Assignment #8
Aniket Muluk
2022-04-11
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)
dt = data[ complete.cases(data$rgdppc_2000) , ]
dt$log_gdp_std = dt$log_gdp/ mean(dt$log_gdp)
dt$rugged_std = dt$rugged/max(dt$rugged)
dt$cid=ifelse(dt$cont_africa==1,1,2)
dt_slim = list(
log_gdp_std = dt$log_gdp_std,
rugged_std = dt$rugged_std,
cid = as.integer( dt$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=dt_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=dt_slim , chains=4, cores = 4 )## Running MCMC with 4 parallel chains, with 1 thread(s) per chain...
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# Has no noticeable impact on the sigma posterior distribution.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=dt_slim , chains=4, cores = 4 )## Running MCMC with 4 parallel chains, with 1 thread(s) per chain...
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#When compared to the uniform prior, the posterior distribution appears to be slightly right skewed.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_mdl = 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=dt_slim , warmup = 100, iter = 4000, chains=4, cores = 4 )## Running MCMC with 4 parallel chains...
##
## Chain 1 WARNING: There aren't enough warmup iterations to fit the
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## Total execution time: 1.9 seconds.precis(Stan_mdl,depth=2)## mean sd 5.5% 94.5% n_eff Rhat4
## a[1] 0.8865641 0.016164727 0.86098878 0.91223011 18250.151 0.9999544
## a[2] 1.0505726 0.010122749 1.03427890 1.06673000 17305.892 0.9999888
## b[1] 0.1314256 0.075328389 0.01198821 0.25135516 6813.325 1.0001960
## b[2] -0.1417086 0.055620160 -0.23046777 -0.05341351 9271.960 1.0001516
## sigma 0.1116105 0.006184875 0.10234689 0.12193006 9454.661 1.0004681
## mu[1] 0.8764894 0.016261351 0.85084200 0.90248372 18598.253 1.0000394
## mu[2] 1.0027370 0.020610139 0.97006894 1.03576110 10468.516 1.0001771
## mu[3] 1.0634692 0.011672009 1.04512945 1.08212055 14277.363 0.9999748
## mu[4] 1.0633321 0.011645087 1.04505000 1.08196000 14313.612 0.9999746
## mu[5] 1.0196222 0.015182162 0.99547367 1.04398055 11732.047 1.0001563
## mu[6] 1.0809029 0.016227776 1.05507000 1.10691110 11289.937 1.0000222
## mu[7] 1.0777726 0.015284630 1.05350945 1.10229055 11620.037 1.0000133
## mu[8] 1.0007720 0.021285996 0.96704879 1.03498055 10385.443 1.0001780
## mu[9] 1.0428367 0.010334555 1.02612945 1.05938000 16764.944 1.0000371
## mu[10] 0.8960273 0.017863231 0.86753896 0.92460044 13964.804 0.9999342
## mu[11] 1.0721746 0.013711232 1.05046890 1.09415110 12403.402 0.9999969
## mu[12] 0.8612956 0.019881734 0.82943778 0.89325822 12264.133 1.0001741
## mu[13] 0.8633087 0.019215328 0.83259717 0.89416037 12909.550 1.0001608
## mu[14] 1.0767901 0.014997132 1.05299890 1.10081055 11737.983 1.0000105
## mu[15] 1.0472465 0.010103118 1.03093945 1.06338000 17293.948 1.0000069
## mu[16] 1.0757619 0.014701107 1.05246890 1.09929055 11869.696 1.0000075
## mu[17] 1.0797833 0.015885954 1.05448945 1.10528000 11400.592 1.0000191
## mu[18] 1.0282363 0.012853945 1.00764945 1.04878055 13093.148 1.0001284
## mu[19] 1.0772928 0.015143721 1.05327000 1.10158110 11676.860 1.0000120
## mu[20] 1.0671707 0.012465662 1.04755890 1.08700055 13379.299 0.9999829
## mu[21] 1.0615499 0.011312134 1.04378000 1.07970000 14899.321 0.9999720
## mu[22] 1.0755562 0.014642497 1.05233000 1.09900000 11897.185 1.0000069
## mu[23] 1.0590365 0.010901813 1.04179000 1.07659000 15799.176 0.9999703
## mu[24] 0.8621432 0.019595624 0.83080678 0.89364161 12525.595 1.0001687
## mu[25] 0.8624822 0.019483395 0.83135856 0.89378006 12634.325 1.0001665
## mu[26] 1.0633321 0.011645087 1.04505000 1.08196000 14313.612 0.9999746
## mu[27] 0.9722566 0.031589408 0.92219762 1.02267275 9750.207 1.0001800
## mu[28] 1.0243520 0.013850199 1.00222000 1.04661055 12389.856 1.0001432
## mu[29] 1.0381298 0.010884084 1.02057000 1.05557110 15614.413 1.0000713
## mu[30] 0.8630544 0.019296968 0.83220962 0.89406628 12823.310 1.0001626
## mu[31] 0.8692209 0.017552901 0.84119318 0.89738849 15300.844 1.0001125
## mu[32] 0.8676952 0.017935430 0.83902583 0.89639232 14616.873 1.0001262
## mu[33] 0.8615287 0.019802275 0.82980613 0.89338905 12334.576 1.0001727
## mu[34] 1.0608187 0.011185405 1.04318945 1.07879055 15156.659 0.9999713
## mu[35] 0.9288308 0.031223848 0.87931147 0.97854155 8024.273 1.0000377
## mu[36] 0.9084664 0.022012919 0.87334756 0.94356033 10095.376 0.9999682
## mu[37] 1.0327832 0.011831700 1.01382945 1.05151055 14135.119 1.0001054
## mu[38] 1.0189368 0.015384222 0.99450723 1.04354055 11651.834 1.0001580
## mu[39] 1.0608416 0.011189258 1.04320000 1.07883000 15147.810 0.9999713
## mu[40] 1.0673992 0.012518463 1.04767945 1.08733055 13327.990 0.9999836
## mu[41] 0.9098438 0.022564215 0.87384094 0.94574249 9843.446 0.9999734
## mu[42] 1.0809714 0.016248773 1.05509945 1.10701110 11283.287 1.0000223
## mu[43] 1.0767215 0.014977244 1.05294945 1.10072000 11746.484 1.0000103
## mu[44] 1.0435450 0.010278172 1.02692000 1.06004055 16882.098 1.0000320
## mu[45] 0.8691150 0.017578331 0.84103162 0.89733016 15252.132 1.0001134
## mu[46] 1.0518391 0.010174337 1.03547945 1.06810000 17217.017 0.9999835
## mu[47] 0.8736286 0.016657673 0.84702184 0.90049006 17389.822 1.0000690
## mu[48] 0.9108821 0.022988996 0.87416784 0.94754788 9669.417 0.9999773
## mu[49] 1.0424483 0.010368571 1.02569000 1.05906055 16696.279 1.0000399
## mu[50] 1.0782295 0.015419810 1.05374945 1.10297055 11567.734 1.0000146
## mu[51] 0.8915773 0.016869712 0.86485789 0.91837944 15976.148 0.9999362
## mu[52] 1.0735455 0.014081111 1.05118000 1.09605110 12184.992 1.0000008
## mu[53] 1.0491429 0.010093616 1.03287000 1.06526055 17345.065 0.9999961
## mu[54] 1.0559519 0.010503508 1.03928000 1.07280000 16642.935 0.9999725
## mu[55] 0.8629273 0.019338069 0.83201678 0.89400621 12780.718 1.0001635
## mu[56] 1.0680618 0.012674158 1.04807945 1.08822000 13183.545 0.9999853
## mu[57] 0.9974360 0.022447439 0.96186995 1.03371110 10262.121 1.0001792
## mu[58] 0.8631392 0.019269670 0.83235094 0.89408298 12851.899 1.0001620
## mu[59] 0.8739889 0.016599377 0.84762201 0.90070405 17555.658 1.0000653
## mu[60] 0.8657880 0.018460770 0.83632450 0.89527983 13823.983 1.0001422
## mu[61] 0.8687124 0.017676528 0.84044956 0.89705310 15068.571 1.0001171
## mu[62] 0.8701533 0.017336704 0.84255556 0.89796022 15735.865 1.0001037
## mu[63] 1.0101400 0.018132417 0.98133594 1.03917055 10874.599 1.0001716
## mu[64] 1.0333316 0.011720719 1.01454000 1.05192110 14276.389 1.0001024
## mu[65] 1.0397521 0.010661804 1.02251945 1.05681110 16163.746 1.0000596
## mu[66] 1.0748022 0.014429489 1.05186000 1.09789000 12000.885 1.0000046
## mu[67] 1.0238950 0.013973650 1.00157000 1.04634000 12317.330 1.0001447
## mu[68] 1.0319150 0.012013166 1.01270945 1.05100000 13917.874 1.0001102
## mu[69] 1.0520905 0.010187351 1.03571000 1.06840000 17193.730 0.9999825
## mu[70] 1.0270710 0.013142188 1.00608945 1.04816000 12864.816 1.0001333
## mu[71] 1.0731343 0.013969039 1.05096890 1.09546000 12248.638 0.9999998
## mu[72] 1.0589451 0.010888343 1.04171945 1.07650000 15832.073 0.9999704
## mu[73] 1.0578941 0.010740099 1.04093945 1.07520055 16213.360 0.9999706
## mu[74] 1.0693185 0.012978556 1.04886945 1.09001000 12924.578 0.9999887
## mu[75] 1.0251745 0.013631057 1.00341945 1.04708055 12525.059 1.0001404
## mu[76] 1.0474065 0.010100195 1.03108000 1.06353055 17302.822 1.0000060
## mu[77] 1.0420827 0.010402530 1.02526000 1.05874110 16629.879 1.0000426
## mu[78] 1.0248775 0.013709743 1.00300890 1.04688110 12475.373 1.0001414
## mu[79] 1.0401862 0.010608011 1.02308000 1.05714110 16254.574 1.0000564
## mu[80] 1.0559748 0.010506005 1.03930945 1.07283000 16638.748 0.9999724
## mu[81] 1.0323262 0.011926298 1.01321945 1.05122110 14019.967 1.0001080
## mu[82] 1.0725402 0.013808793 1.05067000 1.09462000 12343.541 0.9999980
## mu[83] 0.8724843 0.016858300 0.84556058 0.89963144 16849.949 1.0000807
## mu[84] 0.9830869 0.027594654 0.93929456 1.02746055 9906.952 1.0001807
## mu[85] 1.0687016 0.012827643 1.04848945 1.08914000 13049.109 0.9999870
## mu[86] 1.0809029 0.016227776 1.05507000 1.10691110 11289.937 1.0000222
## mu[87] 1.0362562 0.011180542 1.01827000 1.05403055 15074.556 1.0000839
## mu[88] 1.0754420 0.014610048 1.05227000 1.09884110 11912.526 1.0000066
## mu[89] 1.0227069 0.014300208 0.99989162 1.04563000 12138.319 1.0001484
## mu[90] 0.9851433 0.026845048 0.94255724 1.02829055 9945.140 1.0001807
## mu[91] 1.0319835 0.011998588 1.01278945 1.05103055 13934.427 1.0001099
## mu[92] 1.0661882 0.012243203 1.04688890 1.08565055 13603.238 0.9999805
## mu[93] 0.9897332 0.063775851 0.88875803 1.09180165 7003.856 1.0001257
## mu[94] 1.0770414 0.015070238 1.05318670 1.10121000 11707.136 1.0000112
## mu[95] 1.0609101 0.011200856 1.04324945 1.07890000 15123.759 0.9999713
## mu[96] 1.0775212 0.015210641 1.05336945 1.10192000 11649.440 1.0000127
## mu[97] 1.0643146 0.011842250 1.04565945 1.08325000 14059.189 0.9999763
## mu[98] 0.9094411 0.022401595 0.87373112 0.94511482 9914.492 0.9999719
## mu[99] 1.0627837 0.011539154 1.04471000 1.08128055 14478.794 0.9999737
## mu[100] 0.8830797 0.015963647 0.85809290 0.90839111 19252.031 0.9999777
## mu[101] 1.0414657 0.010464028 1.02453000 1.05817110 16512.803 1.0000471
## mu[102] 1.0201478 0.015028652 0.99630094 1.04425110 11795.886 1.0001552
## mu[103] 0.8614227 0.019838320 0.82964178 0.89334316 12302.412 1.0001733
## mu[104] 1.0461498 0.010133626 1.02974000 1.06233000 17211.850 1.0000138
## mu[105] 1.0568887 0.010611445 1.04009000 1.07396055 16467.939 0.9999712
## mu[106] 0.8712764 0.017094841 0.84401467 0.89873344 16270.837 1.0000928
## mu[107] 0.8607446 0.020071825 0.82855572 0.89305477 12101.928 1.0001775
## mu[108] 0.8784178 0.016083251 0.85329878 0.90401544 19130.499 1.0000198
## mu[109] 0.8800706 0.015989942 0.85517778 0.90554522 19371.751 1.0000038
## mu[110] 1.0579855 0.010752500 1.04098945 1.07531055 16180.779 0.9999705
## mu[111] 0.8776549 0.016144900 0.85233607 0.90339400 18947.990 1.0000275
## mu[112] 1.0448017 0.010195825 1.02832890 1.06112000 17063.251 1.0000230
## mu[113] 0.8620796 0.019616810 0.83069094 0.89361566 12505.459 1.0001691
## mu[114] 0.8649192 0.018716374 0.83501329 0.89490227 13488.272 1.0001490
## mu[115] 1.0583739 0.010806071 1.04131000 1.07580055 16039.440 0.9999705
## mu[116] 1.0801945 0.016010951 1.05472000 1.10587055 11358.964 1.0000201
## mu[117] 1.0259971 0.013416163 1.00454000 1.04752055 12667.736 1.0001374
## mu[118] 0.9658132 0.033995492 0.91194340 1.01989055 9682.762 1.0001793
## mu[119] 1.0344740 0.011499119 1.01598000 1.05275000 14580.225 1.0000954
## mu[120] 1.0557006 0.010476569 1.03906000 1.07248055 16687.718 0.9999729
## mu[121] 1.0365076 0.011138392 1.01854945 1.05423000 15145.706 1.0000823
## mu[122] 1.0458527 0.010144991 1.02940000 1.06208000 17183.604 1.0000159
## mu[123] 1.0502626 0.010113798 1.03397000 1.06643055 17319.797 0.9999903
## mu[124] 1.0347025 0.011456359 1.01630780 1.05292055 14642.106 1.0000939
## mu[125] 1.0447332 0.010199779 1.02825000 1.06106000 17054.137 1.0000234
## mu[126] 1.0742539 0.014276407 1.05156945 1.09705055 12079.482 1.0000030
## mu[127] 1.0479777 0.010092972 1.03165000 1.06408110 17327.607 1.0000026
## mu[128] 1.0512451 0.010147135 1.03491945 1.06740055 17264.824 0.9999858
## mu[129] 1.0754420 0.014610048 1.05227000 1.09884110 11912.526 1.0000066
## mu[130] 1.0336971 0.011648395 1.01497945 1.05219000 14372.432 1.0001001
## mu[131] 1.0520905 0.010187351 1.03571000 1.06840000 17193.730 0.9999825
## mu[132] 1.0595621 0.010981569 1.04221945 1.07718000 15607.822 0.9999706
## mu[133] 0.9284281 0.031025680 0.87921507 0.97774622 8044.944 1.0000366
## mu[134] 1.0601561 0.011075629 1.04265000 1.07795055 15393.314 0.9999709
## mu[135] 0.8676740 0.017940988 0.83899556 0.89638188 14607.664 1.0001264
## mu[136] 0.8634782 0.019161325 0.83282900 0.89423699 12967.824 1.0001596
## mu[137] 1.0806744 0.016157569 1.05495945 1.10658055 11311.916 1.0000214
## mu[138] 1.0458527 0.010144991 1.02940000 1.06208000 17183.604 1.0000159
## mu[139] 0.8688607 0.017640062 0.84067673 0.89716949 15135.902 1.0001158
## mu[140] 1.0410545 0.010507927 1.02406000 1.05785055 16431.908 1.0000501
## mu[141] 1.0453958 0.010165084 1.02891945 1.06161055 17134.903 1.0000190
## mu[142] 1.0240093 0.013942674 1.00172945 1.04641000 12335.164 1.0001444
## mu[143] 1.0647031 0.011922750 1.04593000 1.08375055 13961.238 0.9999772
## mu[144] 0.9232152 0.028504390 0.87794094 0.96857522 8359.514 1.0000210
## mu[145] 0.9618249 0.048453701 0.88518962 1.03895220 7203.332 1.0000999
## mu[146] 1.0637662 0.011731063 1.04529945 1.08251000 14199.977 0.9999754
## mu[147] 0.8671866 0.018070546 0.83826572 0.89614199 14398.214 1.0001306
## mu[148] 0.8642411 0.018922572 0.83399567 0.89461800 13237.686 1.0001541
## mu[149] 1.0570258 0.010628278 1.04020945 1.07412055 16441.331 0.9999711
## mu[150] 0.9599182 0.036211514 0.90246556 1.01761110 9632.716 1.0001786
## mu[151] 1.0745509 0.014359145 1.05172000 1.09752055 12036.405 1.0000039
## mu[152] 1.0633778 0.011653989 1.04508000 1.08200000 14301.645 0.9999747
## mu[153] 1.0672621 0.012486739 1.04760000 1.08713000 13358.693 0.9999831
## mu[154] 0.8736922 0.016647218 0.84712396 0.90050527 17419.259 1.0000683
## mu[155] 1.0211760 0.014732054 0.99774112 1.04482055 11926.944 1.0001526
## mu[156] 0.8726538 0.016827121 0.84576889 0.89975621 16930.823 1.0000790
## mu[157] 0.8776549 0.016144900 0.85233607 0.90339400 18947.990 1.0000275
## mu[158] 1.0715348 0.013542526 1.05009945 1.09321055 12512.570 0.9999950
## mu[159] 1.0711007 0.013429484 1.04985000 1.09257000 12588.676 0.9999938
## mu[160] 1.0565231 0.010567970 1.03977945 1.07351000 16537.626 0.9999717
## mu[161] 1.0690214 0.012905528 1.04870945 1.08959055 12983.355 0.9999878
## mu[162] 1.0108711 0.017894692 0.98245439 1.03948055 10924.782 1.0001708
## mu[163] 1.0665538 0.012325010 1.04715000 1.08613000 13519.181 0.9999814
## mu[164] 1.0341084 0.011568571 1.01552000 1.05246110 14481.498 1.0000976
## mu[165] 1.0415343 0.010456890 1.02461000 1.05823000 16526.238 1.0000467
## mu[166] 1.0427224 0.010344381 1.02599000 1.05928000 16745.165 1.0000380
## mu[167] 1.0279621 0.012920909 1.00726000 1.04863000 13038.070 1.0001295
## mu[168] 0.8956247 0.017760561 0.86729100 0.92394466 14141.182 0.9999339
## mu[169] 0.8696024 0.017462803 0.84175696 0.89770017 15477.508 1.0001089
## mu[170] 0.8836095 0.015978298 0.85859361 0.90897339 19152.829 0.9999737Stan_mdl_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=dt_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: 1.1 seconds.
## Total execution time: 1.3 seconds.Stan_mdl_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=dt_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: 1.1 seconds.
## Total execution time: 1.2 seconds.Stan_mdl_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=dt_slim , warmup = 400, iter = 4000, chains=4, cores = 4 )## Running MCMC with 4 parallel chains...
##
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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=dt_slim , warmup = 500, iter = 4000, chains=4, cores = 4 )## Running MCMC with 4 parallel chains...
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## Total execution time: 1.1 seconds.#After warmup is increased,n_eff getting closer to number of iterations. 400 warmup iterations are enough9-4. Run the model below and then inspect the posterior distribution and explain what it is accomplishing.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?
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)
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 adt 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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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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## All 4 chains finished successfully.
## Mean chain execution time: 0.2 seconds.
## Total execution time: 0.2 seconds.set.seed(77)
compare( m5.1 , m5.2 , m5.3 , func=WAIC )## WAIC SE dWAIC dSE pWAIC weight
## m5.1 125.6456 12.569835 0.00000 NA 3.574349 0.6883684931
## m5.3 127.2358 12.586636 1.59025 0.733407 4.562220 0.3108153883
## m5.2 139.1206 9.577032 13.47504 9.159353 2.806663 0.0008161186# Model m5.1 performs best when only median age at marriage is used as a predictor, however it is difficult to distinguish from model m5.3. However, the model with only the marriage rate, m5.2, obviously outperforms both.