.62 1336.45,1364.74 1338.09,1364.34 1339.73,1394.1 1341.36,1380.8 1343,1389.93 \n  1344.64,1411.34 1346.27,1434.96 1347.91,1459.54 1349.55,1402.33 1351.18,1459.69 1352.82,1535.7 1354.46,1459.19 1356.09,1475.14 1357.73,1478.18 1359.37,1488.18 \n  1361,1469.31 1362.64,1509.28 1364.27,1546.14 1365.91,1513.89 1367.55,1494.92 1369.18,1467.31 1370.82,1535.53 1372.46,1549.9 1374.09,1563.78 1375.73,1482.58 \n  1377.37,1437.51 1379,1474.48 1380.64,1430.79 1382.28,1398.53 1383.91,1384.44 1385.55,1377.29 1387.19,1413.36 1388.82,1433.95 1390.46,1526.41 1392.1,1557.45 \n  1393.73,1543.5 1395.37,1495.31 1397.01,1591.94 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     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Plot and save results\n",
    "p = plot(chn)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "70ed24a1",
   "metadata": {},
   "outputs": [
    {
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     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "\n",
    "# Set the true probability of heads in a coin.\n",
    "p_true = 0.5\n",
    "\n",
    "# Iterate from having seen 0 observations to 100 observations.\n",
    "Ns = 0:100\n",
    "\n",
    "# Draw data from a Bernoulli distribution, i.e. draw heads or tails.\n",
    "Random.seed!(12)\n",
    "data = rand(Bernoulli(p_true), last(Ns))\n",
    "\n",
    "# Declare our Turing model.\n",
    "@model function coinflip(y)\n",
    "    # Our prior belief about the probability of heads in a coin.\n",
    "    p ~ Beta(1, 1)\n",
    "\n",
    "    # The number of observations.\n",
    "    N = length(y)\n",
    "    for n in 1:N\n",
    "        # Heads or tails of a coin are drawn from a Bernoulli distribution.\n",
    "        y[n] ~ Bernoulli(p)\n",
    "    end\n",
    "end\n",
    "\n",
    "# Settings of the Hamiltonian Monte Carlo (HMC) sampler.\n",
    "iterations = 1000\n",
    "ϵ = 0.05\n",
    "τ = 10\n",
    "\n",
    "# Start sampling.\n",
    "chain = sample(coinflip(data), HMC(ϵ, τ), iterations)\n",
    "\n",
    "# Plot a summary of the sampling process for the parameter p, i.e. the probability of heads in a coin.\n",
    "histogram(chain[:p])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "8410650b",
   "metadata": {},
   "outputs": [
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     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "\n",
    "\n",
    "# Visualize a blue density plot of the approximate posterior distribution using HMC (see Chain 1 in the legend).\n",
    "density(chain; xlim=(0, 1), legend=:best, w=2, c=:blue)\n",
    "\n",
    "# # Visualize a green density plot of the posterior distribution in closed-form.\n",
    "# plot!(\n",
    "#     0:0.01:1,\n",
    "#     pdf.(updated_belief(prior_belief, data), 0:0.01:1);\n",
    "#     xlabel=\"probability of heads\",\n",
    "#     ylabel=\"\",\n",
    "#     title=\"\",\n",
    "#     xlim=(0, 1),\n",
    "#     label=\"Closed-form\",\n",
    "#     fill=0,\n",
    "#     α=0.3,\n",
    "#     w=3,\n",
    "#     c=:lightgreen,\n",
    "# )\n",
    "\n",
    "# Visualize the true probability of heads in red.\n",
    "vline!([p_true]; label=\"True probability\", c=:red)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "7e63207e",
   "metadata": {},
   "outputs": [],
   "source": [
    "d = Truncated(Normal(40,20), 0, 1000) # Truncated Normal\n",
    "demand = round.(rand(d, 15));"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "18485db0",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5-element Vector{Float64}:\n",
       " 55.0\n",
       " 56.0\n",
       " 42.0\n",
       " 56.0\n",
       " 59.0"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "demand_seen = demand[1:5]; demand_unseen = demand[6:15];\n",
    "demand_seen"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "id": "2929b7d1",
   "metadata": {},
   "source": [
    "Let's use Bayesian Stats\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "\\sigma & \\sim \\text{HalfStudentT}(10, 20) \\\\\n",
    "\\mu & \\sim \\mathcal{N}(\\bar{x}, 20) \\\\\n",
    "\\text{Demand} & \\sim \\text{TruncatedNormal}(\\mu, \\sigma, 0)\\\\\n",
    "\\end{aligned}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "id": "a7980148",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "newspaper (generic function with 4 methods)"
      ]
     },
     "execution_count": 57,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "@model function newspaper(demand)\n",
    "    σ ~ 10 + Truncated(TDist(20), 0, 100)\n",
    "    μ ~ Normal(mean(demand_seen), 20)\n",
    "    for i in 1:length(demand)\n",
    "        demand[i] ~ Truncated(Normal(μ, σ), 0, 100)\n",
    "    end\n",
    "    return demand\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "id": "a621faf8",
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "┌ Info: Found initial step size\n",
      "│   ϵ = 1.6\n",
      "└ @ Turing.Inference /Users/nelsontang/.julia/packages/Turing/0KpGp/src/inference/hmc.jl:188\n",
      "┌ Warning: The current proposal will be rejected due to numerical error(s).\n",
      "│   isfinite.((θ, r, ℓπ, ℓκ)) = (true, false, false, false)\n",
      "└ @ AdvancedHMC /Users/nelsontang/.julia/packages/AdvancedHMC/51xgc/src/hamiltonian.jl:47\n",
      "┌ Warning: The current proposal will be rejected due to numerical error(s).\n",
      "│   isfinite.((θ, r, ℓπ, ℓκ)) = (true, false, false, false)\n",
      "└ @ AdvancedHMC /Users/nelsontang/.julia/packages/AdvancedHMC/51xgc/src/hamiltonian.jl:47\n",
      "┌ Warning: The current proposal will be rejected due to numerical error(s).\n",
      "│   isfinite.((θ, r, ℓπ, ℓκ)) = (true, false, false, false)\n",
      "└ @ AdvancedHMC /Users/nelsontang/.julia/packages/AdvancedHMC/51xgc/src/hamiltonian.jl:47\n",
      "┌ Warning: The current proposal will be rejected due to numerical error(s).\n",
      "│   isfinite.((θ, r, ℓπ, ℓκ)) = (true, false, false, false)\n",
      "└ @ AdvancedHMC /Users/nelsontang/.julia/packages/AdvancedHMC/51xgc/src/hamiltonian.jl:47\n",
      "┌ Warning: The current proposal will be rejected due to numerical error(s).\n",
      "│   isfinite.((θ, r, ℓπ, ℓκ)) = (true, false, false, false)\n",
      "└ @ AdvancedHMC /Users/nelsontang/.julia/packages/AdvancedHMC/51xgc/src/hamiltonian.jl:47\n",
      "┌ Warning: The current proposal will be rejected due to numerical error(s).\n",
      "│   isfinite.((θ, r, ℓπ, ℓκ)) = (true, false, false, false)\n",
      "└ @ AdvancedHMC /Users/nelsontang/.julia/packages/AdvancedHMC/51xgc/src/hamiltonian.jl:47\n",
      "┌ Warning: The current proposal will be rejected due to numerical error(s).\n",
      "│   isfinite.((θ, r, ℓπ, ℓκ)) = (true, false, false, false)\n",
      "└ @ AdvancedHMC /Users/nelsontang/.julia/packages/AdvancedHMC/51xgc/src/hamiltonian.jl:47\n",
      "┌ Warning: The current proposal will be rejected due to numerical error(s).\n",
      "│   isfinite.((θ, r, ℓπ, ℓκ)) = (true, false, false, false)\n",
      "└ @ AdvancedHMC /Users/nelsontang/.julia/packages/AdvancedHMC/51xgc/src/hamiltonian.jl:47\n",
      "┌ Warning: The current proposal will be rejected due to numerical error(s).\n",
      "│   isfinite.((θ, r, ℓπ, ℓκ)) = (true, false, false, false)\n",
      "└ @ AdvancedHMC /Users/nelsontang/.julia/packages/AdvancedHMC/51xgc/src/hamiltonian.jl:47\n",
      "┌ Warning: The current proposal will be rejected due to numerical error(s).\n",
      "│   isfinite.((θ, r, ℓπ, ℓκ)) = (true, false, false, false)\n",
      "└ @ AdvancedHMC /Users/nelsontang/.julia/packages/AdvancedHMC/51xgc/src/hamiltonian.jl:47\n",
      "┌ Warning: The current proposal will be rejected due to numerical error(s).\n",
      "│   isfinite.((θ, r, ℓπ, ℓκ)) = (true, false, false, false)\n",
      "└ @ AdvancedHMC /Users/nelsontang/.julia/packages/AdvancedHMC/51xgc/src/hamiltonian.jl:47\n",
      "┌ Warning: The current proposal will be rejected due to numerical error(s).\n",
      "│   isfinite.((θ, r, ℓπ, ℓκ)) = (true, false, false, false)\n",
      "└ @ AdvancedHMC /Users/nelsontang/.julia/packages/AdvancedHMC/51xgc/src/hamiltonian.jl:47\n",
      "┌ Warning: The current proposal will be rejected due to numerical error(s).\n",
      "│   isfinite.((θ, r, ℓπ, ℓκ)) = (true, false, false, false)\n",
      "└ @ AdvancedHMC /Users/nelsontang/.julia/packages/AdvancedHMC/51xgc/src/hamiltonian.jl:47\n",
      "┌ Warning: The current proposal will be rejected due to numerical error(s).\n",
      "│   isfinite.((θ, r, ℓπ, ℓκ)) = (true, false, false, false)\n",
      "└ @ AdvancedHMC /Users/nelsontang/.julia/packages/AdvancedHMC/51xgc/src/hamiltonian.jl:47\n",
      "┌ Warning: The current proposal will be rejected due to numerical error(s).\n",
      "│   isfinite.((θ, r, ℓπ, ℓκ)) = (true, false, false, false)\n",
      "└ @ AdvancedHMC /Users/nelsontang/.julia/packages/AdvancedHMC/51xgc/src/hamiltonian.jl:47\n",
      "┌ Warning: The current proposal will be rejected due to numerical error(s).\n",
      "│   isfinite.((θ, r, ℓπ, ℓκ)) = (true, false, false, false)\n",
      "└ @ AdvancedHMC /Users/nelsontang/.julia/packages/AdvancedHMC/51xgc/src/hamiltonian.jl:47\n",
      "┌ Warning: The current proposal will be rejected due to numerical error(s).\n",
      "│   isfinite.((θ, r, ℓπ, ℓκ)) = (true, false, false, false)\n",
      "└ @ AdvancedHMC /Users/nelsontang/.julia/packages/AdvancedHMC/51xgc/src/hamiltonian.jl:47\n",
      "┌ Warning: The current proposal will be rejected due to numerical error(s).\n",
      "│   isfinite.((θ, r, ℓπ, ℓκ)) = (true, false, false, false)\n",
      "└ @ AdvancedHMC /Users/nelsontang/.julia/packages/AdvancedHMC/51xgc/src/hamiltonian.jl:47\n",
      "┌ Warning: The current proposal will be rejected due to numerical error(s).\n",
      "│   isfinite.((θ, r, ℓπ, ℓκ)) = (true, false, false, false)\n",
      "└ @ AdvancedHMC /Users/nelsontang/.julia/packages/AdvancedHMC/51xgc/src/hamiltonian.jl:47\n",
      "┌ Warning: The current proposal will be rejected due to numerical error(s).\n",
      "│   isfinite.((θ, r, ℓπ, ℓκ)) = (true, false, false, false)\n",
      "└ @ AdvancedHMC /Users/nelsontang/.julia/packages/AdvancedHMC/51xgc/src/hamiltonian.jl:47\n",
      "┌ Warning: The current proposal will be rejected due to numerical error(s).\n",
      "│   isfinite.((θ, r, ℓπ, ℓκ)) = (true, false, false, false)\n",
      "└ @ AdvancedHMC /Users/nelsontang/.julia/packages/AdvancedHMC/51xgc/src/hamiltonian.jl:47\n",
      "┌ Warning: The current proposal will be rejected due to numerical error(s).\n",
      "│   isfinite.((θ, r, ℓπ, ℓκ)) = (true, false, false, false)\n",
      "└ @ AdvancedHMC /Users/nelsontang/.julia/packages/AdvancedHMC/51xgc/src/hamiltonian.jl:47\n",
      "┌ Warning: The current proposal will be rejected due to numerical error(s).\n",
      "│   isfinite.((θ, r, ℓπ, ℓκ)) = (true, false, false, false)\n",
      "└ @ AdvancedHMC /Users/nelsontang/.julia/packages/AdvancedHMC/51xgc/src/hamiltonian.jl:47\n",
      "┌ Warning: The current proposal will be rejected due to numerical error(s).\n",
      "│   isfinite.((θ, r, ℓπ, ℓκ)) = (true, false, false, false)\n",
      "└ @ AdvancedHMC /Users/nelsontang/.julia/packages/AdvancedHMC/51xgc/src/hamiltonian.jl:47\n",
      "┌ Warning: The current proposal will be rejected due to numerical error(s).\n",
      "│   isfinite.((θ, r, ℓπ, ℓκ)) = (true, false, false, false)\n",
      "└ @ AdvancedHMC /Users/nelsontang/.julia/packages/AdvancedHMC/51xgc/src/hamiltonian.jl:47\n",
      "┌ Warning: The current proposal will be rejected due to numerical error(s).\n",
      "│   isfinite.((θ, r, ℓπ, ℓκ)) = (true, false, false, false)\n",
      "└ @ AdvancedHMC /Users/nelsontang/.julia/packages/AdvancedHMC/51xgc/src/hamiltonian.jl:47\n",
      "┌ Warning: The current proposal will be rejected due to numerical error(s).\n",
      "│   isfinite.((θ, r, ℓπ, ℓκ)) = (true, false, false, false)\n",
      "└ @ AdvancedHMC /Users/nelsontang/.julia/packages/AdvancedHMC/51xgc/src/hamiltonian.jl:47\n",
      "┌ Warning: The current proposal will be rejected due to numerical error(s).\n",
      "│   isfinite.((θ, r, ℓπ, ℓκ)) = (true, false, false, false)\n",
      "└ @ AdvancedHMC /Users/nelsontang/.julia/packages/AdvancedHMC/51xgc/src/hamiltonian.jl:47\n",
      "┌ Warning: The current proposal will be rejected due to numerical error(s).\n",
      "│   isfinite.((θ, r, ℓπ, ℓκ)) = (true, false, false, false)\n",
      "└ @ AdvancedHMC /Users/nelsontang/.julia/packages/AdvancedHMC/51xgc/src/hamiltonian.jl:47\n",
      "\u001b[32mSampling: 100%|█████████████████████████████████████████| Time: 0:00:00\u001b[39m\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "Chains MCMC chain (10000×14×1 Array{Float64, 3}):\n",
       "\n",
       "Iterations        = 1001:1:11000\n",
       "Number of chains  = 1\n",
       "Samples per chain = 10000\n",
       "Wall duration     = 1.31 seconds\n",
       "Compute duration  = 1.31 seconds\n",
       "parameters        = σ, μ\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   10.7407    0.5890     0.0059    0.0067   5980.1285    0.9999    ⋯\n",
       "           μ   53.5572    4.6688     0.0467    0.0646   4640.8277    0.9999    ⋯\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   10.0283   10.2897   10.6054   11.0437   12.2103\n",
       "           μ   44.3721   50.3877   53.6252   56.7150   62.6577\n"
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "idata = sample(newspaper(demand_seen), NUTS(), 10000)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "id": "e0d6be44",
   "metadata": {},
   "outputs": [
    {
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