Supervised regression.ipynb

      "Epoch 294/500  total: 3.70312, -LL: 3.62281, prior: 0.83122, aleatoric unc.: 9.76033\n",
      "Epoch 295/500  total: 3.70929, -LL: 3.64419, prior: 0.83021, aleatoric unc.: 9.76300\n",
      "Epoch 296/500  total: 3.70208, -LL: 3.64338, prior: 0.83049, aleatoric unc.: 9.76104\n",
      "Epoch 297/500  total: 3.70218, -LL: 3.61723, prior: 0.83026, aleatoric unc.: 9.75453\n",
      "Epoch 298/500  total: 3.70252, -LL: 3.64148, prior: 0.83147, aleatoric unc.: 9.74977\n",
      "Epoch 299/500  total: 3.70108, -LL: 3.68672, prior: 0.83107, aleatoric unc.: 9.74179\n",
      "Epoch 300/500  total: 3.70634, -LL: 3.67040, prior: 0.82994, aleatoric unc.: 9.75107\n",
      "Epoch 301/500  total: 3.69355, -LL: 3.60485, prior: 0.82971, aleatoric unc.: 9.73195\n",
      "Epoch 302/500  total: 3.70322, -LL: 3.63580, prior: 0.83051, aleatoric unc.: 9.73157\n",
      "Epoch 303/500  total: 3.70026, -LL: 3.70596, prior: 0.82929, aleatoric unc.: 9.72599\n",
      "Epoch 304/500  total: 3.70397, -LL: 3.65822, prior: 0.83103, aleatoric unc.: 9.72934\n",
      "Epoch 305/500  total: 3.70864, -LL: 3.56731, prior: 0.83211, aleatoric unc.: 9.74369\n",
      "Epoch 306/500  total: 3.70876, -LL: 3.59099, prior: 0.83176, aleatoric unc.: 9.75274\n",
      "Epoch 307/500  total: 3.70129, -LL: 3.63883, prior: 0.82881, aleatoric unc.: 9.74951\n",
      "Epoch 308/500  total: 3.70983, -LL: 3.63237, prior: 0.82886, aleatoric unc.: 9.75686\n",
      "Epoch 309/500  total: 3.70741, -LL: 3.60030, prior: 0.83055, aleatoric unc.: 9.76448\n",
      "Epoch 310/500  total: 3.70709, -LL: 3.59931, prior: 0.83434, aleatoric unc.: 9.76765\n",
      "Epoch 311/500  total: 3.69747, -LL: 3.63639, prior: 0.83337, aleatoric unc.: 9.74833\n",
      "Epoch 312/500  total: 3.71152, -LL: 3.62594, prior: 0.83660, aleatoric unc.: 9.76304\n",
      "Epoch 313/500  total: 3.70416, -LL: 3.64673, prior: 0.83517, aleatoric unc.: 9.76183\n",
      "Epoch 314/500  total: 3.69919, -LL: 3.58278, prior: 0.83536, aleatoric unc.: 9.75043\n",
      "Epoch 315/500  total: 3.70675, -LL: 3.60159, prior: 0.83583, aleatoric unc.: 9.75515\n",
      "Epoch 316/500  total: 3.70445, -LL: 3.64417, prior: 0.83832, aleatoric unc.: 9.75135\n",
      "Epoch 317/500  total: 3.70155, -LL: 3.64957, prior: 0.83247, aleatoric unc.: 9.74824\n",
      "Epoch 318/500  total: 3.70251, -LL: 3.61336, prior: 0.83256, aleatoric unc.: 9.74411\n",
      "Epoch 319/500  total: 3.70864, -LL: 3.61781, prior: 0.83391, aleatoric unc.: 9.75591\n",
      "Epoch 320/500  total: 3.69483, -LL: 3.62589, prior: 0.83023, aleatoric unc.: 9.73673\n",
      "Epoch 321/500  total: 3.70477, -LL: 3.65918, prior: 0.83147, aleatoric unc.: 9.73801\n",
      "Epoch 322/500  total: 3.71006, -LL: 3.68025, prior: 0.83056, aleatoric unc.: 9.75054\n",
      "Epoch 323/500  total: 3.70612, -LL: 3.63204, prior: 0.82882, aleatoric unc.: 9.75776\n",
      "Epoch 324/500  total: 3.70220, -LL: 3.63502, prior: 0.82885, aleatoric unc.: 9.75287\n",
      "Epoch 325/500  total: 3.70991, -LL: 3.67634, prior: 0.82949, aleatoric unc.: 9.75935\n",
      "Epoch 326/500  total: 3.70031, -LL: 3.67920, prior: 0.83210, aleatoric unc.: 9.74968\n",
      "Epoch 327/500  total: 3.71157, -LL: 3.65393, prior: 0.83033, aleatoric unc.: 9.76567\n",
      "Epoch 328/500  total: 3.70492, -LL: 3.64040, prior: 0.82936, aleatoric unc.: 9.76520\n",
      "Epoch 329/500  total: 3.70176, -LL: 3.60859, prior: 0.83023, aleatoric unc.: 9.75874\n",
      "Epoch 330/500  total: 3.70091, -LL: 3.61544, prior: 0.83437, aleatoric unc.: 9.74670\n",
      "Epoch 331/500  total: 3.70303, -LL: 3.65750, prior: 0.83323, aleatoric unc.: 9.74530\n",
      "Epoch 332/500  total: 3.70879, -LL: 3.59750, prior: 0.82948, aleatoric unc.: 9.75656\n",
      "Epoch 333/500  total: 3.70656, -LL: 3.65474, prior: 0.82868, aleatoric unc.: 9.76092\n",
      "Epoch 334/500  total: 3.69965, -LL: 3.63443, prior: 0.83218, aleatoric unc.: 9.74758\n",
      "Epoch 335/500  total: 3.70277, -LL: 3.63288, prior: 0.83110, aleatoric unc.: 9.74762\n",
      "Epoch 336/500  total: 3.70189, -LL: 3.66539, prior: 0.83201, aleatoric unc.: 9.74239\n",
      "Epoch 337/500  total: 3.69537, -LL: 3.65592, prior: 0.82954, aleatoric unc.: 9.72731\n",
      "Epoch 338/500  total: 3.69802, -LL: 3.60642, prior: 0.82822, aleatoric unc.: 9.72182\n",
      "Epoch 339/500  total: 3.71005, -LL: 3.67063, prior: 0.82849, aleatoric unc.: 9.73660\n",
      "Epoch 340/500  total: 3.70104, -LL: 3.60019, prior: 0.83237, aleatoric unc.: 9.73406\n",
      "Epoch 341/500  total: 3.70567, -LL: 3.62259, prior: 0.83000, aleatoric unc.: 9.74208\n",
      "Epoch 342/500  total: 3.70195, -LL: 3.65865, prior: 0.82917, aleatoric unc.: 9.73657\n",
      "Epoch 343/500  total: 3.70123, -LL: 3.62708, prior: 0.83080, aleatoric unc.: 9.73533\n",
      "Epoch 344/500  total: 3.70437, -LL: 3.61342, prior: 0.82911, aleatoric unc.: 9.73596\n",
      "Epoch 345/500  total: 3.70366, -LL: 3.60328, prior: 0.83045, aleatoric unc.: 9.74204\n",
      "Epoch 346/500  total: 3.70356, -LL: 3.62689, prior: 0.83147, aleatoric unc.: 9.73873\n",
      "Epoch 347/500  total: 3.69629, -LL: 3.64269, prior: 0.82647, aleatoric unc.: 9.72898\n",
      "Epoch 348/500  total: 3.69996, -LL: 3.65626, prior: 0.82871, aleatoric unc.: 9.72221\n",
      "Epoch 349/500  total: 3.70417, -LL: 3.63705, prior: 0.82662, aleatoric unc.: 9.72849\n",
      "Epoch 350/500  total: 3.70177, -LL: 3.65497, prior: 0.82323, aleatoric unc.: 9.72953\n",
      "Epoch 351/500  total: 3.70254, -LL: 3.67832, prior: 0.82772, aleatoric unc.: 9.73065\n",
      "Epoch 352/500  total: 3.70162, -LL: 3.63990, prior: 0.83119, aleatoric unc.: 9.72532\n",
      "Epoch 353/500  total: 3.70578, -LL: 3.60402, prior: 0.83188, aleatoric unc.: 9.73709\n",
      "Epoch 354/500  total: 3.70272, -LL: 3.62950, prior: 0.82930, aleatoric unc.: 9.73685\n",
      "Epoch 355/500  total: 3.70043, -LL: 3.62746, prior: 0.82995, aleatoric unc.: 9.73355\n",
      "Epoch 356/500  total: 3.69946, -LL: 3.61545, prior: 0.83072, aleatoric unc.: 9.72545\n",
      "Epoch 357/500  total: 3.70071, -LL: 3.67021, prior: 0.83015, aleatoric unc.: 9.72398\n",
      "Epoch 358/500  total: 3.70548, -LL: 3.63582, prior: 0.82771, aleatoric unc.: 9.73216\n",
      "Epoch 359/500  total: 3.69811, -LL: 3.60202, prior: 0.82758, aleatoric unc.: 9.72475\n",
      "Epoch 360/500  total: 3.69686, -LL: 3.64973, prior: 0.82581, aleatoric unc.: 9.71591\n",
      "Epoch 361/500  total: 3.70717, -LL: 3.65881, prior: 0.82941, aleatoric unc.: 9.72619\n",
      "Epoch 362/500  total: 3.70385, -LL: 3.65302, prior: 0.82832, aleatoric unc.: 9.73025\n",
      "Epoch 363/500  total: 3.70743, -LL: 3.64934, prior: 0.82749, aleatoric unc.: 9.74160\n",
      "Epoch 364/500  total: 3.70326, -LL: 3.58934, prior: 0.82753, aleatoric unc.: 9.74316\n",
      "Epoch 365/500  total: 3.70402, -LL: 3.62665, prior: 0.82742, aleatoric unc.: 9.74414\n",
      "Epoch 366/500  total: 3.70585, -LL: 3.68176, prior: 0.82632, aleatoric unc.: 9.74891\n",
      "Epoch 367/500  total: 3.70568, -LL: 3.62934, prior: 0.82647, aleatoric unc.: 9.75294\n",
      "Epoch 368/500  total: 3.69735, -LL: 3.63394, prior: 0.82584, aleatoric unc.: 9.73833\n",
      "Epoch 369/500  total: 3.70665, -LL: 3.60627, prior: 0.82508, aleatoric unc.: 9.74506\n",
      "Epoch 370/500  total: 3.70283, -LL: 3.64191, prior: 0.82496, aleatoric unc.: 9.74463\n",
      "Epoch 371/500  total: 3.69990, -LL: 3.63656, prior: 0.82942, aleatoric unc.: 9.73714\n",
      "Epoch 372/500  total: 3.70170, -LL: 3.65815, prior: 0.82560, aleatoric unc.: 9.73377\n",
      "Epoch 373/500  total: 3.70135, -LL: 3.66173, prior: 0.82487, aleatoric unc.: 9.73149\n",
      "Epoch 374/500  total: 3.70539, -LL: 3.60903, prior: 0.82451, aleatoric unc.: 9.74010\n",
      "Epoch 375/500  total: 3.69980, -LL: 3.63325, prior: 0.82713, aleatoric unc.: 9.73296\n",
      "Epoch 376/500  total: 3.69921, -LL: 3.60023, prior: 0.82868, aleatoric unc.: 9.72724\n",
      "Epoch 377/500  total: 3.69869, -LL: 3.66106, prior: 0.82731, aleatoric unc.: 9.72021\n",
      "Epoch 378/500  total: 3.69702, -LL: 3.62505, prior: 0.82719, aleatoric unc.: 9.71488\n",
      "Epoch 379/500  total: 3.69819, -LL: 3.60678, prior: 0.82717, aleatoric unc.: 9.71085\n",
      "Epoch 380/500  total: 3.69677, -LL: 3.62688, prior: 0.82746, aleatoric unc.: 9.70245\n",
      "Epoch 381/500  total: 3.70501, -LL: 3.60229, prior: 0.82751, aleatoric unc.: 9.71364\n",
      "Epoch 382/500  total: 3.69922, -LL: 3.57855, prior: 0.83100, aleatoric unc.: 9.71044\n",
      "Epoch 383/500  total: 3.69865, -LL: 3.63590, prior: 0.82836, aleatoric unc.: 9.70802\n",
      "Epoch 384/500  total: 3.70344, -LL: 3.59795, prior: 0.82755, aleatoric unc.: 9.71421\n",
      "Epoch 385/500  total: 3.70533, -LL: 3.64661, prior: 0.82573, aleatoric unc.: 9.72334\n",
      "Epoch 386/500  total: 3.70198, -LL: 3.61014, prior: 0.82611, aleatoric unc.: 9.72450\n",
      "Epoch 387/500  total: 3.70406, -LL: 3.64215, prior: 0.82555, aleatoric unc.: 9.73054\n",
      "Epoch 388/500  total: 3.70490, -LL: 3.63847, prior: 0.82456, aleatoric unc.: 9.73703\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Epoch 389/500  total: 3.69739, -LL: 3.60615, prior: 0.82374, aleatoric unc.: 9.72495\n",
      "Epoch 390/500  total: 3.70646, -LL: 3.59768, prior: 0.82653, aleatoric unc.: 9.73402\n",
      "Epoch 391/500  total: 3.69914, -LL: 3.65293, prior: 0.82449, aleatoric unc.: 9.72576\n",
      "Epoch 392/500  total: 3.70214, -LL: 3.62112, prior: 0.82444, aleatoric unc.: 9.72820\n",
      "Epoch 393/500  total: 3.70022, -LL: 3.57499, prior: 0.82906, aleatoric unc.: 9.72694\n",
      "Epoch 394/500  total: 3.69050, -LL: 3.60771, prior: 0.82554, aleatoric unc.: 9.70561\n",
      "Epoch 395/500  total: 3.69285, -LL: 3.68116, prior: 0.82755, aleatoric unc.: 9.69149\n",
      "Epoch 396/500  total: 3.70537, -LL: 3.61828, prior: 0.82465, aleatoric unc.: 9.70394\n",
      "Epoch 397/500  total: 3.69655, -LL: 3.61455, prior: 0.82317, aleatoric unc.: 9.70052\n",
      "Epoch 398/500  total: 3.70530, -LL: 3.62109, prior: 0.82612, aleatoric unc.: 9.71183\n",
      "Epoch 399/500  total: 3.70086, -LL: 3.61040, prior: 0.82481, aleatoric unc.: 9.71438\n",
      "Epoch 400/500  total: 3.70211, -LL: 3.64791, prior: 0.82474, aleatoric unc.: 9.71751\n",
      "Epoch 401/500  total: 3.70449, -LL: 3.66220, prior: 0.82469, aleatoric unc.: 9.72471\n",
      "Epoch 402/500  total: 3.69559, -LL: 3.68538, prior: 0.82521, aleatoric unc.: 9.71284\n",
      "Epoch 403/500  total: 3.70262, -LL: 3.61727, prior: 0.82641, aleatoric unc.: 9.71845\n",
      "Epoch 404/500  total: 3.70541, -LL: 3.65800, prior: 0.82866, aleatoric unc.: 9.72628\n",
      "Epoch 405/500  total: 3.70215, -LL: 3.66135, prior: 0.82954, aleatoric unc.: 9.72948\n",
      "Epoch 406/500  total: 3.69729, -LL: 3.64724, prior: 0.82800, aleatoric unc.: 9.72026\n",
      "Epoch 407/500  total: 3.69719, -LL: 3.63176, prior: 0.82775, aleatoric unc.: 9.71121\n",
      "Epoch 408/500  total: 3.70596, -LL: 3.66872, prior: 0.82940, aleatoric unc.: 9.72126\n",
      "Epoch 409/500  total: 3.70204, -LL: 3.63414, prior: 0.83020, aleatoric unc.: 9.72397\n",
      "Epoch 410/500  total: 3.70021, -LL: 3.64946, prior: 0.83012, aleatoric unc.: 9.72245\n",
      "Epoch 411/500  total: 3.69402, -LL: 3.61718, prior: 0.82774, aleatoric unc.: 9.70823\n",
      "Epoch 412/500  total: 3.70567, -LL: 3.61743, prior: 0.82851, aleatoric unc.: 9.71849\n",
      "Epoch 413/500  total: 3.70361, -LL: 3.64554, prior: 0.82617, aleatoric unc.: 9.72331\n",
      "Epoch 414/500  total: 3.69468, -LL: 3.60278, prior: 0.82620, aleatoric unc.: 9.71184\n",
      "Epoch 415/500  total: 3.70510, -LL: 3.64241, prior: 0.82747, aleatoric unc.: 9.72071\n",
      "Epoch 416/500  total: 3.70178, -LL: 3.59014, prior: 0.83144, aleatoric unc.: 9.72354\n",
      "Epoch 417/500  total: 3.69967, -LL: 3.63795, prior: 0.83206, aleatoric unc.: 9.71923\n",
      "Epoch 418/500  total: 3.69828, -LL: 3.60308, prior: 0.82796, aleatoric unc.: 9.71243\n",
      "Epoch 419/500  total: 3.70307, -LL: 3.66544, prior: 0.82524, aleatoric unc.: 9.71768\n",
      "Epoch 420/500  total: 3.70183, -LL: 3.61180, prior: 0.82254, aleatoric unc.: 9.72192\n",
      "Epoch 421/500  total: 3.69221, -LL: 3.63945, prior: 0.82239, aleatoric unc.: 9.70423\n",
      "Epoch 422/500  total: 3.69980, -LL: 3.61604, prior: 0.82302, aleatoric unc.: 9.70583\n",
      "Epoch 423/500  total: 3.69928, -LL: 3.63499, prior: 0.81947, aleatoric unc.: 9.70428\n",
      "Epoch 424/500  total: 3.70064, -LL: 3.64438, prior: 0.82250, aleatoric unc.: 9.70499\n",
      "Epoch 425/500  total: 3.69506, -LL: 3.61971, prior: 0.82166, aleatoric unc.: 9.69757\n",
      "Epoch 426/500  total: 3.70398, -LL: 3.62597, prior: 0.82278, aleatoric unc.: 9.70802\n",
      "Epoch 427/500  total: 3.70216, -LL: 3.65498, prior: 0.81911, aleatoric unc.: 9.71137\n",
      "Epoch 428/500  total: 3.70190, -LL: 3.64064, prior: 0.81833, aleatoric unc.: 9.71579\n",
      "Epoch 429/500  total: 3.70573, -LL: 3.66083, prior: 0.81893, aleatoric unc.: 9.72434\n",
      "Epoch 430/500  total: 3.69885, -LL: 3.65177, prior: 0.82123, aleatoric unc.: 9.72031\n",
      "Epoch 431/500  total: 3.70198, -LL: 3.67658, prior: 0.81868, aleatoric unc.: 9.72134\n",
      "Epoch 432/500  total: 3.70692, -LL: 3.63311, prior: 0.81943, aleatoric unc.: 9.73138\n",
      "Epoch 433/500  total: 3.70009, -LL: 3.59875, prior: 0.81951, aleatoric unc.: 9.72938\n",
      "Epoch 434/500  total: 3.69864, -LL: 3.63859, prior: 0.82302, aleatoric unc.: 9.72290\n",
      "Epoch 435/500  total: 3.69655, -LL: 3.66607, prior: 0.82137, aleatoric unc.: 9.71333\n",
      "Epoch 436/500  total: 3.69973, -LL: 3.65163, prior: 0.81886, aleatoric unc.: 9.71366\n",
      "Epoch 437/500  total: 3.69769, -LL: 3.68498, prior: 0.81805, aleatoric unc.: 9.70730\n",
      "Epoch 438/500  total: 3.69672, -LL: 3.64131, prior: 0.81733, aleatoric unc.: 9.70155\n",
      "Epoch 439/500  total: 3.69554, -LL: 3.62049, prior: 0.82319, aleatoric unc.: 9.69490\n",
      "Epoch 440/500  total: 3.69749, -LL: 3.62081, prior: 0.82234, aleatoric unc.: 9.69433\n",
      "Epoch 441/500  total: 3.70329, -LL: 3.59532, prior: 0.82169, aleatoric unc.: 9.70249\n",
      "Epoch 442/500  total: 3.69808, -LL: 3.62759, prior: 0.82254, aleatoric unc.: 9.70274\n",
      "Epoch 443/500  total: 3.70494, -LL: 3.65422, prior: 0.82270, aleatoric unc.: 9.71221\n",
      "Epoch 444/500  total: 3.69976, -LL: 3.65673, prior: 0.82426, aleatoric unc.: 9.71188\n",
      "Epoch 445/500  total: 3.70325, -LL: 3.64046, prior: 0.82422, aleatoric unc.: 9.71596\n",
      "Epoch 446/500  total: 3.70594, -LL: 3.65127, prior: 0.82255, aleatoric unc.: 9.72795\n",
      "Epoch 447/500  total: 3.70256, -LL: 3.65205, prior: 0.82299, aleatoric unc.: 9.72941\n",
      "Epoch 448/500  total: 3.70255, -LL: 3.66160, prior: 0.82127, aleatoric unc.: 9.73019\n",
      "Epoch 449/500  total: 3.70477, -LL: 3.65332, prior: 0.82186, aleatoric unc.: 9.73749\n",
      "Epoch 450/500  total: 3.69821, -LL: 3.62343, prior: 0.82277, aleatoric unc.: 9.72872\n",
      "Epoch 451/500  total: 3.69862, -LL: 3.61988, prior: 0.81971, aleatoric unc.: 9.72094\n",
      "Epoch 452/500  total: 3.69956, -LL: 3.64692, prior: 0.82058, aleatoric unc.: 9.71754\n",
      "Epoch 453/500  total: 3.68746, -LL: 3.61281, prior: 0.81960, aleatoric unc.: 9.69424\n",
      "Epoch 454/500  total: 3.69136, -LL: 3.61534, prior: 0.82130, aleatoric unc.: 9.67839\n",
      "Epoch 455/500  total: 3.70947, -LL: 3.63629, prior: 0.82276, aleatoric unc.: 9.70095\n",
      "Epoch 456/500  total: 3.70455, -LL: 3.62145, prior: 0.82456, aleatoric unc.: 9.71287\n",
      "Epoch 457/500  total: 3.69434, -LL: 3.64094, prior: 0.82408, aleatoric unc.: 9.70524\n",
      "Epoch 458/500  total: 3.69588, -LL: 3.62607, prior: 0.82356, aleatoric unc.: 9.69614\n",
      "Epoch 459/500  total: 3.69965, -LL: 3.67129, prior: 0.82353, aleatoric unc.: 9.69692\n",
      "Epoch 460/500  total: 3.69278, -LL: 3.64340, prior: 0.82238, aleatoric unc.: 9.68522\n",
      "Epoch 461/500  total: 3.69769, -LL: 3.66663, prior: 0.82365, aleatoric unc.: 9.68477\n",
      "Epoch 462/500  total: 3.70174, -LL: 3.64461, prior: 0.82374, aleatoric unc.: 9.69469\n",
      "Epoch 463/500  total: 3.69422, -LL: 3.62371, prior: 0.82634, aleatoric unc.: 9.68775\n",
      "Epoch 464/500  total: 3.70086, -LL: 3.65318, prior: 0.82430, aleatoric unc.: 9.69298\n",
      "Epoch 465/500  total: 3.69787, -LL: 3.66150, prior: 0.81959, aleatoric unc.: 9.69327\n",
      "Epoch 466/500  total: 3.69348, -LL: 3.64051, prior: 0.81754, aleatoric unc.: 9.68517\n",
      "Epoch 467/500  total: 3.70298, -LL: 3.65390, prior: 0.81789, aleatoric unc.: 9.69441\n",
      "Epoch 468/500  total: 3.69817, -LL: 3.65901, prior: 0.81541, aleatoric unc.: 9.69768\n",
      "Epoch 469/500  total: 3.69301, -LL: 3.63411, prior: 0.81612, aleatoric unc.: 9.68669\n",
      "Epoch 470/500  total: 3.70037, -LL: 3.61972, prior: 0.81542, aleatoric unc.: 9.69086\n",
      "Epoch 471/500  total: 3.70139, -LL: 3.60899, prior: 0.81560, aleatoric unc.: 9.69815\n",
      "Epoch 472/500  total: 3.70118, -LL: 3.68662, prior: 0.81407, aleatoric unc.: 9.70126\n",
      "Epoch 473/500  total: 3.70042, -LL: 3.64964, prior: 0.81481, aleatoric unc.: 9.70599\n",
      "Epoch 474/500  total: 3.70516, -LL: 3.62561, prior: 0.81684, aleatoric unc.: 9.71484\n",
      "Epoch 475/500  total: 3.69569, -LL: 3.62760, prior: 0.81538, aleatoric unc.: 9.70852\n",
      "Epoch 476/500  total: 3.69344, -LL: 3.63575, prior: 0.81317, aleatoric unc.: 9.69601\n",
      "Epoch 477/500  total: 3.69725, -LL: 3.66014, prior: 0.81203, aleatoric unc.: 9.69300\n",
      "Epoch 478/500  total: 3.70594, -LL: 3.65619, prior: 0.81437, aleatoric unc.: 9.70621\n",
      "Epoch 479/500  total: 3.69992, -LL: 3.61516, prior: 0.81284, aleatoric unc.: 9.70868\n",
      "Epoch 480/500  total: 3.69582, -LL: 3.64135, prior: 0.81684, aleatoric unc.: 9.70021\n",
      "Epoch 481/500  total: 3.69739, -LL: 3.60619, prior: 0.81517, aleatoric unc.: 9.70025\n",
      "Epoch 482/500  total: 3.70488, -LL: 3.63401, prior: 0.81520, aleatoric unc.: 9.70766\n",
      "Epoch 483/500  total: 3.69736, -LL: 3.61727, prior: 0.81478, aleatoric unc.: 9.70594\n",
      "Epoch 484/500  total: 3.70235, -LL: 3.63803, prior: 0.81578, aleatoric unc.: 9.71074\n",
      "Epoch 485/500  total: 3.70031, -LL: 3.64106, prior: 0.81726, aleatoric unc.: 9.71344\n"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Epoch 486/500  total: 3.70281, -LL: 3.65334, prior: 0.81765, aleatoric unc.: 9.71675\n",
      "Epoch 487/500  total: 3.69665, -LL: 3.62427, prior: 0.81798, aleatoric unc.: 9.71068\n",
      "Epoch 488/500  total: 3.69354, -LL: 3.60938, prior: 0.81778, aleatoric unc.: 9.69865\n",
      "Epoch 489/500  total: 3.69664, -LL: 3.66371, prior: 0.81562, aleatoric unc.: 9.69401\n",
      "Epoch 490/500  total: 3.70017, -LL: 3.61261, prior: 0.81859, aleatoric unc.: 9.69709\n",
      "Epoch 491/500  total: 3.70181, -LL: 3.64116, prior: 0.81858, aleatoric unc.: 9.70037\n",
      "Epoch 492/500  total: 3.69507, -LL: 3.62122, prior: 0.81482, aleatoric unc.: 9.69759\n",
      "Epoch 493/500  total: 3.69890, -LL: 3.65301, prior: 0.81908, aleatoric unc.: 9.69581\n",
      "Epoch 494/500  total: 3.70058, -LL: 3.65166, prior: 0.81909, aleatoric unc.: 9.69850\n",
      "Epoch 495/500  total: 3.69788, -LL: 3.62267, prior: 0.81817, aleatoric unc.: 9.70015\n",
      "Epoch 496/500  total: 3.69267, -LL: 3.64171, prior: 0.81855, aleatoric unc.: 9.68762\n",
      "Epoch 497/500  total: 3.70465, -LL: 3.66362, prior: 0.81682, aleatoric unc.: 9.70045\n",
      "Epoch 498/500  total: 3.70106, -LL: 3.66015, prior: 0.81733, aleatoric unc.: 9.70227\n",
      "Epoch 499/500  total: 3.69741, -LL: 3.64281, prior: 0.81737, aleatoric unc.: 9.70153\n"
     ]
    }
   ],
   "source": [
    "epochs = 500\n",
    "# for each epoch\n",
    "for epoch in range(epochs):\n",
    "    losses = list()\n",
    "    # for each mini-batch given by the loader:\n",
    "    for batch in loader:\n",
    "        # get the input in the mini-batch\n",
    "        # this has size (B, C)\n",
    "        # where B is the mini-batch size\n",
    "        # C is the number of features (1 in this case)\n",
    "        features = batch[\"data\"]\n",
    "        # get the targets in the mini-batch (there shall be B of them)\n",
    "        target = batch[\"target\"]\n",
    "        # get the output of the neural network:\n",
    "        prediction = b_network(features)\n",
    "        \n",
    "        # calculate the loss function being minimized\n",
    "        # in this case, it is the mean-squared error between the prediction and the target values added\n",
    "        # to the Kullback-Leibler divergence between the current weight Gaussian and\n",
    "        # the prior Gaussian, set to the unit Normal distribution\n",
    "        nll = b_network.nll(prediction, target)\n",
    "        prior = kl_loss(b_network)\n",
    "        loss = nll + weight_kl * prior\n",
    "\n",
    "        # clean the optimizer temporary gradient storage\n",
    "        optimizer.zero_grad()\n",
    "        # calculate the gradient of the loss function as a function of the gradients\n",
    "        loss.backward()\n",
    "        # ask the Adam optimizer to change the parameters in the direction of - gradient\n",
    "        # Adam scales the gradient by a constant which is adaptively tuned\n",
    "        # take a look at the Adam paper for more details: https://arxiv.org/abs/1412.6980\n",
    "        optimizer.step()\n",
    "        \n",
    "        ale = b_network.aleatoric_uncertainty().detach().numpy()\n",
    "\n",
    "        losses.append(loss.detach().cpu().item())\n",
    "    avg_loss = np.mean(np.array(losses))\n",
    "    print(f\"Epoch {epoch}/{epochs}  total: {avg_loss:.5f}, -LL: {nll.item():.5f}, prior: {prior.item():.5f}, aleatoric unc.: {ale:.5f}\")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "15f3f633",
   "metadata": {},
   "source": [
    "To evaluate the effect of the uncertainty, we perform the prediction many times for the same data and take the average and root-mean-squared-error of the predictions, since each prediction performed with the Bayesian Neural Network leads to a different results, using a different weight, selected from the final Gaussian."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "a3d244e2",
   "metadata": {},
   "outputs": [],
   "source": [
    "b_predicted = list()\n",
    "for k in range(10):\n",
    "    p = b_network(torch.from_numpy(test_data[:,0:1])).detach().numpy()\n",
    "    b_predicted.append(p[:,0])\n",
    "b_predicted = np.stack(b_predicted, axis=1)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "85fed034",
   "metadata": {},
   "source": [
    "We can now take the average result for each sample, and their root-mean-squared-error as an estimate of the mean and epistemic uncertainty for the results.\n",
    "\n",
    "The aleatoric uncertainty is fitted as an independent parameter. Since we assume the aleatoric uncertainty is independent, we can calculate the total uncertainty as the sum of squares of the epistemic and aleatoric uncertainty."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "4a56960d",
   "metadata": {},
   "outputs": [],
   "source": [
    "b_mean = np.mean(b_predicted, axis=1)\n",
    "b_sigma = np.std(b_predicted, axis=1)\n",
    "aleatoric_uncertainty = b_network.aleatoric_uncertainty().detach().numpy()\n",
    "\n",
    "total_uncertainty = (b_sigma**2 + aleatoric_uncertainty**2)**0.5"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6ef3d430",
   "metadata": {},
   "source": [
    "Let's check how big are those uncertainties found:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "4d01c41f",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Average epistemic uncertainty:  0.83653456\n"
     ]
    }
   ],
   "source": [
    "print(\"Average epistemic uncertainty: \", np.mean(b_sigma))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "67d456a1",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Aleatoric uncertainty:  9.7015295\n"
     ]
    }
   ],
   "source": [
    "print(\"Aleatoric uncertainty: \", aleatoric_uncertainty)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "364bdcd7",
   "metadata": {},
   "source": [
    "Note that the aleatoric uncertainty is very close to the standard deviation of the $\\epsilon$ component of the model we created in the beginning! Clearly the model could fit the uncertainty coming from that component of the noise.\n",
    "\n",
    "It is not easy to estimate the effect of the epistemic uncertainty, as it is different for every data point (as it is scaled by $x^2$), but we can plot it to take a look at its effect.\n",
    "\n",
    "Note that the uncertainties are the standard deviations of Gaussian models and therefore they correspond to a $1\\sigma$ quantile band, which is a 67% confidence band. The quantile corresponding to $2\\sigma$ corresponds to a 95% confidence band in a Gaussian model."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "8b9142e8",
   "metadata": {
    "scrolled": false
   },
   "outputs": [
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       "        button.addEventListener('mouseover', on_mouseover_closure(tooltip));\n",
       "\n",
       "        var icon_img = document.createElement('img');\n",
       "        icon_img.src = '_images/' + image + '.png';\n",
       "        icon_img.srcset = '_images/' + image + '_large.png 2x';\n",
       "        icon_img.alt = tooltip;\n",
       "        button.appendChild(icon_img);\n",
       "\n",
       "        buttonGroup.appendChild(button);\n",
       "    }\n",
       "\n",
       "    if (buttonGroup.hasChildNodes()) {\n",
       "        toolbar.appendChild(buttonGroup);\n",
       "    }\n",
       "\n",
       "    var fmt_picker = document.createElement('select');\n",
       "    fmt_picker.classList = 'mpl-widget';\n",
       "    toolbar.appendChild(fmt_picker);\n",
       "    this.format_dropdown = fmt_picker;\n",
       "\n",
       "    for (var ind in mpl.extensions) {\n",
       "        var fmt = mpl.extensions[ind];\n",
       "        var option = document.createElement('option');\n",
       "        option.selected = fmt === mpl.default_extension;\n",
       "        option.innerHTML = fmt;\n",
       "        fmt_picker.appendChild(option);\n",
       "    }\n",
       "\n",
       "    var status_bar = document.createElement('span');\n",
       "    status_bar.classList = 'mpl-message';\n",
       "    toolbar.appendChild(status_bar);\n",
       "    this.message = status_bar;\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.request_resize = function (x_pixels, y_pixels) {\n",
       "    // Request matplotlib to resize the figure. Matplotlib will then trigger a resize in the client,\n",
       "    // which will in turn request a refresh of the image.\n",
       "    this.send_message('resize', { width: x_pixels, height: y_pixels });\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.send_message = function (type, properties) {\n",
       "    properties['type'] = type;\n",
       "    properties['figure_id'] = this.id;\n",
       "    this.ws.send(JSON.stringify(properties));\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.send_draw_message = function () {\n",
       "    if (!this.waiting) {\n",
       "        this.waiting = true;\n",
       "        this.ws.send(JSON.stringify({ type: 'draw', figure_id: this.id }));\n",
       "    }\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.handle_save = function (fig, _msg) {\n",
       "    var format_dropdown = fig.format_dropdown;\n",
       "    var format = format_dropdown.options[format_dropdown.selectedIndex].value;\n",
       "    fig.ondownload(fig, format);\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.handle_resize = function (fig, msg) {\n",
       "    var size = msg['size'];\n",
       "    if (size[0] !== fig.canvas.width || size[1] !== fig.canvas.height) {\n",
       "        fig._resize_canvas(size[0], size[1], msg['forward']);\n",
       "        fig.send_message('refresh', {});\n",
       "    }\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.handle_rubberband = function (fig, msg) {\n",
       "    var x0 = msg['x0'] / fig.ratio;\n",
       "    var y0 = (fig.canvas.height - msg['y0']) / fig.ratio;\n",
       "    var x1 = msg['x1'] / fig.ratio;\n",
       "    var y1 = (fig.canvas.height - msg['y1']) / fig.ratio;\n",
       "    x0 = Math.floor(x0) + 0.5;\n",
       "    y0 = Math.floor(y0) + 0.5;\n",
       "    x1 = Math.floor(x1) + 0.5;\n",
       "    y1 = Math.floor(y1) + 0.5;\n",
       "    var min_x = Math.min(x0, x1);\n",
       "    var min_y = Math.min(y0, y1);\n",
       "    var width = Math.abs(x1 - x0);\n",
       "    var height = Math.abs(y1 - y0);\n",
       "\n",
       "    fig.rubberband_context.clearRect(\n",
       "        0,\n",
       "        0,\n",
       "        fig.canvas.width / fig.ratio,\n",
       "        fig.canvas.height / fig.ratio\n",
       "    );\n",
       "\n",
       "    fig.rubberband_context.strokeRect(min_x, min_y, width, height);\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.handle_figure_label = function (fig, msg) {\n",
       "    // Updates the figure title.\n",
       "    fig.header.textContent = msg['label'];\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.handle_cursor = function (fig, msg) {\n",
       "    var cursor = msg['cursor'];\n",
       "    switch (cursor) {\n",
       "        case 0:\n",
       "            cursor = 'pointer';\n",
       "            break;\n",
       "        case 1:\n",
       "            cursor = 'default';\n",
       "            break;\n",
       "        case 2:\n",
       "            cursor = 'crosshair';\n",
       "            break;\n",
       "        case 3:\n",
       "            cursor = 'move';\n",
       "            break;\n",
       "    }\n",
       "    fig.rubberband_canvas.style.cursor = cursor;\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.handle_message = function (fig, msg) {\n",
       "    fig.message.textContent = msg['message'];\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.handle_draw = function (fig, _msg) {\n",
       "    // Request the server to send over a new figure.\n",
       "    fig.send_draw_message();\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.handle_image_mode = function (fig, msg) {\n",
       "    fig.image_mode = msg['mode'];\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.handle_history_buttons = function (fig, msg) {\n",
       "    for (var key in msg) {\n",
       "        if (!(key in fig.buttons)) {\n",
       "            continue;\n",
       "        }\n",
       "        fig.buttons[key].disabled = !msg[key];\n",
       "        fig.buttons[key].setAttribute('aria-disabled', !msg[key]);\n",
       "    }\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.handle_navigate_mode = function (fig, msg) {\n",
       "    if (msg['mode'] === 'PAN') {\n",
       "        fig.buttons['Pan'].classList.add('active');\n",
       "        fig.buttons['Zoom'].classList.remove('active');\n",
       "    } else if (msg['mode'] === 'ZOOM') {\n",
       "        fig.buttons['Pan'].classList.remove('active');\n",
       "        fig.buttons['Zoom'].classList.add('active');\n",
       "    } else {\n",
       "        fig.buttons['Pan'].classList.remove('active');\n",
       "        fig.buttons['Zoom'].classList.remove('active');\n",
       "    }\n",
       "};\n",
       "\n",
       "mpl.figure.prototype.updated_canvas_event = function () {\n",
       "    // Called whenever the canvas gets updated.\n",
       "    this.send_message('ack', {});\n",
       "};\n",
       "\n",
       "// A function to construct a web socket function for onmessage handling.\n",
       "// Called in the figure constructor.\n",
       "mpl.figure.prototype._make_on_message_function = function (fig) {\n",
       "    return function socket_on_message(evt) {\n",
       "        if (evt.data instanceof Blob) {\n",
       "            var img = evt.data;\n",
       "            if (img.type !== 'image/png') {\n",
       "                /* FIXME: We get \"Resource interpreted as Image but\n",
       "                 * transferred with MIME type text/plain:\" errors on\n",
       "                 * Chrome.  But how to set the MIME type?  It doesn't seem\n",
       "                 * to be part of the websocket stream */\n",
       "                img.type = 'image/png';\n",
       "            }\n",
       "\n",
       "            /* Free the memory for the previous frames */\n",
       "            if (fig.imageObj.src) {\n",
       "                (window.URL || window.webkitURL).revokeObjectURL(\n",
       "                    fig.imageObj.src\n",
       "                );\n",
       "            }\n",
       "\n",
       "            fig.imageObj.src = (window.URL || window.webkitURL).createObjectURL(\n",
       "                img\n",
       "            );\n",
       "            fig.updated_canvas_event();\n",
       "            fig.waiting = false;\n",
       "            return;\n",
       "        } else if (\n",
       "            typeof evt.data === 'string' &&\n",
       "            evt.data.slice(0, 21) === 'data:image/png;base64'\n",
       "        ) {\n",
       "            fig.imageObj.src = evt.data;\n",
       "            fig.updated_canvas_event();\n",
       "            fig.waiting = false;\n",
       "            return;\n",
       "        }\n",
       "\n",
       "        var msg = JSON.parse(evt.data);\n",
       "        var msg_type = msg['type'];\n",
       "\n",
       "        // Call the  \"handle_{type}\" callback, which takes\n",
       "        // the figure and JSON message as its only arguments.\n",
       "        try {\n",
       "            var callback = fig['handle_' + msg_type];\n",
       "        } catch (e) {\n",
       "            console.log(\n",
       "                \"No handler for the '\" + msg_type + \"' message type: \",\n",
       "                msg\n",
       "            );\n",
       "            return;\n",
       "        }\n",
       "\n",
       "        if (callback) {\n",
       "            try {\n",
       "                // console.log(\"Handling '\" + msg_type + \"' message: \", msg);\n",
       "                callback(fig, msg);\n",
       "            } catch (e) {\n",
       "                console.log(\n",
       "                    \"Exception inside the 'handler_\" + msg_type + \"' callback:\",\n",
       "                    e,\n",
       "                    e.stack,\n",
       "                    msg\n",
       "                );\n",
       "            }\n",
       "        }\n",
       "    };\n",
       "};\n",
       "\n",
       "// from http://stackoverflow.com/questions/1114465/getting-mouse-location-in-canvas\n",
       "mpl.findpos = function (e) {\n",
       "    //this section is from http://www.quirksmode.org/js/events_properties.html\n",
       "    var targ;\n",
       "    if (!e) {\n",
       "        e = window.event;\n",
       "    }\n",
       "    if (e.target) {\n",
       "        targ = e.target;\n",
       "    } else if (e.srcElement) {\n",
       "        targ = e.srcElement;\n",
       "    }\n",
       "    if (targ.nodeType === 3) {\n",
       "        // defeat Safari bug\n",
       "        targ = targ.parentNode;\n",
       "    }\n",
       "\n",
       "    // pageX,Y are the mouse positions relative to the document\n",
       "    var boundingRect = targ.getBoundingClientRect();\n",
       "    var x = e.pageX - (boundingRect.left + document.body.scrollLeft);\n",
       "    var y = e.pageY - (boundingRect.top + document.body.scrollTop);\n",
       "\n",
       "    return { x: x, y: y };\n",
       "};\n",
       "\n",
       "/*\n",
       " * return a copy of an object with only non-object keys\n",
       " * we need this to avoid circular references\n",
       " * http://stackoverflow.com/a/24161582/3208463\n",
       " */\n",
       "function simpleKeys(original) {\n",
       "    return Object.keys(original).reduce(function (obj, key) {\n",
       "        if (typeof original[key] !== 'object') {\n",
       "            obj[key] = original[key];\n",
       "        }\n",
       "        return obj;\n",
       "    }, {});\n",
       "}\n",
       "\n",
       "mpl.figure.prototype.mouse_event = function (event, name) {\n",
       "    var canvas_pos = mpl.findpos(event);\n",
       "\n",
       "    if (name === 'button_press') {\n",
       "        this.canvas.focus();\n",
       "        this.canvas_div.focus();\n",
       "    }\n",
       "\n",
       "    var x = canvas_pos.x * this.ratio;\n",
       "    var y = canvas_pos.y * this.ratio;\n",
       "\n",
       "    this.send_message(name, {\n",
       "        x: x,\n",
       "        y: y,\n",
       "        button: event.button,\n",