Assignment #8
Shruti Jose
2022-04-09
Chapter 9 - Markov Chain Monte Carlo
This chapter has been an informal introduction to Markov chain Monte Carlo (MCMC) estimation. The goal has been to introduce the purpose and approach MCMC algorithms. The major algorithms introduced were the Metropolis, Gibbs sampling, and Hamiltonian Monte Carlo algorithms. Each has its advantages and disadvantages. The ulam function in the rethinking package was introduced. It uses the Stan (mc-stan.org) Hamiltonian Monte Carlo engine to fit models as they are defined in this book. General advice about diagnosing poor MCMC fits was introduced by the use of a couple of pathological examples.
Place each answer inside the code chunk (grey box). The code chunks should contain a text response or a code that completes/answers the question or activity requested. Make sure to include plots if the question requests them.
Finally, upon completion, name your final output .html file as: YourName_ANLY505-Year-Semester.html and publish the assignment to your R Pubs account and submit the link to Canvas. Each question is worth 5 points.
Questions
9-1. Re-estimate the terrain ruggedness model from the chapter, but now using a uniform prior for the standard deviation, sigma. The uniform prior should be dunif(0,1). Visualize the priors. Use ulam to estimate the posterior. Visualize the posteriors for both models. Does the different prior have any detectible influence on the posterior distribution of sigma? Why or why not?
library(rethinking)
data(rugged)
data= rugged
data$log_gdp = log(data$rgdppc_2000)
dd = data[ complete.cases(data$rgdppc_2000) , ]
dd$log_gdp_std = dd$log_gdp/ mean(dd$log_gdp)
dd$rugged_std = dd$rugged/max(dd$rugged)
dd$cid=ifelse(dd$cont_africa==1,1,2)
dat_slim = list(
log_gdp_std = dd$log_gdp_std,
rugged_std = dd$rugged_std,
cid = as.integer( dd$cid )
)
model1 = ulam(
alist(
log_gdp_std ~ dnorm( mu , sigma ) ,
mu <- a[cid] + b[cid]*( rugged_std - 0.215 ) ,
a[cid] ~ dnorm( 1 , 0.1 ) ,
b[cid] ~ dnorm( 0 , 0.3 ) ,
sigma ~ dexp( 1 )
), data=dat_slim , chains=4, cores = 4)## Running MCMC with 4 parallel chains, with 1 thread(s) per chain...
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model2 = ulam(
alist(
log_gdp_std ~ dnorm( mu , sigma ) ,
mu <- a[cid] + b[cid]*( rugged_std - 0.215 ) ,
a[cid] ~ dnorm( 1 , 0.1 ) ,
b[cid] ~ dnorm( 0 , 0.3 ) ,
sigma ~ dunif(0, 1 )
), data=dat_slim , chains=4, cores = 4 )## Running MCMC with 4 parallel chains, with 1 thread(s) per chain...
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# From the results we infer that the different prior does not have any detectable influence on the posterior distribution of sigma9-2. Modify the terrain ruggedness model again. This time, change the prior for b[cid] to dexp(0.3). What does this do to the posterior distribution? Can you explain it?
model3 = ulam(
alist(
log_gdp_std ~ dnorm( mu , sigma ) ,
mu <- a[cid] + b[cid]*( rugged_std - 0.215 ) ,
a[cid] ~ dnorm( 1 , 0.1 ) ,
b[cid] ~ dexp(0.3) ,
sigma ~ dexp( 1 )
), data=dat_slim , chains=4, cores = 4 )## Running MCMC with 4 parallel chains, with 1 thread(s) per chain...
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# From the results of model 3, we conclude that there us no difference in the posterior distribution.9-3. Re-estimate one of the Stan models from the chapter, but at different numbers of warmup iterations. Be sure to use the same number of sampling iterations in each case. Compare the n_eff values. How much warmup is enough?
Stan_model_100 = map2stan(
alist(
log_gdp_std ~ dnorm( mu , sigma ) ,
mu <- a[cid] + b[cid]*( rugged_std - 0.215 ) ,
a[cid] ~ dnorm( 1 , 0.1 ) ,
b[cid] ~ dnorm( 0 , 0.3 ) ,
sigma ~ dexp( 1 )
), data=dat_slim , warmup = 100, iter = 4000, chains=4, cores = 4 )## Running MCMC with 4 parallel chains...
##
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## Total execution time: 2.3 seconds.precis(Stan_model_100,depth=2)## mean sd 5.5% 94.5% n_eff Rhat4
## a[1] 0.8866094 0.016063938 0.860953780 0.91246350 18233.168 0.9998570
## a[2] 1.0506544 0.010085211 1.034710000 1.06674000 18454.811 0.9998351
## b[1] 0.1328020 0.076577545 0.008237304 0.25449806 6583.272 1.0006782
## b[2] -0.1425408 0.056713451 -0.232031440 -0.05185553 9853.031 0.9999820
## sigma 0.1117333 0.006248562 0.102228670 0.12210344 9429.928 0.9999782
## mu[1] 0.8764292 0.016152121 0.850529780 0.90243155 17954.163 0.9999664
## mu[2] 1.0025378 0.020981712 0.969426725 1.03595000 10990.710 0.9999352
## mu[3] 1.0636267 0.011657451 1.045090000 1.08229000 15264.942 0.9998832
## mu[4] 1.0634888 0.011629787 1.044990000 1.08212000 15303.883 0.9998827
## mu[5] 1.0195223 0.015395911 0.995221615 1.04431000 12244.511 0.9999036
## mu[6] 1.0811628 0.016334039 1.055369450 1.10749165 12065.819 0.9999395
## mu[7] 1.0780141 0.015366334 1.053799450 1.10273055 12419.395 0.9999323
## mu[8] 1.0005613 0.021675185 0.966453735 1.03510055 10909.224 0.9999377
## mu[9] 1.0428730 0.010333014 1.026560000 1.05941000 17687.495 0.9998335
## mu[10] 0.8961717 0.017836827 0.867647725 0.92492304 14198.443 0.9999345
## mu[11] 1.0723832 0.013751073 1.050400000 1.09455055 13256.935 0.9999165
## mu[12] 0.8610762 0.019887687 0.829344560 0.89248255 11642.844 1.0003087
## mu[13] 0.8631104 0.019200872 0.832622560 0.89362493 12283.711 1.0002675
## mu[14] 1.0770258 0.015071303 1.053269450 1.10131000 12545.505 0.9999298
## mu[15] 1.0473088 0.010076584 1.031359450 1.06340055 18502.944 0.9998310
## mu[16] 1.0759915 0.014767382 1.052590000 1.09981110 12686.560 0.9999271
## mu[17] 1.0800366 0.015983384 1.054820000 1.10578000 12184.316 0.9999371
## mu[18] 1.0281869 0.012982848 1.007500000 1.04897055 13625.202 0.9998777
## mu[19] 1.0775314 0.015221741 1.053560000 1.10208055 12480.031 0.9999311
## mu[20] 1.0673499 0.012471870 1.047419450 1.08726000 14295.501 0.9998986
## mu[21] 1.0616961 0.011288695 1.043740000 1.07974110 15826.566 0.9998750
## mu[22] 1.0757847 0.014707276 1.052480000 1.09952110 12715.660 0.9999266
## mu[23] 1.0591680 0.010869134 1.041900000 1.07664055 16593.768 0.9998639
## mu[24] 0.8619327 0.019592874 0.830716900 0.89291596 11901.059 1.0002917
## mu[25] 0.8622753 0.019477206 0.831308515 0.89310338 12008.987 1.0002848
## mu[26] 1.0634888 0.011629787 1.044990000 1.08212000 15303.883 0.9998827
## mu[27] 0.9718784 0.032222301 0.920963395 1.02344055 10291.768 0.9999587
## mu[28] 1.0242798 0.014017526 1.002048350 1.04680055 12907.472 0.9998902
## mu[29] 1.0381385 0.010919253 1.020759450 1.05545055 16284.733 0.9998443
## mu[30] 0.8628535 0.019285042 0.832218835 0.89347838 12197.404 1.0002729
## mu[31] 0.8690846 0.017485368 0.841365835 0.89696793 14773.799 1.0001327
## mu[32] 0.8675429 0.017880358 0.839248955 0.89595200 14040.599 1.0001689
## mu[33] 0.8613118 0.019805821 0.829743625 0.89259206 11712.234 1.0003041
## mu[34] 1.0609606 0.011158860 1.043219450 1.07880055 16047.186 0.9998716
## mu[35] 0.9293187 0.031578605 0.879034625 0.97979888 8018.012 1.0003550
## mu[36] 0.9087410 0.022128028 0.873962615 0.94399928 10595.175 1.0001286
## mu[37] 1.0327605 0.011916058 1.013769450 1.05177055 14706.526 0.9998620
## mu[38] 1.0188328 0.015604641 0.994181670 1.04390055 12164.243 0.9999053
## mu[39] 1.0609837 0.011162822 1.043230000 1.07883000 16040.490 0.9998719
## mu[40] 1.0675798 0.012526201 1.047608900 1.08757165 14241.337 0.9998995
## mu[41] 0.9101329 0.022695837 0.874413790 0.94631005 10269.763 1.0001488
## mu[42] 1.0812317 0.016355687 1.055400000 1.10759110 12058.712 0.9999395
## mu[43] 1.0769569 0.015050889 1.053220000 1.10121055 12554.750 0.9999296
## mu[44] 1.0435855 0.010271830 1.027400000 1.06008055 17866.161 0.9998325
## mu[45] 0.8689775 0.017511625 0.841226890 0.89689781 14720.947 1.0001352
## mu[46] 1.0519283 0.010134594 1.035929450 1.06806055 18319.283 0.9998380
## mu[47] 0.8735384 0.016560837 0.847094340 0.89999927 16762.590 1.0000283
## mu[48] 0.9111821 0.023133068 0.874737680 0.94807822 10046.571 1.0001636
## mu[49] 1.0424823 0.010369669 1.026110000 1.05909055 17583.478 0.9998342
## mu[50] 1.0784737 0.015505101 1.053990000 1.10341055 12363.379 0.9999334
## mu[51] 0.8916751 0.016802245 0.864858890 0.91868510 16144.388 0.9998813
## mu[52] 1.0737622 0.014130937 1.051269450 1.09648055 13023.359 0.9999208
## mu[53] 1.0492164 0.010059946 1.033320000 1.06532055 18532.194 0.9998327
## mu[54] 1.0560653 0.010463903 1.039548900 1.07293000 17503.314 0.9998513
## mu[55] 0.8627250 0.019327413 0.832023945 0.89339999 12154.856 1.0002756
## mu[56] 1.0682463 0.012686066 1.048068900 1.08855000 14087.448 0.9999020
## mu[57] 0.9972057 0.022866161 0.961041230 1.03372165 10788.645 0.9999415
## mu[58] 0.8629391 0.019256901 0.832344890 0.89354911 12225.983 1.0002711
## mu[59] 0.8739025 0.016500656 0.847562000 0.90027483 16922.797 1.0000201
## mu[60] 0.8656157 0.018422583 0.836415460 0.89492465 13212.727 1.0002131
## mu[61] 0.8685707 0.017613032 0.840666725 0.89661199 14522.661 1.0001448
## mu[62] 0.8700267 0.017262073 0.842746945 0.89754028 15194.889 1.0001103
## mu[63] 1.0099843 0.018436407 0.980923780 1.03949000 11390.812 0.9999240
## mu[64] 1.0333121 0.011799787 1.014490000 1.05213000 14855.119 0.9998601
## mu[65] 1.0397703 0.010683377 1.022870000 1.05684055 16791.382 0.9998399
## mu[66] 1.0750263 0.014488586 1.052028350 1.09840055 12826.512 0.9999245
## mu[67] 1.0238201 0.014145506 1.001360000 1.04655055 12834.168 0.9998916
## mu[68] 1.0318872 0.012105887 1.012570000 1.05124000 14478.924 0.9998650
## mu[69] 1.0521812 0.010147361 1.036148350 1.06836000 18285.410 0.9998386
## mu[70] 1.0270147 0.013282619 1.005869450 1.04818055 13391.264 0.9998816
## mu[71] 1.0733485 0.014015810 1.050980000 1.09588165 13091.397 0.9999195
## mu[72] 1.0590760 0.010855288 1.041849450 1.07652000 16621.692 0.9998636
## mu[73] 1.0580188 0.010704269 1.041030000 1.07514055 16941.013 0.9998591
## mu[74] 1.0695103 0.012998603 1.048869450 1.09033000 13811.825 0.9999067
## mu[75] 1.0251071 0.013790256 1.003139450 1.04719055 13045.265 0.9998877
## mu[76] 1.0474696 0.010072956 1.031529450 1.06356000 18511.242 0.9998310
## mu[77] 1.0421146 0.010406206 1.025698900 1.05872000 17483.299 0.9998348
## mu[78] 1.0248084 0.013871888 1.002720000 1.04707055 12994.606 0.9998886
## mu[79] 1.0402070 0.010626095 1.023370000 1.05718220 16924.848 0.9998388
## mu[80] 1.0560883 0.010466431 1.039568900 1.07295055 17497.207 0.9998514
## mu[81] 1.0323008 0.012015046 1.013148900 1.05147055 14585.778 0.9998636
## mu[82] 1.0727510 0.013851258 1.050619450 1.09504000 13192.779 0.9999177
## mu[83] 0.8723822 0.016767983 0.845763835 0.89909628 16245.642 1.0000548
## mu[84] 0.9827724 0.028137000 0.938215835 1.02779000 10443.008 0.9999531
## mu[85] 1.0688899 0.012843647 1.048460000 1.08946000 13944.688 0.9999044
## mu[86] 1.0811628 0.016334039 1.055369450 1.10749165 12065.819 0.9999395
## mu[87] 1.0362539 0.011232240 1.018320000 1.05408055 15704.757 0.9998502
## mu[88] 1.0756699 0.014673912 1.052419450 1.09934220 12732.207 0.9999262
## mu[89] 1.0226250 0.014483807 0.999748395 1.04590000 12653.271 0.9998951
## mu[90] 0.9848409 0.027370014 0.941525670 1.02859055 10479.962 0.9999517
## mu[91] 1.0319561 0.012090632 1.012650000 1.05129000 14496.998 0.9998648
## mu[92] 1.0663617 0.012243546 1.046858900 1.08587000 14536.284 0.9998946
## mu[93] 0.9908590 0.064772318 0.885847635 1.09331055 6834.700 1.0005635
## mu[94] 1.0772786 0.015146352 1.053438900 1.10169055 12512.658 0.9999305
## mu[95] 1.0610526 0.011174825 1.043280000 1.07892055 16019.592 0.9998721
## mu[96] 1.0777613 0.015290430 1.053688350 1.10239000 12451.051 0.9999317
## mu[97] 1.0644771 0.011832057 1.045609450 1.08335000 15029.158 0.9998869
## mu[98] 0.9097260 0.022528389 0.874323670 0.94561650 10361.252 1.0001430
## mu[99] 1.0629372 0.011521251 1.044619450 1.08142000 15461.638 0.9998803
## mu[100] 0.8830885 0.015850436 0.857727285 0.90863916 19012.563 0.9998708
## mu[101] 1.0414940 0.010472198 1.024920000 1.05817055 17307.624 0.9998359
## mu[102] 1.0200509 0.015237318 0.995976670 1.04457055 12308.577 0.9999022
## mu[103] 0.8612047 0.019842958 0.829560010 0.89253444 11680.546 1.0003061
## mu[104] 1.0462056 0.010112191 1.030229450 1.06236055 18378.616 0.9998308
## mu[105] 1.0570076 0.010573401 1.040260000 1.07399000 17238.747 0.9998550
## mu[106] 0.8711616 0.017012271 0.844276945 0.89827911 15697.198 1.0000834
## mu[107] 0.8605195 0.020083501 0.828451955 0.89227111 11483.590 1.0003194
## mu[108] 0.8783777 0.015969155 0.852919945 0.90403338 18577.478 0.9999304
## mu[109] 0.8800479 0.015874070 0.854640835 0.90552232 18926.915 0.9999043
## mu[110] 1.0581107 0.010716825 1.041120000 1.07527000 16913.873 0.9998595
## mu[111] 0.8776069 0.016032376 0.852022780 0.90337133 18354.988 0.9999440
## mu[112] 1.0448496 0.010181827 1.028750000 1.06119000 18145.944 0.9998313
## mu[113] 0.8618685 0.019614707 0.830593450 0.89288838 11881.122 1.0002930
## mu[114] 0.8647378 0.018686299 0.835154780 0.89446266 12868.647 1.0002326
## mu[115] 1.0585014 0.010771415 1.041410000 1.07577110 16795.846 0.9998612
## mu[116] 1.0804503 0.016111650 1.055028900 1.10642000 12139.706 0.9999380
## mu[117] 1.0259346 0.013567198 1.004309450 1.04758055 13190.112 0.9998850
## mu[118] 0.9653972 0.034681499 0.910635945 1.02082110 10227.005 0.9999613
## mu[119] 1.0344612 0.011567325 1.016000000 1.05292055 15176.576 0.9998561
## mu[120] 1.0558125 0.010436642 1.039350000 1.07263000 17571.467 0.9998503
## mu[121] 1.0365067 0.011187835 1.018669450 1.05425055 15781.723 0.9998494
## mu[122] 1.0459068 0.010125089 1.029899450 1.06208110 18334.066 0.9998308
## mu[123] 1.0503425 0.010076917 1.034410000 1.06640165 18478.550 0.9998345
## mu[124] 1.0346911 0.011522484 1.016310000 1.05305055 15242.361 0.9998554
## mu[125] 1.0447807 0.010186092 1.028670000 1.06113055 18132.361 0.9998314
## mu[126] 1.0744747 0.014331482 1.051660000 1.09758110 12910.691 0.9999229
## mu[127] 1.0480442 0.010063391 1.032090000 1.06411055 18531.901 0.9998315
## mu[128] 1.0513308 0.010108273 1.035340000 1.06743000 18390.267 0.9998365
## mu[129] 1.0756699 0.014673912 1.052419450 1.09934220 12732.207 0.9999262
## mu[130] 1.0336798 0.011723999 1.014970000 1.05233110 14956.577 0.9998588
## mu[131] 1.0521812 0.010147361 1.036148350 1.06836000 18285.410 0.9998386
## mu[132] 1.0596966 0.010950541 1.042268900 1.07728000 16432.340 0.9998662
## mu[133] 0.9289119 0.031375884 0.878915340 0.97905336 8041.762 1.0003519
## mu[134] 1.0602941 0.011046652 1.042719450 1.07801110 16249.757 0.9998688
## mu[135] 0.8675215 0.017886091 0.839212945 0.89591758 14030.866 1.0001694
## mu[136] 0.8632817 0.019145196 0.832893780 0.89369416 12342.149 1.0002639
## mu[137] 1.0809329 0.016262026 1.055248900 1.10714000 12089.205 0.9999389
## mu[138] 1.0459068 0.010125089 1.029899450 1.06208110 18334.066 0.9998308
## mu[139] 0.8687206 0.017575372 0.840864615 0.89671882 14595.248 1.0001413
## mu[140] 1.0410803 0.010519134 1.024389450 1.05785055 17186.584 0.9998368
## mu[141] 1.0454470 0.010147674 1.029350000 1.06167000 18257.973 0.9998310
## mu[142] 1.0239350 0.014113393 1.001510000 1.04660110 12852.417 0.9998913
## mu[143] 1.0648678 0.011914676 1.045859450 1.08386000 14923.576 0.9998885
## mu[144] 0.9236443 0.028794592 0.877777890 0.96971350 8405.154 1.0003072
## mu[145] 0.9626584 0.049165575 0.883226965 1.04111165 7071.446 1.0005092
## mu[146] 1.0639255 0.011717999 1.045250000 1.08269000 15181.009 0.9998845
## mu[147] 0.8670290 0.018019849 0.838478780 0.89570821 13810.062 1.0001809
## mu[148] 0.8640526 0.018898989 0.834100395 0.89407166 12614.132 1.0002475
## mu[149] 1.0571455 0.010590391 1.040340000 1.07416000 17198.590 0.9998556
## mu[150] 0.9594676 0.036945707 0.901208835 1.01856165 10179.156 0.9999632
## mu[151] 1.0747735 0.014416322 1.051859450 1.09801055 12864.786 0.9999238
## mu[152] 1.0635348 0.011639072 1.045019450 1.08217055 15290.909 0.9998829
## mu[153] 1.0674419 0.012493567 1.047500000 1.08739000 14273.645 0.9998989
## mu[154] 0.8736027 0.016550041 0.847176560 0.90002811 16790.979 1.0000268
## mu[155] 1.0210851 0.014930668 0.997499505 1.04512000 12440.302 0.9998994
## mu[156] 0.8725535 0.016735784 0.845986890 0.89920060 16322.677 1.0000508
## mu[157] 0.8776069 0.016032376 0.852022780 0.90337133 18354.988 0.9999440
## mu[158] 1.0717397 0.013577850 1.050069450 1.09354000 13373.150 0.9999144
## mu[159] 1.0713030 0.013461705 1.049869450 1.09290055 13454.371 0.9999130
## mu[160] 1.0566399 0.010529183 1.040009450 1.07354055 17343.743 0.9998536
## mu[161] 1.0692116 0.012923646 1.048679450 1.08992055 13875.377 0.9999056
## mu[162] 1.0107198 0.018191876 0.982019470 1.03986000 11440.500 0.9999227
## mu[163] 1.0667294 0.012327539 1.047049450 1.08637055 14445.204 0.9998961
## mu[164] 1.0340935 0.011640252 1.015519450 1.05262110 15072.085 0.9998574
## mu[165] 1.0415630 0.010464622 1.024999450 1.05823000 17327.548 0.9998359
## mu[166] 1.0427581 0.010343574 1.026430000 1.05932055 17657.550 0.9998337
## mu[167] 1.0279110 0.013052477 1.007138900 1.04876110 13568.714 0.9998786
## mu[168] 0.8957649 0.017730106 0.867412000 0.92431966 14363.061 0.9999289
## mu[169] 0.8694700 0.017392321 0.841934505 0.89719339 14953.512 1.0001235
## mu[170] 0.8836239 0.015866350 0.858195560 0.90917505 18950.627 0.9998670Stan_model_200 = map2stan(
alist(
log_gdp_std ~ dnorm( mu , sigma ) ,
mu <- a[cid] + b[cid]*( rugged_std - 0.215 ) ,
a[cid] ~ dnorm( 1 , 0.1 ) ,
b[cid] ~ dnorm( 0 , 0.3 ) ,
sigma ~ dexp( 1 )
), data=dat_slim , warmup = 200, iter = 4000, chains=4, cores = 4 )## Running MCMC with 4 parallel chains...
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##
## All 4 chains finished successfully.
## Mean chain execution time: 1.2 seconds.
## Total execution time: 1.5 seconds.Stan_model_300 = map2stan(
alist(
log_gdp_std ~ dnorm( mu , sigma ) ,
mu <- a[cid] + b[cid]*( rugged_std - 0.215 ) ,
a[cid] ~ dnorm( 1 , 0.1 ) ,
b[cid] ~ dnorm( 0 , 0.3 ) ,
sigma ~ dexp( 1 )
), data=dat_slim , warmup = 300, iter = 4000, chains=4, cores = 4 )## Running MCMC with 4 parallel chains...
##
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##
## All 4 chains finished successfully.
## Mean chain execution time: 1.2 seconds.
## Total execution time: 1.4 seconds.Stan_model_400 = map2stan(
alist(
log_gdp_std ~ dnorm( mu , sigma ) ,
mu <- a[cid] + b[cid]*( rugged_std - 0.215 ) ,
a[cid] ~ dnorm( 1 , 0.1 ) ,
b[cid] ~ dnorm( 0 , 0.3 ) ,
sigma ~ dexp( 1 )
), data=dat_slim , warmup = 400, iter = 4000, chains=4, cores = 4 )## Running MCMC with 4 parallel chains...
##
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## Total execution time: 1.6 seconds.Stan_model_500 = map2stan(
alist(
log_gdp_std ~ dnorm( mu , sigma ) ,
mu <- a[cid] + b[cid]*( rugged_std - 0.215 ) ,
a[cid] ~ dnorm( 1 , 0.1 ) ,
b[cid] ~ dnorm( 0 , 0.3 ) ,
sigma ~ dexp( 1 )
), data=dat_slim , warmup = 500, iter = 4000, chains=4, cores = 4 )## Running MCMC with 4 parallel chains...
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## Total execution time: 1.6 seconds.# From the results shown, we note that as warmup is increased,n_eff gets closer to the number of iterations and specifically 400 warmup iterations are enough for our model.9-4. Run the model below and then inspect the posterior distribution and explain what it is accomplishing.
mp <- ulam(
alist(
a ~ dnorm(0,1),
b ~ dcauchy(0,1)
), data=list(y=1) , chains=1 )## Running MCMC with 1 chain, with 1 thread(s) per chain...
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traceplot(mp)
#From the results, we note that Figure 'a' should be a normal distribution as the prior is around 0 and spread is in the range between 2 and -2. Figure 'b' shows Cauchy distribution which contains some extreme values going that lie beyond the values 50 and -50.
Compare the samples for the parameters a and b. Can you explain the different trace plots? If you are unfamiliar with the Cauchy distribution, you should look it up. The key feature to attend to is that it has no expected value. Can you connect this fact to the trace plot?
9-5. Recall the divorce rate example from Chapter 5. Repeat that analysis, using ulam this time, fitting models m5.1, m5.2, and m5.3. Use compare to compare the models on the basis of WAIC or PSIS. To use WAIC or PSIS with ulam, you need add the argument log_log=TRUE. Explain the model comparison results.
library(tidybayes)
data(WaffleDivorce)
data = WaffleDivorce
data$Divorce_sd=standardize(data$Divorce)
data$Marriage_sd=standardize(data$Marriage)
data$MedianAgeMarriage_sd=standardize(data$MedianAgeMarriage)
d_trim = list(D = data$Divorce_sd, M = data$Marriage_sd, A = data$MedianAgeMarriage_sd)
m5.1 = ulam(
alist(
D ~ dnorm(mu, sigma),
mu <- a + bA * A,
a ~ dnorm(0, 0.2),
bA ~ dnorm(0, 0.5),
sigma ~ dexp(1)
),
data = d_trim,
chains = 4,
cores = 4,
log_lik = TRUE
)## Running MCMC with 4 parallel chains, with 1 thread(s) per chain...
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alist(
D ~ dnorm(mu, sigma),
mu <- a + bM * M,
a ~ dnorm(0, 0.2),
bM ~ dnorm(0, 0.5),
sigma ~ dexp(1)
),
data = d_trim,
chains = 4,
cores = 4,
log_lik = TRUE
)## Running MCMC with 4 parallel chains, with 1 thread(s) per chain...
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## Total execution time: 0.5 seconds.m5.3 = ulam(
alist(
D ~ dnorm(mu, sigma),
mu <- a + bA * A + bM * M,
a ~ dnorm(0, 0.2),
bA ~ dnorm(0, 0.5),
bM ~ dnorm(0, 0.5),
sigma ~ dexp(1)
),
data = d_trim,
chains = 4,
cores = 4,
log_lik = TRUE
)## Running MCMC with 4 parallel chains, with 1 thread(s) per chain...
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compare( m5.1 , m5.2 , m5.3 , func=WAIC )## WAIC SE dWAIC dSE pWAIC weight
## m5.1 125.8064 12.634168 0.000000 NA 3.676068 0.6863825852
## m5.3 127.3782 12.697816 1.571744 0.750632 4.630438 0.3127996914
## m5.2 139.2717 9.758319 13.465333 9.251377 2.938041 0.0008177234# From the results shown, we infer that the model m5.1 with only median age at marriage as a predictor performs best, but is not really much different compared to the model m5.3. However, the model with marriage rate only, model m5.2 clearly under performs compared to the other two models.