In model m12.3
in Statistical Rethinking Edition 1, representative values for the tadpole data are generated.
The multilevel binomial model is defined as
import Random
using DataFrames
using StatsFuns: logistic
using Turing
Random.seed!(1)
μ = 1.4
σ = 1.5
nponds = 60
ni = repeat([5,10,25,35], inner=15)
a_pond = rand(Normal(μ, σ), nponds)
dsim = DataFrame(pond = 1:nponds, ni = ni, true_a = a_pond)
prob = logistic.(dsim.true_a)
dsim.s = [rand(Binomial(ni[i], prob[i])) for i in 1:nponds]
dsim.p_nopool = dsim.s ./ dsim.ni;
@model function m12_3(pond, s, ni)
σ ~ truncated(Cauchy(0, 1), 0, Inf)
α ~ Normal(0, 1)
N_ponds = length(pond)
α_pond ~ filldist(Normal(α, σ), N_ponds)
logitp = α_pond[pond]
s .~ BinomialLogit.(ni, logitp)
end;
chns = sample(
m12_3(dsim.pond, dsim.s, dsim.ni),
NUTS(),
1000
)
Chains MCMC chain (1000×74×1 Array{Float64, 3}):
Iterations = 501:1:1500
Number of chains = 1
Samples per chain = 1000
Wall duration = 15.36 seconds
Compute duration = 15.36 seconds
parameters = σ, α, α_pond[1], α_pond[2], α_pond[3], α_pond[4], α_pond[5], α_pond[6], α_pond[7], α_pond[8], α_pond[9], α_pond[10], α_pond[11], α_pond[12], α_pond[13], α_pond[14], α_pond[15], α_pond[16], α_pond[17], α_pond[18], α_pond[19], α_pond[20], α_pond[21], α_pond[22], α_pond[23], α_pond[24], α_pond[25], α_pond[26], α_pond[27], α_pond[28], α_pond[29], α_pond[30], α_pond[31], α_pond[32], α_pond[33], α_pond[34], α_pond[35], α_pond[36], α_pond[37], α_pond[38], α_pond[39], α_pond[40], α_pond[41], α_pond[42], α_pond[43], α_pond[44], α_pond[45], α_pond[46], α_pond[47], α_pond[48], α_pond[49], α_pond[50], α_pond[51], α_pond[52], α_pond[53], α_pond[54], α_pond[55], α_pond[56], α_pond[57], α_pond[58], α_pond[59], α_pond[60]
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
Summary Statistics
parameters mean std naive_se mcse ess rhat ess_per_sec
Symbol Float64 Float64 Float64 Float64 Float64 Float64 Float64
σ 1.8009 0.2546 0.0081 0.0135 384.2791 1.0010 25.0214
α 1.5445 0.2587 0.0082 0.0091 571.5669 1.0034 37.2162
α_pond[1] 2.9456 1.3747 0.0435 0.0477 856.9236 0.9992 55.7966
α_pond[2] 2.9122 1.3460 0.0426 0.0337 929.2347 1.0033 60.5049
α_pond[3] 1.6562 1.0080 0.0319 0.0303 1061.6547 0.9995 69.1271
α_pond[4] 2.8702 1.3196 0.0417 0.0431 1048.0109 1.0009 68.2388
α_pond[5] 2.9209 1.3346 0.0422 0.0475 823.2436 0.9991 53.6036
α_pond[6] 2.8979 1.2955 0.0410 0.0329 1237.6350 0.9990 80.5857
α_pond[7] 2.8859 1.3473 0.0426 0.0327 1317.3384 1.0007 85.7754
α_pond[8] -0.7466 0.8979 0.0284 0.0262 1254.0289 1.0010 81.6531
α_pond[9] 1.6310 0.9941 0.0314 0.0326 852.6673 0.9991 55.5194
α_pond[10] 0.7736 0.8738 0.0276 0.0225 1430.4703 0.9992 93.1417
α_pond[11] 1.6287 1.0356 0.0327 0.0288 1206.4127 0.9990 78.5527
α_pond[12] 1.6440 1.0025 0.0317 0.0227 1436.5007 0.9990 93.5344
α_pond[13] 2.9310 1.3123 0.0415 0.0520 692.6341 0.9990 45.0992
α_pond[14] 2.9884 1.3382 0.0423 0.0467 848.4459 1.0004 55.2446
α_pond[15] 2.9607 1.3207 0.0418 0.0487 735.7561 1.0049 47.9070
α_pond[16] 3.3630 1.2441 0.0393 0.0407 754.5678 1.0005 49.1319
α_pond[17] 3.3882 1.2444 0.0394 0.0383 832.6468 1.0013 54.2158
α_pond[18] 3.3242 1.2384 0.0392 0.0398 941.6705 1.0018 61.3147
α_pond[19] 1.5157 0.7267 0.0230 0.0211 984.0112 0.9990 64.0716
α_pond[20] 2.2549 0.9146 0.0289 0.0312 1003.9205 1.0033 65.3679
α_pond[21] 2.1919 0.9270 0.0293 0.0295 1108.0611 1.0019 72.1488
α_pond[22] -0.5924 0.6676 0.0211 0.0223 699.9895 1.0011 45.5782
α_pond[23] 1.0377 0.6964 0.0220 0.0191 1653.2369 0.9990 107.6466
α_pond[24] 3.3448 1.2732 0.0403 0.0421 1026.3001 0.9997 66.8251
α_pond[25] 0.6058 0.6269 0.0198 0.0182 970.8337 1.0001 63.2135
α_pond[26] 3.4360 1.3507 0.0427 0.0433 892.4738 1.0004 58.1113
α_pond[27] 3.3562 1.1970 0.0379 0.0316 1148.5416 0.9990 74.7846
α_pond[28] 3.3755 1.2703 0.0402 0.0462 752.3470 1.0017 48.9873
α_pond[29] -0.1797 0.5872 0.0186 0.0132 1771.5999 0.9993 115.3536
α_pond[30] -0.5753 0.6247 0.0198 0.0156 1341.5381 0.9990 87.3511
α_pond[31] 3.8807 1.0604 0.0335 0.0400 838.5016 0.9990 54.5971
α_pond[32] 1.0401 0.4468 0.0141 0.0111 1209.1466 0.9990 78.7307
α_pond[33] 3.9757 1.1786 0.0373 0.0311 1121.2720 1.0003 73.0090
α_pond[34] 2.4749 0.6767 0.0214 0.0245 860.9840 0.9990 56.0609
α_pond[35] 0.6438 0.4074 0.0129 0.0103 2039.2275 0.9992 132.7795
α_pond[36] 2.4406 0.7111 0.0225 0.0229 996.8285 1.0030 64.9061
α_pond[37] 0.8283 0.4398 0.0139 0.0116 1411.0920 1.0000 91.8799
α_pond[38] 1.7294 0.5332 0.0169 0.0198 660.2453 1.0030 42.9903
α_pond[39] 1.2423 0.4584 0.0145 0.0112 1279.2957 0.9993 83.2983
α_pond[40] -1.2411 0.5151 0.0163 0.0131 1259.7507 0.9996 82.0257
α_pond[41] 1.0367 0.4556 0.0144 0.0158 738.5686 0.9991 48.0902
α_pond[42] 3.0647 0.9037 0.0286 0.0277 1260.6724 0.9997 82.0857
α_pond[43] -0.1574 0.4179 0.0132 0.0110 1615.2679 0.9990 105.1744
α_pond[44] 0.6613 0.4128 0.0131 0.0135 786.5108 1.0008 51.2118
α_pond[45] 1.4483 0.5242 0.0166 0.0145 1209.3514 0.9995 78.7441
α_pond[46] 1.0988 0.3802 0.0120 0.0128 708.9005 0.9992 46.1584
α_pond[47] 2.4128 0.5905 0.0187 0.0189 941.7316 0.9990 61.3186
α_pond[48] 2.7997 0.6385 0.0202 0.0208 935.7683 1.0009 60.9304
α_pond[49] 2.8003 0.6622 0.0209 0.0194 1076.5409 1.0001 70.0964
α_pond[50] -1.2775 0.3896 0.0123 0.0092 1951.7548 1.0002 127.0839
α_pond[51] 0.4757 0.3611 0.0114 0.0087 1800.3932 1.0011 117.2284
α_pond[52] -0.9645 0.3651 0.0115 0.0143 1053.1285 0.9991 68.5720
α_pond[53] 1.4315 0.4124 0.0130 0.0114 1241.0271 0.9991 80.8066
α_pond[54] -0.5752 0.3387 0.0107 0.0127 816.5744 0.9993 53.1693
α_pond[55] -1.4491 0.4380 0.0139 0.0115 1393.2439 1.0039 90.7178
α_pond[56] -2.3787 0.5605 0.0177 0.0221 850.4970 1.0006 55.3781
α_pond[57] 2.8014 0.6389 0.0202 0.0194 1062.9600 0.9993 69.2121
α_pond[58] 3.3421 0.8725 0.0276 0.0299 1022.8664 0.9991 66.6015
α_pond[59] 0.2279 0.3378 0.0107 0.0099 985.1045 1.0005 64.1428
α_pond[60] 1.2862 0.3872 0.0122 0.0102 1305.9014 0.9990 85.0307
Quantiles
parameters 2.5% 25.0% 50.0% 75.0% 97.5%
Symbol Float64 Float64 Float64 Float64 Float64
σ 1.3557 1.6183 1.7856 1.9661 2.3300
α 1.0713 1.3577 1.5417 1.7192 2.0382
α_pond[1] 0.5568 1.9688 2.8001 3.8156 6.0519
α_pond[2] 0.5924 1.9462 2.7717 3.6928 5.9424
α_pond[3] -0.1537 0.9668 1.5982 2.3438 3.6991
α_pond[4] 0.7078 1.9286 2.7376 3.6813 5.6706
α_pond[5] 0.7441 1.9642 2.7829 3.7098 5.8995
α_pond[6] 0.7198 1.9699 2.7845 3.7469 5.7244
α_pond[7] 0.6369 1.9186 2.7893 3.7337 5.7659
α_pond[8] -2.6227 -1.2896 -0.7124 -0.1582 0.8904
α_pond[9] -0.1302 0.8959 1.5833 2.2647 3.6451
α_pond[10] -0.8590 0.2036 0.7510 1.3598 2.6028
α_pond[11] -0.1340 0.9038 1.5460 2.3231 3.7764
α_pond[12] -0.2087 0.9158 1.6396 2.3287 3.7049
α_pond[13] 0.7129 2.0685 2.8336 3.6838 5.9100
α_pond[14] 0.7384 2.0262 2.8672 3.8032 5.9406
α_pond[15] 0.6733 2.0174 2.9014 3.7628 5.7862
α_pond[16] 1.2601 2.4578 3.2418 4.1804 6.0656
α_pond[17] 1.2886 2.5407 3.2223 4.1354 6.2083
α_pond[18] 1.3123 2.4339 3.1828 4.1064 5.9721
α_pond[19] 0.1554 1.0021 1.4874 1.9905 3.0045
α_pond[20] 0.6853 1.6019 2.1769 2.7844 4.3713
α_pond[21] 0.6222 1.5499 2.1386 2.7733 4.0952
α_pond[22] -1.9984 -0.9900 -0.5767 -0.1719 0.6546
α_pond[23] -0.2023 0.5691 0.9948 1.4978 2.4355
α_pond[24] 1.2337 2.4523 3.2124 4.1475 5.9780
α_pond[25] -0.6484 0.1921 0.5854 1.0095 1.9133
α_pond[26] 1.2560 2.4876 3.2679 4.2234 6.4553
α_pond[27] 1.3658 2.5129 3.2522 4.0746 6.0685
α_pond[28] 1.3053 2.4913 3.2514 4.1248 6.2159
α_pond[29] -1.2736 -0.5874 -0.1826 0.2360 0.9325
α_pond[30] -1.8882 -0.9621 -0.5655 -0.1456 0.5820
α_pond[31] 2.1720 3.0881 3.7838 4.4961 6.2932
α_pond[32] 0.2137 0.7549 1.0347 1.2970 1.9924
α_pond[33] 2.0444 3.1100 3.8427 4.7508 6.5654
α_pond[34] 1.2903 1.9685 2.4322 2.9291 3.9418
α_pond[35] -0.1048 0.3764 0.6252 0.8907 1.5131
α_pond[36] 1.2439 1.9123 2.3936 2.8881 4.0584
α_pond[37] 0.0168 0.5422 0.8201 1.1027 1.6976
α_pond[38] 0.7950 1.3525 1.6991 2.0844 2.8302
α_pond[39] 0.4156 0.9262 1.2258 1.5304 2.2246
α_pond[40] -2.3607 -1.5852 -1.2061 -0.8680 -0.3567
α_pond[41] 0.2185 0.7347 0.9945 1.2861 2.0946
α_pond[42] 1.6087 2.4099 2.9706 3.5998 5.1271
α_pond[43] -0.9936 -0.4250 -0.1627 0.1165 0.6477
α_pond[44] -0.1432 0.3822 0.6423 0.9272 1.4871
α_pond[45] 0.4655 1.0780 1.4200 1.7619 2.5498
α_pond[46] 0.3486 0.8620 1.0853 1.3332 1.8817
α_pond[47] 1.3069 2.0037 2.3858 2.8219 3.6859
α_pond[48] 1.7151 2.3510 2.7574 3.2094 4.1283
α_pond[49] 1.6422 2.3094 2.7463 3.2274 4.2894
α_pond[50] -2.0730 -1.5289 -1.2659 -0.9992 -0.5879
α_pond[51] -0.2350 0.2337 0.4698 0.7013 1.1762
α_pond[52] -1.7125 -1.2197 -0.9661 -0.7033 -0.2851
α_pond[53] 0.6613 1.1326 1.4126 1.6987 2.2572
α_pond[54] -1.2222 -0.8091 -0.5755 -0.3349 0.0740
α_pond[55] -2.3760 -1.7280 -1.4480 -1.1444 -0.6256
α_pond[56] -3.5959 -2.7393 -2.3199 -1.9973 -1.4114
α_pond[57] 1.7441 2.3458 2.7264 3.1951 4.2290
α_pond[58] 1.8799 2.7485 3.2505 3.8229 5.4142
α_pond[59] -0.4061 -0.0079 0.2399 0.4633 0.8684
α_pond[60] 0.5417 1.0222 1.2756 1.5405 2.0490
using StatsPlots
StatsPlots.plot(chns)
"/home/runner/work/TuringModels.jl/TuringModels.jl/__site/assets/models/partial-pooling-estimates/code/output/chns.svg"
m123rethinking = "
mean sd 5.5% 94.5% n_eff Rhat
a 1.30 0.23 0.94 1.67 8064 1
sigma 1.55 0.21 1.24 1.92 3839 1
a_pond[1] 2.57 1.17 0.85 4.57 9688 1
a_pond[2] 2.58 1.19 0.83 4.56 9902 1
a_pond[3] 2.56 1.16 0.84 4.57 12841 1
a_pond[4] 1.49 0.92 0.12 3.03 15532 1
a_pond[5] 1.51 0.95 0.07 3.09 14539 1
a_pond[6] 0.72 0.84 -0.59 2.08 13607 1
a_pond[7] 2.56 1.16 0.86 4.51 12204 1
a_pond[8] 1.50 0.93 0.07 3.05 19903 1
a_pond[9] 2.56 1.15 0.86 4.51 11054 1
a_pond[10] 1.49 0.95 0.05 3.09 14134 1
a_pond[11] -0.64 0.86 -2.06 0.70 15408 1
a_pond[12] 2.56 1.16 0.86 4.53 11512 1
a_pond[13] 1.49 0.95 0.05 3.10 16270 1
a_pond[14] 0.71 0.84 -0.59 2.07 17077 1
a_pond[15] 1.50 0.93 0.10 3.05 16996 1
a_pond[16] 2.98 1.07 1.45 4.84 9033 1
a_pond[17] 2.09 0.84 0.85 3.54 14636 1
a_pond[18] 1.01 0.66 0.00 2.10 12971 1
a_pond[19] 1.01 0.68 -0.03 2.13 12598 1
a_pond[20] 1.48 0.72 0.38 2.67 15500 1
a_pond[21] 2.96 1.09 1.42 4.87 11204 1
a_pond[22] -2.04 0.87 -3.53 -0.75 9065 1
a_pond[23] 0.99 0.67 -0.04 2.11 15365 1
a_pond[24] 1.48 0.72 0.41 2.67 14879 1
a_pond[25] 2.10 0.85 0.85 3.53 13298 1
a_pond[26] 1.00 0.65 0.01 2.06 18583 1
a_pond[27] 3.00 1.08 1.44 4.86 9312 1
a_pond[28] 0.98 0.66 -0.03 2.09 14703 1
a_pond[29] 0.21 0.61 -0.76 1.19 15554 1
a_pond[30] 2.95 1.05 1.45 4.73 9816 1
a_pond[31] 1.70 0.53 0.89 2.59 19148 1
a_pond[32] 0.82 0.42 0.17 1.51 13556 1
a_pond[33] 0.32 0.40 -0.33 0.96 19388 1
a_pond[34] -0.15 0.40 -0.79 0.48 18684 1
a_pond[35] 3.57 0.98 2.19 5.26 8769 1
a_pond[36] 0.16 0.40 -0.46 0.80 17595 1
a_pond[37] 2.00 0.58 1.13 2.99 14669 1
a_pond[38] -1.41 0.49 -2.22 -0.65 12957 1
a_pond[39] 1.21 0.46 0.49 1.97 14185 1
a_pond[40] -1.18 0.46 -1.95 -0.48 16142 1
a_pond[41] 2.86 0.78 1.73 4.18 10508 1
a_pond[42] 0.00 0.39 -0.61 0.63 16138 1
a_pond[43] 1.43 0.48 0.70 2.24 17100 1
a_pond[44] 2.86 0.77 1.75 4.15 12002 1
a_pond[45] -1.40 0.49 -2.21 -0.66 14292 1
a_pond[46] 0.12 0.33 -0.40 0.66 20425 1
a_pond[47] -0.56 0.36 -1.14 0.00 18981 1
a_pond[48] 1.11 0.38 0.52 1.73 14176 1
a_pond[49] 3.81 0.95 2.47 5.45 8841 1
a_pond[50] 2.05 0.50 1.31 2.88 15898 1
a_pond[51] -1.40 0.41 -2.08 -0.76 17188 1
a_pond[52] -0.11 0.34 -0.65 0.43 17158 1
a_pond[53] 1.61 0.44 0.94 2.36 15132 1
a_pond[54] 2.05 0.50 1.30 2.89 15799 1
a_pond[55] 3.14 0.75 2.08 4.40 12702 1
a_pond[56] 3.13 0.74 2.07 4.41 11143 1
a_pond[57] 1.26 0.40 0.65 1.92 14587 1
a_pond[58] 1.11 0.38 0.51 1.74 21740 1
a_pond[59] 2.33 0.56 1.50 3.25 13116 1
a_pond[60] 1.27 0.40 0.66 1.91 15611 1
";