Assignment #8

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)

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pairs(model1)

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 )

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pairs(model2)

# Does not have detectible influence on the posterior distribution of sigma

9-2. 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 )

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pairs(model3)

# There is not differences 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 )

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precis(Stan_model_100,depth=2)

##               mean          sd        5.5%       94.5%     n_eff     Rhat4
## a[1]     0.8865697 0.016053562  0.86106756  0.91210676 17515.448 0.9998771
## a[2]     1.0505813 0.010063725  1.03440945  1.06655055 16752.181 0.9998515
## b[1]     0.1328875 0.075847978  0.01279902  0.25552628  6711.880 1.0000552
## b[2]    -0.1426609 0.056161535 -0.23277309 -0.05239599  9700.626 1.0001119
## sigma    0.1115139 0.006162754  0.10210094  0.12179044  9469.505 1.0002201
## mu[1]    0.8763829 0.016162036  0.85067467  0.90216888 17089.682 0.9999813
## mu[2]    1.0024242 0.020792315  0.96912462  1.03590055 10778.117 0.9999569
## mu[3]    1.0635646 0.011621117  1.04518945  1.08217055 13934.052 0.9999976
## mu[4]    1.0634266 0.011593813  1.04509000  1.08198000 13964.410 0.9999962
## mu[5]    1.0194230 0.015276880  0.99495934  1.04396000 11941.514 0.9998862
## mu[6]    1.0811154 0.016240559  1.05556945  1.10712055 11485.267 1.0001095
## mu[7]    1.0779641 0.015284416  1.05394945  1.10250000 11756.697 1.0000978
## mu[8]    1.0004460 0.021477643  0.96613050  1.03506110 10701.759 0.9999630
## mu[9]    1.0427934 0.010300016  1.02624945  1.05900000 16733.241 0.9998013
## mu[10]   0.8961382 0.017776819  0.86783245  0.92455922 12955.664 0.9998309
## mu[11]   1.0723285 0.013688822  1.05074000  1.09437000 12393.823 1.0000690
## mu[12]   0.8610200 0.019858341  0.82908861  0.89265706 11079.868 1.0001080
## mu[13]   0.8630556 0.019180071  0.83230589  0.89357205 11601.237 1.0000974
## mu[14]   1.0769750 0.014992965  1.05332000  1.10106055 11852.583 1.0000936
## mu[15]   1.0472329 0.010052030  1.03110945  1.06300000 17047.402 0.9998229
## mu[16]   1.0759399 0.014692734  1.05273945  1.09957000 11959.873 1.0000887
## mu[17]   1.0799883 0.015894046  1.05500000  1.10543055 11578.475 1.0001057
## mu[18]   1.0280949 0.012899435  1.00742890  1.04864110 13197.395 0.9998394
## mu[19]   1.0774811 0.015141548  1.05366945  1.10176055 11802.950 1.0000958
## mu[20]   1.0672910 0.012425489  1.04772000  1.08712000 13187.110 1.0000322
## mu[21]   1.0616324 0.011256780  1.04380945  1.07974000 14372.212 0.9999773
## mu[22]   1.0757329 0.014633249  1.05262945  1.09929110 11981.974 1.0000878
## mu[23]   1.0591021 0.010842034  1.04184000  1.07658000 14982.012 0.9999486
## mu[24]   0.8618771 0.019567209  0.83052056  0.89305721 11291.243 1.0001038
## mu[25]   0.8622199 0.019452983  0.83099795  0.89320606 11379.070 1.0001020
## mu[26]   1.0634266 0.011593813  1.04509000  1.08198000 13964.410 0.9999962
## mu[27]   0.9717390 0.031909127  0.92102061  1.02293055 10119.874 1.0000207
## mu[28]   1.0241845 0.013918155  1.00187945  1.04647055 12548.003 0.9998608
## mu[29]   1.0380549 0.010872765  1.02059835  1.05528055 15540.937 0.9997994
## mu[30]   0.8627985 0.019263196  0.83187289  0.89345071 11531.642 1.0000988
## mu[31]   0.8690336 0.017484795  0.84103446  0.89698616 13568.313 1.0000549
## mu[32]   0.8674909 0.017875383  0.83890383  0.89626238 12995.855 1.0000675
## mu[33]   0.8612557 0.019777501  0.82947594  0.89274405 11136.832 1.0001069
## mu[34]   1.0608963 0.011128517  1.04323000  1.07883000 14545.832 0.9999692
## mu[35]   0.9293065 0.031303222  0.87944351  0.97900217  8039.491 0.9998944
## mu[36]   0.9087156 0.021986065  0.87353020  0.94392617  9888.984 0.9998406
## mu[37]   1.0326723 0.011850511  1.01367000  1.05152000 14162.723 0.9998165
## mu[38]   1.0187329 0.015482709  0.99393667  1.04356000 11867.514 0.9998898
## mu[39]   1.0609193 0.011132503  1.04324945  1.07886000 14541.177 0.9999693
## mu[40]   1.0675210 0.012479109  1.04787000  1.08742000 13145.650 1.0000342
## mu[41]   0.9101083 0.022544599  0.87389156  0.94620739  9678.534 0.9998439
## mu[42]   1.0811844 0.016261911  1.05559945  1.10722055 11479.696 1.0001097
## mu[43]   1.0769060 0.014972748  1.05328945  1.10097000 11859.519 1.0000932
## mu[44]   1.0435065 0.010240537  1.02702000  1.05956000 16882.356 0.9998035
## mu[45]   0.8689264 0.017510768  0.84086790  0.89691228 13527.440 1.0000558
## mu[46]   1.0518563 0.010113424  1.03556945  1.06783000 16567.701 0.9998645
## mu[47]   0.8734903 0.016568914  0.84701006  0.89976964 15415.614 1.0000124
## mu[48]   0.9111582 0.022974862  0.87423468  0.94802915  9532.346 0.9998465
## mu[49]   1.0424024 0.010335750  1.02584000  1.05865165 16646.009 0.9998004
## mu[50]   1.0784241 0.015421508  1.05420945  1.10315000 11714.167 1.0000998
## mu[51]   0.8916386 0.016768130  0.86491378  0.91851205 14614.891 0.9998449
## mu[52]   1.0737087 0.014063981  1.05147945  1.09634055 12215.856 1.0000772
## mu[53]   1.0491421 0.010037527  1.03304000  1.06498000 16926.177 0.9998380
## mu[54]   1.0559968 0.010440961  1.03932000  1.07266000 15737.633 0.9999119
## mu[55]   0.8626699 0.019305047  0.83168246  0.89341011 11497.270 1.0000995
## mu[56]   1.0681881 0.012636984  1.04830945  1.08833000 13027.887 1.0000397
## mu[57]   0.9970876 0.022654922  0.96078445  1.03354165 10588.443 0.9999726
## mu[58]   0.8628842 0.019235407  0.83202873  0.89349577 11554.717 1.0000983
## mu[59]   0.8738546 0.016509143  0.84755978  0.90006816 15634.046 1.0000086
## mu[60]   0.8655625 0.018411257  0.83607794  0.89495611 12338.155 1.0000817
## mu[61]   0.8685193 0.017611062  0.84027640  0.89676688 13373.664 1.0000592
## mu[62]   0.8699764 0.017263865  0.84224862  0.89756861 13933.587 1.0000466
## mu[63]   1.0098770 0.018277708  0.98079367  1.03938055 11151.764 0.9999302
## mu[64]   1.0332244 0.011736354  1.01444000  1.05188000 14293.798 0.9998141
## mu[65]   1.0396881 0.010641987  1.02258835  1.05649000 15975.350 0.9997979
## mu[66]   1.0749738 0.014417260  1.05219000  1.09817000 12066.247 1.0000840
## mu[67]   1.0237244 0.014044275  1.00117890  1.04618055 12481.207 0.9998633
## mu[68]   1.0317983 0.012037016  1.01251000  1.05096110 13961.044 0.9998205
## mu[69]   1.0521094 0.010126174  1.03578890  1.06810055 16531.817 0.9998672
## mu[70]   1.0269217 0.013194471  1.00583945  1.04795055 12986.439 0.9998458
## mu[71]   1.0732946 0.013950228  1.05121945  1.09574110 12267.418 1.0000748
## mu[72]   1.0590101 0.010828410  1.04177000  1.07646000 15004.887 0.9999476
## mu[73]   1.0579520 0.010678948  1.04089835  1.07509000 15265.185 0.9999351
## mu[74]   1.0694532 0.012945637  1.04907000  1.09012055 12817.089 1.0000495
## mu[75]   1.0250126 0.013694331  1.00309945  1.04691110 12672.682 0.9998563
## mu[76]   1.0473939 0.010048629  1.03128945  1.06318055 17041.847 0.9998240
## mu[77]   1.0420343 0.010371329  1.02540000  1.05837055 16561.570 0.9997997
## mu[78]   1.0247135 0.013774700  1.00264835  1.04674110 12627.144 0.9998579
## mu[79]   1.0401251 0.010585992  1.02312945  1.05679055 16089.206 0.9997978
## mu[80]   1.0560198 0.010443468  1.03934000  1.07269055 15732.463 0.9999120
## mu[81]   1.0322124 0.011947830  1.01304945  1.05122055 14055.596 0.9998186
## mu[82]   1.0726965 0.013787767  1.05091945  1.09487000 12344.563 1.0000712
## mu[83]   0.8723333 0.016774450  0.84539184  0.89909771 14874.827 1.0000242
## mu[84]   0.9826421 0.027867396  0.93832056  1.02733165 10262.934 1.0000041
## mu[85]   1.0688322 0.012792650  1.04871000  1.08928055 12918.860 1.0000448
## mu[86]   1.0811154 0.016240559  1.05556945  1.10712055 11485.267 1.0001095
## mu[87]   1.0361687 0.011179481  1.01821945  1.05393110 15038.785 0.9998036
## mu[88]   1.0756178 0.014600393  1.05256945  1.09911110 11994.365 1.0000872
## mu[89]   1.0225283 0.014377675  0.99933578  1.04553055 12316.348 0.9998698
## mu[90]   0.9847123 0.027108650  0.94163240  1.02811055 10297.841 1.0000004
## mu[91]   1.0318673 0.012022062  1.01259000  1.05101055 13976.419 0.9998202
## mu[92]   1.0663019 0.012199957  1.04712945  1.08581000 13371.993 1.0000238
## mu[93]   0.9908864 0.064134309  0.88956130  1.09384110  6975.024 0.9999777
## mu[94]   1.0772280 0.015067034  1.05348945  1.10139055 11827.282 1.0000947
## mu[95]   1.0609883 0.011144237  1.04329945  1.07894055 14524.173 0.9999702
## mu[96]   1.0777111 0.015209412  1.05380000  1.10212000 11780.674 1.0000968
## mu[97]   1.0644157 0.011793600  1.04581945  1.08322055 13751.393 1.0000061
## mu[98]   0.9097012 0.022379860  0.87377507  0.94554977  9738.033 0.9998429
## mu[99]   1.0628745 0.011486513  1.04473945  1.08126055 14086.633 0.9999905
## mu[100]  0.8830466 0.015851725  0.85795661  0.90834605 18447.721 0.9999091
## mu[101]  1.0414133 0.010435729  1.02468000  1.05781000 16413.249 0.9997987
## mu[102]  1.0199520 0.015120454  0.99570989  1.04420000 12000.207 0.9998835
## mu[103]  0.8611486 0.019814177  0.82932806  0.89267854 11110.828 1.0001074
## mu[104]  1.0461288 0.010086040  1.02990000  1.06198000 17092.472 0.9998157
## mu[105]  1.0569399 0.010549438  1.04006000  1.07386055 15512.315 0.9999230
## mu[106]  0.8711120 0.017016544  0.84375867  0.89834122 14384.701 1.0000361
## mu[107]  0.8604630 0.020051708  0.82809573  0.89239006 10948.631 1.0001106
## mu[108]  0.8783327 0.015978432  0.85288200  0.90380333 18013.989 0.9999597
## mu[109]  0.8800040 0.015881555  0.85473467  0.90523005 18334.998 0.9999412
## mu[110]  1.0580440 0.010691385  1.04097000  1.07520000 15242.674 0.9999362
## mu[111]  0.8775614 0.016042095  0.85196673  0.90313322 17679.961 0.9999683
## mu[112]  1.0447716 0.010153146  1.02844000  1.06071000 17113.383 0.9998084
## mu[113]  0.8618128 0.019588771  0.83044191  0.89301711 11274.975 1.0001041
## mu[114]  0.8646840 0.018671796  0.83478690  0.89437326 12067.789 1.0000875
## mu[115]  1.0584350 0.010745460  1.04126945  1.07569000 15146.239 0.9999407
## mu[116]  1.0804023 0.016020760  1.05517945  1.10606055 11544.809 1.0001071
## mu[117]  1.0258406 0.013474670  1.00429000  1.04734055 12804.467 0.9998517
## mu[118]  0.9652523 0.034342539  0.91069356  1.02053165 10058.482 1.0000285
## mu[119]  1.0343745 0.011508153  1.01599945  1.05268055 14576.917 0.9998095
## mu[120]  1.0557437 0.010413903  1.03910000  1.07236000 15797.443 0.9999087
## mu[121]  1.0364218 0.011135969  1.01854000  1.05409055 15105.687 0.9998029
## mu[122]  1.0458297 0.010098449  1.02956945  1.06167000 17107.783 0.9998140
## mu[123]  1.0502692 0.010055345  1.03411945  1.06622055 16796.176 0.9998484
## mu[124]  1.0346046 0.011464094  1.01631000  1.05284055 14634.674 0.9998086
## mu[125]  1.0447026 0.010157319  1.02837000  1.06063055 17102.197 0.9998082
## mu[126]  1.0744218 0.014262033  1.05186890  1.09740055 12129.868 1.0000811
## mu[127]  1.0479690 0.010039775  1.03183000  1.06374055 17015.712 0.9998283
## mu[128]  1.0512583 0.010087025  1.03502890  1.06723055 16649.170 0.9998583
## mu[129]  1.0756178 0.014600393  1.05256945  1.09911110 11994.365 1.0000872
## mu[130]  1.0335925 0.011661916  1.01494000  1.05213055 14382.959 0.9998126
## mu[131]  1.0521094 0.010126174  1.03578890  1.06810055 16531.817 0.9998672
## mu[132]  1.0596312 0.010922516  1.04225000  1.07718000 14852.482 0.9999547
## mu[133]  0.9288994 0.031103039  0.87928557  0.97830438  8059.099 0.9998934
## mu[134]  1.0602292 0.011017581  1.04270890  1.07793055 14707.180 0.9999616
## mu[135]  0.8674694 0.017881053  0.83887667  0.89625311 12988.160 1.0000677
## mu[136]  0.8632270 0.019125076  0.83255751  0.89366422 11648.231 1.0000964
## mu[137]  1.0808854 0.016169431  1.05545945  1.10679000 11504.508 1.0001088
## mu[138]  1.0458297 0.010098449  1.02956945  1.06167000 17107.783 0.9998140
## mu[139]  0.8686693 0.017573820  0.84050079  0.89682011 13430.043 1.0000580
## mu[140]  1.0409992 0.010481582  1.02417945  1.05747055 16311.184 0.9997983
## mu[141]  1.0453697 0.010120192  1.02907000  1.06123055 17124.883 0.9998115
## mu[142]  1.0238394 0.014012648  1.00134945  1.04626000 12497.548 0.9998627
## mu[143]  1.0648067 0.011875149  1.04610890  1.08375000 13670.134 1.0000099
## mu[144]  0.9236285 0.028555094  0.87779383  0.96914982  8356.477 0.9998802
## mu[145]  0.9626677 0.048689718  0.88593650  1.04018000  7201.167 0.9999519
## mu[146]  1.0638636 0.011680882  1.04538000  1.08252000 13869.683 1.0000007
## mu[147]  0.8669766 0.018013269  0.83819812  0.89600166 12813.228 1.0000715
## mu[148]  0.8639983 0.018881894  0.83380089  0.89403209 11865.888 1.0000918
## mu[149]  1.0570779 0.010566312  1.04015000  1.07400055 15479.014 0.9999246
## mu[150]  0.9593177 0.036583196  0.90117202  1.01811330 10013.085 1.0000347
## mu[151]  1.0747208 0.014345912  1.05204000  1.09781000 12095.166 1.0000827
## mu[152]  1.0634725 0.011602861  1.04512945  1.08204000 13953.877 0.9999968
## mu[153]  1.0673830 0.012446877  1.04778000  1.08723000 13170.104 1.0000331
## mu[154]  0.8735546 0.016558186  0.84709683  0.89982138 15454.183 1.0000118
## mu[155]  1.0209872 0.014818122  0.99715878  1.04472110 12121.148 0.9998781
## mu[156]  0.8725047 0.016742513  0.84560267  0.89917600 14941.005 1.0000225
## mu[157]  0.8775614 0.016042095  0.85196673  0.90313322 17679.961 0.9999683
## mu[158]  1.0716844 0.013517676  1.05033835  1.09340000 12481.988 1.0000650
## mu[159]  1.0712474 0.013403042  1.05009945  1.09275220 12544.247 1.0000620
## mu[160]  1.0565718 0.010505675  1.03978945  1.07338000 15601.360 0.9999186
## mu[161]  1.0691542 0.012871635  1.04890945  1.08968055 12865.522 1.0000473
## mu[162]  1.0106131 0.018036202  0.98193646  1.03973000 11197.902 0.9999272
## mu[163]  1.0666699 0.012282960  1.04731000  1.08632000 13301.885 1.0000270
## mu[164]  1.0340065 0.011579737  1.01550945  1.05243000 14484.890 0.9998109
## mu[165]  1.0414822 0.010428315  1.02476000  1.05788000 16430.382 0.9997988
## mu[166]  1.0426784 0.010310323  1.02611945  1.05891000 16707.765 0.9998010
## mu[167]  1.0278188 0.012967970  1.00703945  1.04846000 13146.821 0.9998409
## mu[168]  0.8957311 0.017672568  0.86755200  0.92397005 13095.395 0.9998316
## mu[169]  0.8694193 0.017392742  0.84152384  0.89722576 13716.480 1.0000515
## mu[170]  0.8835822 0.015866153  0.85843078  0.90892805 18418.545 0.9999038

Stan_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 )

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## 
## All 4 chains finished successfully.
## Mean chain execution time: 2.2 seconds.
## Total execution time: 3.0 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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## Chain 4 finished in 2.9 seconds.
## 
## All 4 chains finished successfully.
## Mean chain execution time: 2.9 seconds.
## Total execution time: 3.8 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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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 )

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#As warmup is increased,n_eff got closer to number of iterations. 400 warmup iterations are enough

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 )

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p=precis(mp)

traceplot(mp)

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?

#From the plot we can see plot a should be a normal distribution as the prior is aroung 0 and spread in between 2 and -2. Plot b is Cauchy distribution which contains some extreme value go up to over 30 and -50.

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
)

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m5.2 = ulam(
  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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## 
## All 4 chains finished successfully.
## Mean chain execution time: 0.3 seconds.
## Total execution time: 1.0 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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## Chain 4 finished in 0.7 seconds.
## 
## All 4 chains finished successfully.
## Mean chain execution time: 0.7 seconds.
## Total execution time: 1.8 seconds.

set.seed(77)
compare( m5.1 , m5.2 , m5.3 , func=WAIC )

##          WAIC        SE    dWAIC       dSE    pWAIC       weight
## m5.1 125.5194 12.553225  0.00000        NA 3.530059 0.7636320968
## m5.3 127.8714 12.737362  2.35193 0.6948482 4.876636 0.2355965776
## m5.2 139.3149  9.881935 13.79546 9.1406045 2.965815 0.0007713256