using StatisticalRethinking
using Turing
Turing.setadbackend(:reverse_diff)
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.(Vector{Real}(dsim[:true_a]))
dsim[:si] = [rand(Binomial(ni[i], prob[i])) for i = 1:nponds]60-element Array{Int64,1}:
0
5
2
2
4
5
5
4
4
3
⋮
6
32
26
34
22
31
28
28
34Used only in the continuation of this example
dsim[:p_nopool] = dsim[:si] ./ dsim[:ni]
@model m12_3(pond, si, ni) = beginSeparate priors on μ and σ for each pond
σ ~ Truncated(Cauchy(0, 1), 0, Inf)
μ ~ Normal(0, 1)Number of ponds in the data set
N_ponds = length(pond)vector for the priors for each pond
a_pond = Vector{Real}(undef, N_ponds)For each pond set a prior. Note the [] around Normal(), i.e.,
a_pond ~ [Normal(μ, σ)]Observation
logitp = [a_pond[pond[i]] for i = 1:N_ponds]
si ~ VecBinomialLogit(ni, logitp)
end
posterior = sample(m12_3(Vector{Int64}(dsim[:pond]), Vector{Int64}(dsim[:si]),
Vector{Int64}(dsim[:ni])), Turing.NUTS(10000, 1000, 0.8))
describe(posterior) Mean SD Naive SE MCSE ESSapond[13] 0.0177228141 0.841956449 0.0084195645 0.00771496194 10000.0000 apond[39] 0.0066110725 0.403509812 0.0040350981 0.00300378764 10000.0000 apond[12] -0.6879306819 0.887142394 0.0088714239 0.00983038264 8144.1511 apond[33] 1.0230070658 0.440802431 0.0044080243 0.00340565683 10000.0000 apond[40] -0.4710107332 0.397151811 0.0039715181 0.00287378746 10000.0000 μ 1.4171959646 0.247164994 0.0024716499 0.00279659983 7811.1193 apond[52] -0.1075137501 0.334424028 0.0033442403 0.00304380214 10000.0000 apond[48] 0.2270466468 0.340522575 0.0034052258 0.00288072684 10000.0000 apond[31] -0.3116943388 0.406497387 0.0040649739 0.00327039672 10000.0000 apond[21] -0.5664379565 0.649981939 0.0064998194 0.00531522292 10000.0000 apond[14] -0.6980794242 0.863848603 0.0086384860 0.00834117683 10000.0000 apond[26] 1.5157188342 0.746372296 0.0074637230 0.00689302178 10000.0000 apond[35] -0.3104176671 0.400246807 0.0040024681 0.00351529487 10000.0000 apond[53] 2.3656224490 0.570317605 0.0057031760 0.00514104769 10000.0000 apond[45] 3.7602403329 1.040583568 0.0104058357 0.01371112136 5759.8076 apond[27] 3.1787683092 1.143238040 0.0114323804 0.01676903058 4647.9069 apond[10] 2.7440414813 1.264526612 0.0126452661 0.01600961219 6238.7032 apond[19] 3.1606479454 1.166461112 0.0116646111 0.01482980989 6186.8468 apond[58] 3.2339626006 0.791995574 0.0079199557 0.00801476020 9764.8245 apond[5] 0.7262878590 0.875745952 0.0087574595 0.00740596596 10000.0000 apond[36] 0.8278459525 0.433666591 0.0043366659 0.00340041538 10000.0000 apond[22] -0.9954707978 0.698527821 0.0069852782 0.00639152189 10000.0000 apond[2] 1.5514073486 0.973915502 0.0097391550 0.01111824521 7673.0859 apond[59] 3.9781779304 1.018961505 0.0101896150 0.01278245028 6354.5870 apond[46] 2.0703897434 0.515233417 0.0051523342 0.00470261856 10000.0000 σ 1.6843877698 0.243610374 0.0024361037 0.00421903982 3333.9901 apond[47] 3.9862598194 1.020488268 0.0102048827 0.01240672087 6765.5329 apond[57] 2.0741920322 0.509608416 0.0050960842 0.00391260407 10000.0000 apond[23] -0.9989820391 0.681962640 0.0068196264 0.00642167767 10000.0000 apond[51] 2.7344759539 0.649587763 0.0064958776 0.00648076189 10000.0000 apond[20] 0.5764910930 0.625086947 0.0062508695 0.00506310281 10000.0000 apond[8] 2.7496005216 1.272989569 0.0127298957 0.01733946864 5389.8679 apond[18] 3.1900259466 1.171811970 0.0117181197 0.01440863531 6614.0909 apond[54] 3.2291234955 0.784255555 0.0078425556 0.00820057334 9145.9069 apond[11] 2.7535202736 1.237297764 0.0123729776 0.01757842164 4954.3730 apond[16] 2.2007440950 0.916943320 0.0091694332 0.00992638854 8533.0137 lp -214.2598610993 7.211089559 0.0721108956 0.13728259285 2759.1215 apond[44] 1.4579671233 0.494609207 0.0049460921 0.00415623790 10000.0000 apond[7] 1.5544027763 0.979190641 0.0097919064 0.01106135909 7836.4168 lfeps 0.0806304543 0.025824042 0.0002582404 0.00087843406 864.2315 apond[50] 1.2646118079 0.396937728 0.0039693773 0.00301671654 10000.0000 apond[1] 2.7358650934 1.256841000 0.0125684100 0.01628364652 5957.4080 epsilon 0.0806304543 0.025824042 0.0002582404 0.00087843406 864.2315 apond[55] 1.6194868499 0.449848512 0.0044984851 0.00337126412 10000.0000 apond[56] 0.4636724052 0.333661394 0.0033366139 0.00317090692 10000.0000 apond[9] -1.5582336653 1.054033998 0.0105403400 0.01325783842 6320.6799 evalnum 140.7574000000 49.461823586 0.4946182359 1.04675890947 2232.7851 apond[38] 2.4213667923 0.669981340 0.0066998134 0.00670341721 9989.2507 apond[41] 2.9441756245 0.805063431 0.0080506343 0.00777409335 10000.0000 lfnum 0.0002000000 0.020000000 0.0002000000 0.00020000000 10000.0000 apond[42] 1.0189318364 0.450896154 0.0045089615 0.00353871779 10000.0000 apond[6] 0.7172274132 0.875913940 0.0087591394 0.00921226138 9040.4568 apond[37] 1.2311086658 0.467581916 0.0046758192 0.00370489273 10000.0000 apond[17] 2.1840208942 0.885232595 0.0088523259 0.01005752459 7746.9827 apond[34] 3.7647116451 1.070577317 0.0107057732 0.01454992135 5413.9568 apond[25] 0.1983088389 0.595635693 0.0059563569 0.00423421680 10000.0000 apond[49] 3.2308198064 0.783408432 0.0078340843 0.00741996001 10000.0000 apond[29] 0.5869977398 0.641031578 0.0064103158 0.00596687017 10000.0000 elapsed 0.2124224895 0.091334338 0.0009133434 0.00277238819 1085.3257 apond[3] 2.7351595176 1.252877992 0.0125287799 0.01650408216 5762.8167 apond[43] 1.4589068954 0.502713246 0.0050271325 0.00419773751 10000.0000 apond[4] -0.6955128978 0.886643420 0.0088664342 0.00892026440 9879.6722 apond[28] 3.1799658392 1.190210605 0.0119021061 0.01598221933 5545.9182 apond[15] 1.5668531946 1.015083021 0.0101508302 0.01066654407 9056.4014 apond[24] 1.0178669471 0.671538554 0.0067153855 0.00534254055 10000.0000 apond[32] 0.4829825423 0.408095990 0.0040809599 0.00290615908 10000.0000 apond[30] 2.1782787743 0.892245915 0.0089224591 0.00878540719 10000.0000 apond[60] 2.7321739924 0.664559051 0.0066455905 0.00632100816 10000.0000 Rethinking mean sd 5.5% 94.5% neff Rhat a 1.30 0.23 0.94 1.67 8064 1 sigma 1.55 0.21 1.24 1.92 3839 1 apond[1] 2.57 1.17 0.85 4.57 9688 1 apond[2] 2.58 1.19 0.83 4.56 9902 1 apond[3] 2.56 1.16 0.84 4.57 12841 1 apond[4] 1.49 0.92 0.12 3.03 15532 1 apond[5] 1.51 0.95 0.07 3.09 14539 1 apond[6] 0.72 0.84 -0.59 2.08 13607 1 apond[7] 2.56 1.16 0.86 4.51 12204 1 apond[8] 1.50 0.93 0.07 3.05 19903 1 apond[9] 2.56 1.15 0.86 4.51 11054 1 apond[10] 1.49 0.95 0.05 3.09 14134 1 apond[11] -0.64 0.86 -2.06 0.70 15408 1 apond[12] 2.56 1.16 0.86 4.53 11512 1 apond[13] 1.49 0.95 0.05 3.10 16270 1 apond[14] 0.71 0.84 -0.59 2.07 17077 1 apond[15] 1.50 0.93 0.10 3.05 16996 1 apond[16] 2.98 1.07 1.45 4.84 9033 1 apond[17] 2.09 0.84 0.85 3.54 14636 1 apond[18] 1.01 0.66 0.00 2.10 12971 1 apond[19] 1.01 0.68 -0.03 2.13 12598 1 apond[20] 1.48 0.72 0.38 2.67 15500 1 apond[21] 2.96 1.09 1.42 4.87 11204 1 apond[22] -2.04 0.87 -3.53 -0.75 9065 1 apond[23] 0.99 0.67 -0.04 2.11 15365 1 apond[24] 1.48 0.72 0.41 2.67 14879 1 apond[25] 2.10 0.85 0.85 3.53 13298 1 apond[26] 1.00 0.65 0.01 2.06 18583 1 apond[27] 3.00 1.08 1.44 4.86 9312 1 apond[28] 0.98 0.66 -0.03 2.09 14703 1 apond[29] 0.21 0.61 -0.76 1.19 15554 1 apond[30] 2.95 1.05 1.45 4.73 9816 1 apond[31] 1.70 0.53 0.89 2.59 19148 1 apond[32] 0.82 0.42 0.17 1.51 13556 1 apond[33] 0.32 0.40 -0.33 0.96 19388 1 apond[34] -0.15 0.40 -0.79 0.48 18684 1 apond[35] 3.57 0.98 2.19 5.26 8769 1 apond[36] 0.16 0.40 -0.46 0.80 17595 1 apond[37] 2.00 0.58 1.13 2.99 14669 1 apond[38] -1.41 0.49 -2.22 -0.65 12957 1 apond[39] 1.21 0.46 0.49 1.97 14185 1 apond[40] -1.18 0.46 -1.95 -0.48 16142 1 apond[41] 2.86 0.78 1.73 4.18 10508 1 apond[42] 0.00 0.39 -0.61 0.63 16138 1 apond[43] 1.43 0.48 0.70 2.24 17100 1 apond[44] 2.86 0.77 1.75 4.15 12002 1 apond[45] -1.40 0.49 -2.21 -0.66 14292 1 apond[46] 0.12 0.33 -0.40 0.66 20425 1 apond[47] -0.56 0.36 -1.14 0.00 18981 1 apond[48] 1.11 0.38 0.52 1.73 14176 1 apond[49] 3.81 0.95 2.47 5.45 8841 1 apond[50] 2.05 0.50 1.31 2.88 15898 1 apond[51] -1.40 0.41 -2.08 -0.76 17188 1 apond[52] -0.11 0.34 -0.65 0.43 17158 1 apond[53] 1.61 0.44 0.94 2.36 15132 1 apond[54] 2.05 0.50 1.30 2.89 15799 1 apond[55] 3.14 0.75 2.08 4.40 12702 1 apond[56] 3.13 0.74 2.07 4.41 11143 1 apond[57] 1.26 0.40 0.65 1.92 14587 1 apond[58] 1.11 0.38 0.51 1.74 21740 1 apond[59] 2.33 0.56 1.50 3.25 13116 1 apond[60] 1.27 0.40 0.66 1.91 15611 1
Mean SD Naive SE MCSE ESS
α -1.43756402 0.167281208 0.0016728121 0.00210877863 6292.63192
σ 0.94572510 0.159509660 0.0015950966 0.00292977115 2964.19373apond[13] -0.96409032 0.695682000 0.0069568200 0.00712009032 9546.63960 apond[39] -2.16432055 0.560230451 0.0056023045 0.00401929327 10000.00000 apond[12] -0.98211221 0.700132015 0.0070013202 0.00719599345 9466.25839 apond[33] -1.89716948 0.507094638 0.0050709464 0.00393759633 10000.00000 apond[40] -2.15927404 0.553498230 0.0055349823 0.00525989968 10000.00000 apond[52] -1.95778572 0.468392293 0.0046839229 0.00295136724 10000.00000 apond[48] -2.17567314 0.487971067 0.0048797107 0.00325138185 10000.00000 apond[31] -1.65553570 0.479752516 0.0047975252 0.00309756598 10000.00000 apond[21] -1.46693217 0.634848584 0.0063484858 0.00548329380 10000.00000 apond[14] -0.51655733 0.697560885 0.0069756089 0.00693184327 10000.00000 apond[26] -1.11415242 0.579221588 0.0057922159 0.00497058099 10000.00000 apond[35] -2.16959658 0.546408662 0.0054640866 0.00487151680 10000.00000 apond[53] -2.17290466 0.499898730 0.0049989873 0.00381882101 10000.00000 apond[45] -1.65831075 0.475684574 0.0047568457 0.00282288126 10000.00000 apond[27] -1.47390856 0.625904201 0.0062590420 0.00597555549 10000.00000 apond[10] -0.06470584 0.709512539 0.0070951254 0.00810899930 7655.71265 apond[19] -1.47611448 0.619367365 0.0061936736 0.00573498517 10000.00000 apond[58] -1.96067155 0.471148496 0.0047114850 0.00310756256 10000.00000 apond[5] -0.51301911 0.693083568 0.0069308357 0.00766059211 8185.52478 apond[36] -2.16398901 0.552362190 0.0055236219 0.00423015168 10000.00000 apond[22] -0.79183001 0.562682452 0.0056268245 0.00395177756 10000.00000 apond[2] -0.07065260 0.723388535 0.0072338853 0.00837009596 7469.34241 apond[59] -3.04846657 0.659338141 0.0065933814 0.00767658576 7377.00686 apond[46] -2.41950226 0.530010130 0.0053001013 0.00447275751 10000.00000 apond[47] -2.41940262 0.529600266 0.0052960027 0.00495650076 10000.00000 apond[57] -2.42196906 0.545199651 0.0054519965 0.00421172829 10000.00000 apond[23] -1.47836307 0.626164879 0.0062616488 0.00546703424 10000.00000 apond[51] -1.95865755 0.453931725 0.0045393173 0.00323782050 10000.00000 apond[20] -1.87882953 0.685389064 0.0068538906 0.00691600262 9821.18842 apond[8] -0.51278872 0.698168747 0.0069816875 0.00674015806 10000.00000 apond[18] -0.79464713 0.570683666 0.0057068367 0.00517526089 10000.00000 apond[54] -2.17663143 0.486692179 0.0048669218 0.00357970522 10000.00000 apond[11] -0.52002272 0.700422488 0.0070042249 0.00758407273 8529.33523 apond[16] -2.35154693 0.748462872 0.0074846287 0.01009234165 5499.92332 apond[44] -2.16190756 0.555077841 0.0055507784 0.00464353670 10000.00000 apond[7] -0.07294353 0.722895184 0.0072289518 0.00933609743 5995.42414 apond[50] -1.76719334 0.433238573 0.0043323857 0.00301606024 10000.00000 apond[1] -0.07479566 0.715498813 0.0071549881 0.00821909124 7578.27275 apond[55] -1.95861084 0.462566383 0.0046256638 0.00326122131 10000.00000 apond[56] -1.95767294 0.451852167 0.0045185217 0.00336244216 10000.00000 apond[9] -0.97174862 0.706466469 0.0070646647 0.00685957974 10000.00000 apond[38] -1.65719327 0.477253975 0.0047725397 0.00357994187 10000.00000 apond[41] -2.47295206 0.610429423 0.0061042942 0.00555558551 10000.00000 apond[42] -1.65879054 0.487850118 0.0048785012 0.00420032610 10000.00000 apond[6] -0.51322197 0.699819483 0.0069981948 0.00837828549 6976.88795 apond[37] -1.89163219 0.514164120 0.0051416412 0.00396572368 10000.00000 apond[17] -0.48373615 0.562778997 0.0056277900 0.00445855972 10000.00000 apond[34] -1.89391017 0.517579732 0.0051757973 0.00449881473 10000.00000 apond[25] -0.48483903 0.564804766 0.0056480477 0.00432604358 10000.00000 apond[49] -1.95877851 0.456040949 0.0045604095 0.00251026103 10000.00000 apond[29] -1.47041681 0.615401982 0.0061540198 0.00516784541 10000.00000 apond[3] -0.52147126 0.688164420 0.0068816442 0.00702386169 9599.14444 apond[43] -1.65667827 0.476433792 0.0047643379 0.00374622975 10000.00000 apond[4] -0.97038449 0.704920197 0.0070492020 0.00798856637 7786.49863 apond[28] -1.11670032 0.594046332 0.0059404633 0.00481138686 10000.00000 apond[15] -0.97643818 0.702064283 0.0070206428 0.00627008271 10000.00000 apond[24] -0.78700355 0.580971213 0.0058097121 0.00463692344 10000.00000 apond[32] -1.66143918 0.481944799 0.0048194480 0.00375554807 10000.00000 apond[30] -0.79075103 0.573147275 0.0057314727 0.00550557273 10000.00000 apond[60] -2.17672820 0.503739378 0.0050373938 0.00376315154 10000.00000
This page was generated using Literate.jl.