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)

# It shows that it does not have detectable 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)

# It looks like there are basically no 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]     0.8864569 0.01586764  0.861203560  0.91206722 17601.045 0.9998999
## a[2]     1.0506122 0.01010668  1.034459450  1.06665000 17805.917 1.0000549
## b[1]     0.1304473 0.07593325  0.007672645  0.25154615  6551.381 1.0001076
## b[2]    -0.1425156 0.05572222 -0.232014770 -0.05360469  9723.058 0.9999043
## sigma    0.1116901 0.00618775  0.102324615  0.12195317  9386.134 1.0002300
## mu[1]    0.8764571 0.01599775  0.850893285  0.90212633 16818.906 1.0000333
## mu[2]    1.0025041 0.02070371  0.969554020  1.03555055 11090.282 0.9997865
## mu[3]    1.0635822 0.01162868  1.045158900  1.08187055 14344.592 1.0001494
## mu[4]    1.0634443 0.01160190  1.045050000  1.08167000 14381.098 1.0001491
## mu[5]    1.0194855 0.01525089  0.995283505  1.04375000 12427.590 0.9997872
## mu[6]    1.0811151 0.01617718  1.055210000  1.10691110 11466.244 1.0001228
## mu[7]    1.0779670 0.01523381  1.053510000  1.10223110 11776.542 1.0001321
## mu[8]    1.0005279 0.02138195  0.966535890  1.03477000 11001.023 0.9997880
## mu[9]    1.0428322 0.01034326  1.026179450  1.05917000 17554.237 0.9999486
## mu[10]   0.8958496 0.01759564  0.867491505  0.92424911 13121.716 0.9998378
## mu[11]   1.0723372 0.01366134  1.050348350  1.09404055 12518.702 1.0001462
## mu[12]   0.8613764 0.01975705  0.829528945  0.89269155 10841.452 1.0001874
## mu[13]   0.8633745 0.01907024  0.832735670  0.89367171 11406.550 1.0001750
## mu[14]   1.0769789 0.01494633  1.052929450  1.10081000 11887.693 1.0001349
## mu[15]   1.0472672 0.01009740  1.030939450  1.06328000 18067.737 1.0000104
## mu[16]   1.0759449 0.01465038  1.052400000  1.09925055 12012.014 1.0001377
## mu[17]   1.0799892 0.01583520  1.054640000  1.10526165 11570.002 1.0001262
## mu[18]   1.0281486 0.01290484  1.007599450  1.04859000 13837.822 0.9998120
## mu[19]   1.0774845 0.01509286  1.053210000  1.10154110 11829.928 1.0001335
## mu[20]   1.0673048 0.01241854  1.047600000  1.08693055 13453.442 1.0001524
## mu[21]   1.0616519 0.01127139  1.043720000  1.07946055 14872.277 1.0001444
## mu[22]   1.0757380 0.01459183  1.052279450  1.09892000 12037.995 1.0001382
## mu[23]   1.0591242 0.01086531  1.041880000  1.07627000 15612.007 1.0001334
## mu[24]   0.8622177 0.01946235  0.830941945  0.89308500 11069.449 1.0001825
## mu[25]   0.8625542 0.01934668  0.831421725  0.89323811 11164.633 1.0001803
## mu[26]   1.0634443 0.01160190  1.045050000  1.08167000 14381.098 1.0001491
## mu[27]   0.9718501 0.03171398  0.921348790  1.02259110 10308.659 0.9998124
## mu[28]   1.0242421 0.01390971  1.002050000  1.04625000 13112.376 0.9997972
## mu[29]   1.0380985 0.01090784  1.020740000  1.05541000 16366.605 0.9998896
## mu[30]   0.8631221 0.01915446  0.832314800  0.89353704 11330.593 1.0001767
## mu[31]   0.8692428 0.01734933  0.841309000  0.89683111 13579.381 1.0001241
## mu[32]   0.8677284 0.01774653  0.839138560  0.89579149 12943.148 1.0001394
## mu[33]   0.8616077 0.01967523  0.829857000  0.89279955 10902.747 1.0001861
## mu[34]   1.0609166 0.01114582  1.043219450  1.07853000 15082.427 1.0001417
## mu[35]   0.9284089 0.03119104  0.877547570  0.97774599  7876.414 0.9999104
## mu[36]   0.9081961 0.02183026  0.872820725  0.94282610  9714.592 0.9998461
## mu[37]   1.0327214 0.01187101  1.013719450  1.05155000 14898.745 0.9998396
## mu[38]   1.0187962 0.01545417  0.994205780  1.04339055 12343.765 0.9997863
## mu[39]   1.0609396 0.01114960  1.043239450  1.07856000 15076.047 1.0001419
## mu[40]   1.0675345 0.01247118  1.047718900  1.08725000 13404.582 1.0001523
## mu[41]   0.9095632 0.02239197  0.873213780  0.94505449  9495.881 0.9998500
## mu[42]   1.0811841 0.01619828  1.055240000  1.10702055 11460.133 1.0001226
## mu[43]   1.0769100 0.01492644  1.052898900  1.10069000 11895.753 1.0001351
## mu[44]   1.0435445 0.01028445  1.026919450  1.05973055 17691.809 0.9999582
## mu[45]   0.8691376 0.01737577  0.841162615  0.89674683 13533.618 1.0001252
## mu[46]   1.0518859 0.01015465  1.035710000  1.06800055 17582.758 1.0000704
## mu[47]   0.8736176 0.01641518  0.847218230  0.89974655 15569.416 1.0000720
## mu[48]   0.9105938 0.02282462  0.873608085  0.94675705  9344.709 0.9998531
## mu[49]   1.0424416 0.01037850  1.025738900  1.05882000 17472.660 0.9999433
## mu[50]   1.0784266 0.01536899  1.053779450  1.10292000 11727.137 1.0001308
## mu[51]   0.8914327 0.01658214  0.864828780  0.91819516 15499.672 0.9998572
## mu[52]   1.0737159 0.01403079  1.051070000  1.09600055 12310.902 1.0001433
## mu[53]   1.0491743 0.01008198  1.032940000  1.06523000 17979.914 1.0000361
## mu[54]   1.0560221 0.01047336  1.039359450  1.07256055 16537.791 1.0001122
## mu[55]   0.8629959 0.01919684  0.832084890  0.89346622 11293.121 1.0001775
## mu[56]   1.0682009 0.01262639  1.048059450  1.08812000 13265.324 1.0001520
## mu[57]   0.9971730 0.02254725  0.961337340  1.03333165 10868.157 0.9997906
## mu[58]   0.8632063 0.01912630  0.832451120  0.89357250 11355.766 1.0001761
## mu[59]   0.8739752 0.01635402  0.847724560  0.89998877 15733.413 1.0000673
## mu[60]   0.8658354 0.01829070  0.836346890  0.89487825 12221.289 1.0001564
## mu[61]   0.8687380 0.01747780  0.840637450  0.89648505 13361.761 1.0001294
## mu[62]   0.8701682 0.01712442  0.842503295  0.89734527 13991.605 1.0001141
## mu[63]   1.0099492 0.01821632  0.981155625  1.03901055 11523.864 0.9997829
## mu[64]   1.0332729 0.01175859  1.014488900  1.05187055 15041.098 0.9998438
## mu[65]   1.0397300 0.01068045  1.022609450  1.05663000 16812.068 0.9999085
## mu[66]   1.0749797 0.01437890  1.051780000  1.09778000 12136.100 1.0001402
## mu[67]   1.0237826 0.01403413  1.001430000  1.04599110 13037.396 0.9997959
## mu[68]   1.0318482 0.01205482  1.012579450  1.05090110 14678.636 0.9998333
## mu[69]   1.0521386 0.01016707  1.035949450  1.06827055 17530.531 1.0000734
## mu[70]   1.0269767 0.01319582  1.005950000  1.04786055 13603.073 0.9998068
## mu[71]   1.0733023 0.01391878  1.050860000  1.09542000 12371.212 1.0001443
## mu[72]   1.0590323 0.01085198  1.041800000  1.07614000 15639.763 1.0001328
## mu[73]   1.0579753 0.01070585  1.040950000  1.07485000 15957.460 1.0001265
## mu[74]   1.0694648 0.01292996  1.048800000  1.08982055 13016.382 1.0001507
## mu[75]   1.0250694 0.01368885  1.003258900  1.04669000 13252.823 0.9997997
## mu[76]   1.0474280 0.01009398  1.031119450  1.06345055 18067.111 1.0000125
## mu[77]   1.0420738 0.01041366  1.025339450  1.05851000 17391.610 0.9999383
## mu[78]   1.0247707 0.01376818  1.002838900  1.04654000 13201.279 0.9997988
## mu[79]   1.0401666 0.01062529  1.023109450  1.05700000 16926.887 0.9999140
## mu[80]   1.0560450 0.01047580  1.039388900  1.07259055 16531.202 1.0001124
## mu[81]   1.0322618 0.01196690  1.013110000  1.05119000 14781.984 0.9998362
## mu[82]   1.0727048 0.01375876  1.050530000  1.09459000 12461.780 1.0001455
## mu[83]   0.8724819 0.01662529  0.845694615  0.89892805 15046.463 1.0000866
## mu[84]   0.9827421 0.02770925  0.938584955  1.02699110 10481.682 0.9998033
## mu[85]   1.0688444 0.01277944  1.048440000  1.08897055 13136.159 1.0001514
## mu[86]   1.0811151 0.01617718  1.055210000  1.10691110 11466.244 1.0001228
## mu[87]   1.0362142 0.01120995  1.018399450  1.05406000 15839.574 0.9998698
## mu[88]   1.0756232 0.01455938  1.052210000  1.09876000 12052.511 1.0001386
## mu[89]   1.0225877 0.01436312  0.999666725  1.04539055 12851.856 0.9997928
## mu[90]   0.9848103 0.02695767  0.941854790  1.02782000 10523.579 0.9998015
## mu[91]   1.0319172 0.01204003  1.012659450  1.05097000 14695.760 0.9998339
## mu[92]   1.0663167 0.01219694  1.046870000  1.08558055 13673.306 1.0001523
## mu[93]   0.9888581 0.06410429  0.884928525  1.09091330  6844.187 1.0000119
## mu[94]   1.0772317 0.01501943  1.053059450  1.10117055 11858.456 1.0001342
## mu[95]   1.0610085 0.01116116  1.043289450  1.07865000 15055.866 1.0001421
## mu[96]   1.0777142 0.01515982  1.053367800  1.10189000 11804.184 1.0001329
## mu[97]   1.0644324 0.01179792  1.045750000  1.08301000 14126.206 1.0001510
## mu[98]   0.9091636 0.02222629  0.873095670  0.94446339  9557.572 0.9998489
## mu[99]   1.0628928 0.01149668  1.044679450  1.08097055 14528.278 1.0001479
## mu[100]  0.8829984 0.01567004  0.858011670  0.90848916 18210.745 0.9999415
## mu[101]  1.0414534 0.01047714  1.024680000  1.05806055 17247.811 0.9999303
## mu[102]  1.0200141 0.01509643  0.995988120  1.04402000 12494.254 0.9997880
## mu[103]  0.8615026 0.01971235  0.829717955  0.89276206 10874.758 1.0001867
## mu[104]  1.0461641 0.01013143  1.029809450  1.06220000 18042.663 0.9999949
## mu[105]  1.0569642 0.01057926  1.040120000  1.07362110 16260.516 1.0001195
## mu[106]  0.8712830 0.01687235  0.844075075  0.89808533 14500.390 1.0001012
## mu[107]  0.8608295 0.01995271  0.828672790  0.89243260 10700.618 1.0001904
## mu[108]  0.8783711 0.01580793  0.853126780  0.90392111 17518.055 1.0000060
## mu[109]  0.8800117 0.01570641  0.854985175  0.90550899 17952.021 0.9999825
## mu[110]  1.0580672 0.01071797  1.041030000  1.07497000 15930.145 1.0001272
## mu[111]  0.8776139 0.01587395  0.852295670  0.90319722 17262.228 1.0000168
## mu[112]  1.0448084 0.01019811  1.028319450  1.06090000 17894.700 0.9999757
## mu[113]  0.8621546 0.01948418  0.830874890  0.89305160 11051.860 1.0001828
## mu[114]  0.8649730 0.01855502  0.835194340  0.89446238 11920.122 1.0001633
## mu[115]  1.0584579 0.01077080  1.041359450  1.07544055 15812.143 1.0001296
## mu[116]  1.0804028 0.01596031  1.054868900  1.10588055 11531.040 1.0001249
## mu[117]  1.0258967 0.01347215  1.004478900  1.04715055 13400.003 0.9998026
## mu[118]  0.9653700 0.03412571  0.911069845  1.01996055 10233.470 0.9998173
## mu[119]  1.0344219 0.01153370  1.016030000  1.05269000 15346.220 0.9998532
## mu[120]  1.0557693 0.01044701  1.039150000  1.07227055 16610.709 1.0001100
## mu[121]  1.0364670 0.01116711  1.018749450  1.05426055 15910.086 0.9998723
## mu[122]  1.0458654 0.01014379  1.029480000  1.06187055 18017.191 0.9999906
## mu[123]  1.0503004 0.01009871  1.034139450  1.06633000 17851.325 1.0000510
## mu[124]  1.0346517 0.01149031  1.016340000  1.05286000 15408.731 0.9998553
## mu[125]  1.0447394 0.01020221  1.028248900  1.06083055 17884.817 0.9999749
## mu[126]  1.0744283 0.01422590  1.051470000  1.09701055 12210.583 1.0001415
## mu[127]  1.0480024 0.01008492  1.031660000  1.06406000 18054.502 1.0000204
## mu[128]  1.0512884 0.01012915  1.035139450  1.06733000 17694.531 1.0000633
## mu[129]  1.0756232 0.01455938  1.052210000  1.09876000 12052.511 1.0001386
## mu[130]  1.0336406 0.01168514  1.014970000  1.05215055 15137.614 0.9998467
## mu[131]  1.0521386 0.01016707  1.035949450  1.06827055 17530.531 1.0000734
## mu[132]  1.0596528 0.01094415  1.042280000  1.07695000 15454.609 1.0001363
## mu[133]  0.9280093 0.03099009  0.877431670  0.97701511  7894.684 0.9999092
## mu[134]  1.0602502 0.01103714  1.042739450  1.07771110 15277.401 1.0001390
## mu[135]  0.8677073 0.01775229  0.839115340  0.89577138 12934.661 1.0001396
## mu[136]  0.8635428 0.01901452  0.833056130  0.89374416 11457.945 1.0001738
## mu[137]  1.0808853 0.01610698  1.055100000  1.10658000 11486.747 1.0001235
## mu[138]  1.0458654 0.01014379  1.029480000  1.06187055 18017.191 0.9999906
## mu[139]  0.8688852 0.01743991  0.840874845  0.89655394 13424.673 1.0001279
## mu[140]  1.0410398 0.01052234  1.024169450  1.05770110 17148.243 0.9999248
## mu[141]  1.0454058 0.01016537  1.028990000  1.06145000 17969.511 0.9999842
## mu[142]  1.0238975 0.01400295  1.001600000  1.04606000 13055.972 0.9997962
## mu[143]  1.0648230 0.01187795  1.045969450  1.08354000 14028.913 1.0001513
## mu[144]  0.9228351 0.02843186  0.876575945  0.96768188  8173.790 0.9998932
## mu[145]  0.9611574 0.04862819  0.881983000  1.03816165  7062.623 0.9999804
## mu[146]  1.0638809 0.01168736  1.045340000  1.08225110 14267.071 1.0001501
## mu[147]  0.8672236 0.01788663  0.838425965  0.89555127 12742.484 1.0001441
## mu[148]  0.8643000 0.01876806  0.834193725  0.89405959 11696.911 1.0001684
## mu[149]  1.0571021 0.01059570  1.040230000  1.07379000 16219.540 1.0001207
## mu[150]  0.9594415 0.03634678  0.901712920  1.01763000 10177.352 0.9998215
## mu[151]  1.0747270 0.01430861  1.051649450  1.09743110 12169.933 1.0001408
## mu[152]  1.0634903 0.01161081  1.045090000  1.08174055 14369.054 1.0001493
## mu[153]  1.0673967 0.01243954  1.047639450  1.08705055 13433.745 1.0001523
## mu[154]  0.8736807 0.01640421  0.847286120  0.89979550 15598.427 1.0000712
## mu[155]  1.0210480 0.01479794  0.997507230  1.04452000 12631.210 0.9997897
## mu[156]  0.8726501 0.01659267  0.845883725  0.89900839 15123.844 1.0000845
## mu[157]  0.8776139 0.01587395  0.852295670  0.90319722 17262.228 1.0000168
## mu[158]  1.0716938 0.01349277  1.049998900  1.09306000 12622.469 1.0001475
## mu[159]  1.0712572 0.01337998  1.049770000  1.09241110 12695.192 1.0001483
## mu[160]  1.0565965 0.01053653  1.039839450  1.07321000 16369.649 1.0001167
## mu[161]  1.0691661 0.01285710  1.048620000  1.08940055 13073.511 1.0001511
## mu[162]  1.0106846 0.01797754  0.982280395  1.03937055 11577.096 0.9997827
## mu[163]  1.0666843 0.01227848  1.047130000  1.08610000 13590.084 1.0001524
## mu[164]  1.0340542 0.01160430  1.015529450  1.05242055 15247.390 0.9998502
## mu[165]  1.0415224 0.01046982  1.024769450  1.05811000 17264.266 0.9999311
## mu[166]  1.0427173 0.01035340  1.026049450  1.05906055 17530.519 0.9999469
## mu[167]  1.0278728 0.01297243  1.007240000  1.04842000 13781.379 0.9998107
## mu[168]  0.8954500 0.01749082  0.867305000  0.92369420 13317.090 0.9998389
## mu[169]  0.8696214 0.01725565  0.841768570  0.89703704 13746.035 1.0001201
## mu[170]  0.8835242 0.01568360  0.858532230  0.90903411 18178.378 0.9999347

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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## 
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## Mean chain execution time: 0.6 seconds.
## Total execution time: 0.7 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 0.6 seconds.
## 
## All 4 chains finished successfully.
## Mean chain execution time: 0.6 seconds.
## Total execution time: 0.6 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 the results show, we can observe that warmup is increased,n_eff gets closer to the number of iterations and specifically 400 warmup iterations are enough for our model.

set_cmdstan_path('/Users/vinnakotanishitha/cmdstan')

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)

#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
)

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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 
)

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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 
)

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

##          WAIC        SE     dWAIC       dSE    pWAIC       weight
## m5.1 125.6551 12.604516  0.000000        NA 3.579695 0.6863253133
## m5.3 127.2261 12.648553  1.570988 0.8079528 4.599220 0.3128918868
## m5.2 139.2076  9.773978 13.552460 9.2753036 2.906573 0.0007827999

#Comparing the WAIC values for the models using divorce data, the first model which used medianage of marriage as predictor performed better. However, the model with marriage as predictor had worse performance compared to the other 2 models.