第 68 章 贝叶斯层级模型
library(tidyverse)
library(tidybayes)
library(rstan)
rstan_options(auto_write = TRUE)
options(mc.cores = parallel::detectCores())
68.1 明尼苏达州房屋中氡的存在
## floor county log_radon log_uranium
## 1 1 AITKIN 0.83290912 -0.6890476
## 2 0 AITKIN 0.83290912 -0.6890476
## 3 0 AITKIN 1.09861229 -0.6890476
## 4 0 AITKIN 0.09531018 -0.6890476
## 5 0 ANOKA 1.16315081 -0.8473129
## 6 0 ANOKA 0.95551145 -0.8473129
数据来源美国明尼苏达州85个县中房屋氡含量测量
-
log_radon
房屋氡含量 (log scale) -
log_uranium
这个县放射性化学元素铀的等级 (log scale) -
floor
房屋楼层 (0 = basement, 1 = first floor) -
county
所在县 (factor)
68.2 任务
估计房屋中的氡含量。
68.2.1 可视化探索
## # A tibble: 85 × 2
## county n
## <fct> <int>
## 1 AITKIN 4
## 2 ANOKA 52
## 3 BECKER 3
## 4 BELTRAMI 7
## 5 BENTON 4
## 6 BIGSTONE 3
## 7 BLUEEARTH 14
## 8 BROWN 4
## 9 CARLTON 10
## 10 CARVER 6
## # ℹ 75 more rows
统计每个县,样本量、氡含量均值、标准差、铀等级的均值、标准误
radon_county <- radon %>%
group_by(county) %>%
summarise(
log_radon_mean = mean(log_radon),
log_radon_sd = sd(log_radon),
log_uranium = mean(log_uranium),
n = length(county)
) %>%
mutate(log_radon_se = log_radon_sd / sqrt(n))
radon_county
## # A tibble: 85 × 6
## county log_radon_mean log_radon_sd log_uranium n log_radon_se
## <fct> <dbl> <dbl> <dbl> <int> <dbl>
## 1 AITKIN 0.715 0.432 -0.689 4 0.216
## 2 ANOKA 0.891 0.718 -0.847 52 0.0995
## 3 BECKER 1.09 0.717 -0.113 3 0.414
## 4 BELTRAMI 1.19 0.894 -0.593 7 0.338
## 5 BENTON 1.28 0.415 -0.143 4 0.207
## 6 BIGSTONE 1.54 0.504 0.387 3 0.291
## 7 BLUEEARTH 1.93 0.542 0.272 14 0.145
## 8 BROWN 1.65 0.595 0.278 4 0.298
## 9 CARLTON 0.977 0.585 -0.332 10 0.185
## 10 CARVER 1.22 1.90 0.0959 6 0.777
## # ℹ 75 more rows
ggplot() +
geom_boxplot(data = radon,
mapping = aes(y = log_radon,
x = fct_reorder(county, log_radon, mean)),
colour = "gray") +
geom_point(data = radon,
mapping = aes(y = log_radon,
x = fct_reorder(county, log_radon, mean)),
colour = "gray") +
geom_point(data = radon_county,
mapping = aes(x = fct_reorder(county, log_radon_mean),
y = log_radon_mean),
colour = "red") +
coord_flip() +
labs(y = "log(radon)", x = "")
68.2.2 pooling model
这是最简单的模型,该模型假定所有的房屋的氡含量来自同一个分布, 估计整体的均值和方差
\[ \begin{aligned}[t] y_i &\sim \operatorname{normal}(\mu, \sigma) \\ \mu &\sim \operatorname{normal}(0, 10) \\ \sigma &\sim \operatorname{exp}(1) \end{aligned} \] 这里我们指定 \(\mu\) 和 \(\sigma\) 较弱的先验信息.
stan_program <- "
data {
int N;
vector[N] y;
}
parameters {
real mu;
real<lower=0> sigma;
}
model {
mu ~ normal(0, 10);
sigma ~ exponential(1);
y ~ normal(mu, sigma);
}
"
stan_data <- list(
N = nrow(radon),
y = radon$log_radon
)
fit_pooling <- stan(model_code = stan_program, data = stan_data)
模型估计了均值和方差两个参数。
summary(fit_pooling)$summary
## mean se_mean sd 2.5% 25%
## mu 1.2642340 0.0005205044 0.02742536 1.2104850 1.2450453
## sigma 0.8195736 0.0003149489 0.01917117 0.7823672 0.8063292
## lp__ -277.9433172 0.0239258806 0.98310328 -280.4665786 -278.3198548
## 50% 75% 97.5% n_eff Rhat
## mu 1.2646388 1.2832237 1.3173825 2776.233 1.0005855
## sigma 0.8193615 0.8321428 0.8580254 3705.248 0.9991691
## lp__ -277.6559459 -277.2370988 -276.9561998 1688.350 1.0010905
68.2.3 no-pooling model
每个县都有独立的均值和方差,又叫 individual model
\[ \begin{aligned}[t] y_i &\sim \operatorname{normal}(\mu_{j[i]}, \sigma) \\ \mu_j &\sim \operatorname{normal}(0, 10) \\ \sigma &\sim \operatorname{exp}(1) \end{aligned} \] 其中, \(j[i]\) 表示观测\(i\)对应的所在县。
stan_program <- "
data {
int<lower=1> N;
int<lower=2> J;
int<lower=1, upper=J> county[N];
vector[N] y;
}
parameters {
vector[J] mu;
real<lower=0> sigma;
}
model {
mu ~ normal(0, 10);
sigma ~ exponential(1);
for(i in 1:N) {
y[i] ~ normal(mu[county[i]], sigma);
}
}
"
stan_data <- list(
N = nrow(radon),
J = length(unique(radon$county)),
county = as.numeric(radon$county),
y = radon$log_radon
)
fit_no_pooling <- stan(model_code = stan_program, data = stan_data)
summary(fit_no_pooling)$summary
## mean se_mean sd 2.5% 25%
## mu[1] 0.7111322 0.0051471628 0.38775685 -0.04625801 0.4439757
## mu[2] 0.8886618 0.0014140589 0.10745053 0.68102047 0.8177827
## mu[3] 1.0851203 0.0057132158 0.44167254 0.21388813 0.7879462
## mu[4] 1.1989189 0.0042106519 0.29541091 0.62725943 0.9967546
## mu[5] 1.2801826 0.0046562256 0.38209277 0.54576821 1.0211879
## mu[6] 1.5398567 0.0054162314 0.43614443 0.69368134 1.2407143
## mu[7] 1.9301271 0.0026665838 0.20376830 1.53004430 1.7914584
## mu[8] 1.6517076 0.0054926203 0.37868107 0.92465085 1.3892574
## mu[9] 0.9736545 0.0030516408 0.24536681 0.50511530 0.8028961
## mu[10] 1.2239658 0.0043937616 0.31112190 0.63343938 1.0043179
## mu[11] 1.4320218 0.0046123848 0.35211394 0.74148794 1.1931877
## mu[12] 1.7477481 0.0047445242 0.37944918 0.98687967 1.4918186
## mu[13] 1.0821507 0.0040408488 0.30904492 0.47233140 0.8767577
## mu[14] 1.8058367 0.0026779801 0.20552421 1.41048441 1.6692026
## mu[15] 1.0262008 0.0054737795 0.37873815 0.27074663 0.7729473
## mu[16] 0.7139751 0.0069030541 0.53120257 -0.34942870 0.3612031
## mu[17] 0.7507313 0.0049128356 0.37722734 0.01536168 0.4913063
## mu[18] 0.9893581 0.0028808150 0.21987538 0.56845473 0.8404206
## mu[19] 1.3281471 0.0012683613 0.09832603 1.13961091 1.2637369
## mu[20] 1.8174455 0.0056020154 0.44054073 0.97569709 1.5178909
## mu[21] 1.6687423 0.0031702736 0.25326974 1.18389205 1.4937584
## mu[22] 0.6625477 0.0041832605 0.31544100 0.03252701 0.4518154
## mu[23] 1.0652039 0.0067220462 0.52677437 0.06136117 0.7024825
## mu[24] 1.9579887 0.0036258767 0.24841943 1.46634307 1.7954081
## mu[25] 1.8707021 0.0027126644 0.20963597 1.44578541 1.7295559
## mu[26] 1.3210896 0.0009618109 0.07411845 1.17683734 1.2723418
## mu[27] 1.5565117 0.0043983579 0.31399421 0.95723852 1.3414527
## mu[28] 0.8670729 0.0045567436 0.34164013 0.21001426 0.6373072
## mu[29] 1.0837716 0.0056298876 0.44265788 0.19966845 0.7879808
## mu[30] 0.9688479 0.0031841270 0.24094585 0.49370297 0.8051168
## mu[31] 2.0326059 0.0046560239 0.33852495 1.37991399 1.8010037
## mu[32] 1.2688706 0.0052809909 0.38303595 0.51662037 1.0041978
## mu[33] 2.0826321 0.0047194418 0.38194714 1.31687221 1.8244039
## mu[34] 1.1461042 0.0056247903 0.44077404 0.27959955 0.8468336
## mu[35] 0.4773884 0.0038767772 0.29049029 -0.08475653 0.2832415
## mu[36] 2.6093020 0.0070756347 0.54593391 1.52289758 2.2403150
## mu[37] 0.4103055 0.0032864692 0.26146579 -0.10644428 0.2353852
## mu[38] 1.5256009 0.0047589937 0.38069600 0.78782530 1.2740028
## mu[39] 1.6152248 0.0046746046 0.33835146 0.94622191 1.3840532
## mu[40] 2.1329294 0.0052197885 0.38264122 1.39062311 1.8792079
## mu[41] 1.8839113 0.0034272437 0.27090411 1.34039395 1.7018162
## mu[42] 1.3960887 0.0101643335 0.75418365 -0.07234397 0.8864505
## mu[43] 1.2581641 0.0033990456 0.25685625 0.76000193 1.0834345
## mu[44] 1.0117496 0.0036148456 0.28986289 0.45411756 0.8117379
## mu[45] 1.1106409 0.0028224274 0.22123883 0.68177335 0.9550247
## mu[46] 1.2449417 0.0043084845 0.34927111 0.57407837 1.0112484
## mu[47] 0.6180657 0.0068944006 0.55366424 -0.41855581 0.2302346
## mu[48] 1.1050019 0.0033578946 0.26317674 0.57465197 0.9324872
## mu[49] 1.6185611 0.0029361068 0.21826980 1.19405429 1.4702212
## mu[50] 2.4963361 0.0106982969 0.77082352 0.98054246 1.9618386
## mu[51] 2.1754868 0.0049363954 0.37835051 1.43585283 1.9183861
## mu[52] 1.9423815 0.0057155960 0.44375440 1.10039750 1.6438087
## mu[53] 1.0476503 0.0064917053 0.45042544 0.15936948 0.7408987
## mu[54] 1.2486106 0.0019289634 0.15766553 0.93177113 1.1413596
## mu[55] 1.3809118 0.0035577204 0.27054764 0.85581300 1.2007481
## mu[56] 0.7312310 0.0061036444 0.44815420 -0.13442477 0.4252391
## mu[57] 0.7001667 0.0040610189 0.31883564 0.07592075 0.4940726
## mu[58] 1.7069052 0.0053585822 0.38112609 0.96583104 1.4469458
## mu[59] 1.4005913 0.0051904390 0.39626860 0.60894228 1.1285244
## mu[60] 1.3205251 0.0073277858 0.54382986 0.25413808 0.9539209
## mu[61] 1.1363239 0.0017498655 0.13789472 0.86283225 1.0436752
## mu[62] 1.8525381 0.0044087540 0.34930638 1.15787773 1.6190477
## mu[63] 1.4513110 0.0058785878 0.45172528 0.57754720 1.1339691
## mu[64] 1.7976104 0.0033272273 0.23982557 1.33660803 1.6387128
## mu[65] 1.3354134 0.0067588265 0.53464544 0.24387360 0.9818927
## mu[66] 1.2907430 0.0025647927 0.20313431 0.89816136 1.1550542
## mu[67] 1.6035086 0.0027810100 0.20805121 1.19647307 1.4611291
## mu[68] 1.1297494 0.0035340659 0.26937701 0.61481023 0.9481912
## mu[69] 1.2704005 0.0051242622 0.38023359 0.51632306 1.0087094
## mu[70] 0.8267275 0.0009324458 0.06990462 0.68877514 0.7796967
## mu[71] 1.4065758 0.0019539633 0.15804281 1.10309026 1.3016637
## mu[72] 1.5957654 0.0033533368 0.24274406 1.12158755 1.4321495
## mu[73] 1.8061553 0.0071898441 0.55810818 0.72863295 1.4182185
## mu[74] 1.0262057 0.0048545001 0.38718572 0.28689061 0.7588639
## mu[75] 1.5016619 0.0062401585 0.44721657 0.62173391 1.1945653
## mu[76] 1.8420065 0.0049120096 0.37550944 1.11088609 1.5828323
## mu[77] 1.7380852 0.0040344069 0.29392294 1.14730674 1.5444603
## mu[78] 1.0369636 0.0043779603 0.33632314 0.38924035 0.8114321
## mu[79] 0.5253026 0.0050279678 0.39549618 -0.23971303 0.2532753
## mu[80] 1.2893947 0.0014496049 0.11212626 1.07102268 1.2155844
## mu[81] 2.2316430 0.0055623710 0.43935034 1.38360796 1.9319273
## mu[82] 2.2336444 0.0101277337 0.79276849 0.70942379 1.6823692
## mu[83] 1.4911503 0.0026868401 0.21063869 1.06638184 1.3558501
## mu[84] 1.6161087 0.0029609333 0.21549187 1.20161166 1.4723507
## mu[85] 1.2032884 0.0067918128 0.53911846 0.16211252 0.8384707
## sigma 0.7673864 0.0002965435 0.01886820 0.73099625 0.7546396
## lp__ -217.8570858 0.1714924991 6.91382182 -232.54899610 -222.3813022
## 50% 75% 97.5% n_eff Rhat
## mu[1] 0.7155564 0.9775561 1.4531034 5675.226 0.9999096
## mu[2] 0.8898304 0.9592479 1.0995124 5774.071 1.0000975
## mu[3] 1.0889972 1.3948524 1.9294730 5976.398 0.9992129
## mu[4] 1.2023084 1.3989847 1.7714679 4922.145 1.0003925
## mu[5] 1.2782922 1.5420297 2.0132644 6733.946 0.9994417
## mu[6] 1.5370370 1.8309648 2.4260684 6484.347 0.9994143
## mu[7] 1.9306151 2.0674181 2.3236100 5839.327 0.9996558
## mu[8] 1.6526241 1.8998967 2.3842923 4753.221 1.0000686
## mu[9] 0.9742991 1.1424265 1.4512131 6464.945 0.9999780
## mu[10] 1.2212041 1.4487543 1.8184475 5014.045 0.9998495
## mu[11] 1.4315552 1.6673601 2.1267590 5827.944 0.9998963
## mu[12] 1.7454853 2.0089611 2.4741879 6396.198 0.9995963
## mu[13] 1.0805497 1.2915529 1.6721105 5849.221 0.9996375
## mu[14] 1.8030615 1.9438805 2.2080809 5889.945 0.9994039
## mu[15] 1.0223787 1.2784182 1.7606803 4787.442 1.0001804
## mu[16] 0.7170419 1.0667224 1.7655231 5921.582 0.9993209
## mu[17] 0.7488441 1.0063966 1.4914413 5895.788 0.9998403
## mu[18] 0.9853665 1.1339890 1.4339258 5825.356 1.0007978
## mu[19] 1.3287771 1.3936563 1.5208747 6009.677 0.9998554
## mu[20] 1.8195409 2.1202614 2.6642656 6184.200 0.9994298
## mu[21] 1.6690996 1.8420279 2.1658601 6382.240 1.0000075
## mu[22] 0.6598504 0.8719498 1.2863307 5685.995 0.9993541
## mu[23] 1.0694350 1.4260722 2.0865418 6141.102 0.9993030
## mu[24] 1.9610652 2.1271211 2.4370884 4694.021 0.9996936
## mu[25] 1.8718374 2.0097454 2.2857911 5972.271 0.9993332
## mu[26] 1.3213884 1.3712629 1.4672615 5938.453 0.9994468
## mu[27] 1.5606270 1.7675847 2.1721663 5096.384 0.9992942
## mu[28] 0.8652949 1.0936519 1.5436288 5621.194 0.9995402
## mu[29] 1.0895220 1.3784285 1.9514462 6182.113 0.9996947
## mu[30] 0.9686812 1.1333977 1.4474919 5726.089 0.9994782
## mu[31] 2.0292778 2.2618732 2.6940782 5286.290 0.9996295
## mu[32] 1.2673343 1.5289493 2.0124666 5260.757 0.9996388
## mu[33] 2.0798590 2.3448912 2.8457371 6549.758 0.9995523
## mu[34] 1.1483419 1.4424870 1.9996309 6140.721 0.9994833
## mu[35] 0.4742873 0.6720013 1.0358752 5614.635 0.9999449
## mu[36] 2.6094378 2.9775232 3.7118337 5953.184 0.9994911
## mu[37] 0.4081408 0.5844288 0.9206740 6329.518 0.9999563
## mu[38] 1.5260735 1.7756984 2.2859770 6399.210 0.9994045
## mu[39] 1.6185435 1.8398961 2.2772923 5238.975 0.9999099
## mu[40] 2.1344733 2.3895204 2.8940944 5373.753 0.9996265
## mu[41] 1.8815681 2.0671652 2.4162116 6248.003 0.9996148
## mu[42] 1.3865004 1.9058120 2.8970757 5505.496 0.9995579
## mu[43] 1.2627883 1.4348890 1.7435877 5710.397 0.9997667
## mu[44] 1.0089675 1.2086387 1.5701991 6429.922 0.9991866
## mu[45] 1.1105842 1.2626544 1.5504506 6144.367 0.9996543
## mu[46] 1.2465994 1.4744446 1.9206925 6571.678 0.9997690
## mu[47] 0.6037456 0.9944272 1.6984953 6449.112 0.9993302
## mu[48] 1.1019891 1.2808217 1.6224471 6142.723 0.9995302
## mu[49] 1.6166843 1.7662238 2.0428573 5526.416 0.9995966
## mu[50] 2.4887651 3.0275730 4.0023267 5191.354 1.0001714
## mu[51] 2.1743210 2.4319182 2.9099611 5874.471 0.9994947
## mu[52] 1.9433468 2.2412317 2.8243361 6027.848 0.9992974
## mu[53] 1.0477685 1.3666372 1.9134915 4814.245 1.0000262
## mu[54] 1.2503657 1.3568692 1.5616711 6680.755 0.9996161
## mu[55] 1.3777845 1.5673235 1.8963204 5782.876 0.9995429
## mu[56] 0.7312639 1.0316395 1.5982235 5391.089 0.9996533
## mu[57] 0.6998760 0.9134282 1.3160133 6164.015 1.0006425
## mu[58] 1.7023028 1.9615772 2.4689360 5058.684 0.9993778
## mu[59] 1.4051968 1.6681424 2.1586638 5828.692 0.9995257
## mu[60] 1.3246783 1.6734359 2.3961102 5507.831 0.9994979
## mu[61] 1.1375652 1.2301085 1.4066298 6209.919 0.9994934
## mu[62] 1.8513848 2.0853264 2.5436950 6277.422 0.9993999
## mu[63] 1.4592603 1.7616656 2.3147412 5904.767 0.9995695
## mu[64] 1.7937917 1.9559008 2.2783407 5195.484 0.9992241
## mu[65] 1.3330864 1.6926897 2.3877719 6257.331 0.9996593
## mu[66] 1.2886188 1.4231365 1.7009422 6272.809 1.0000191
## mu[67] 1.6014488 1.7435780 2.0091619 5596.743 0.9999109
## mu[68] 1.1246574 1.3135023 1.6656720 5809.942 0.9992656
## mu[69] 1.2733570 1.5363095 1.9979378 5506.026 0.9993673
## mu[70] 0.8262328 0.8737956 0.9645315 5620.365 0.9998041
## mu[71] 1.4068492 1.5110129 1.7167406 6542.092 0.9992257
## mu[72] 1.5969947 1.7590166 2.0795952 5240.140 0.9993436
## mu[73] 1.8100368 2.1851456 2.8797325 6025.566 0.9996684
## mu[74] 1.0221999 1.2954157 1.7816269 6361.355 0.9995297
## mu[75] 1.5090498 1.8081668 2.3846559 5136.231 0.9998098
## mu[76] 1.8410447 2.0933981 2.5654723 5844.176 0.9996260
## mu[77] 1.7399940 1.9365581 2.2964017 5307.715 0.9998699
## mu[78] 1.0355756 1.2621507 1.6977685 5901.601 0.9992444
## mu[79] 0.5279796 0.8050489 1.2915812 6187.278 0.9993945
## mu[80] 1.2877908 1.3644046 1.5164391 5982.950 0.9996766
## mu[81] 2.2327981 2.5361098 3.0765737 6238.814 0.9992648
## mu[82] 2.2240799 2.7956624 3.7761321 6127.287 0.9999655
## mu[83] 1.4935295 1.6298332 1.9056314 6146.001 0.9995169
## mu[84] 1.6176148 1.7627565 2.0404349 5296.689 1.0013096
## mu[85] 1.1931093 1.5755865 2.2460814 6300.819 0.9993381
## sigma 0.7672921 0.7800309 0.8047443 4048.408 1.0002868
## lp__ -217.4686179 -213.0385412 -205.4721038 1625.347 1.0017040
有多少县,就有多少个模型,每个模型有一个 \(\mu\),参数\(\sigma\)是共同的。需要注意的是,每组之间彼此独立的,没有共享信息。
68.2.4 partially pooled model
和 “no-pooling model” 模型一样,每个县都有自己的均值,但是,这些县彼此会分享信息,一个县获取的信息可以帮助我们估计其它县的均值。
- 模型同时考虑各个类别数据中的信息以及整个群体中的信息
- 怎么叫共享信息?参数来自同一个分布
- 怎么做到的呢?通过先验
\[ \begin{aligned}[t] y_i &\sim \operatorname{normal}(\mu_{j[i]}, \sigma) \\ \mu_j &\sim \operatorname{normal}(\gamma, \tau) \\ \gamma &\sim \operatorname{normal}(0, 5) \\ \tau &\sim \operatorname{exp}(1) \end{aligned} \]
每个县的氡含量均值\(\mu_j\)都服从均值为 \(\gamma\)、标准差为 \(\tau\) 的正态分布。但先验分布中的参数 \(\gamma\) 和 \(\tau\) 都各自有自己的先验分布,即参数的参数, 通常称之为超参数,这就是多层模型中”层”的来历,\(\mu_j\) 是第一层参数,\(\gamma\) 是第二层参数。
- \(\gamma\) 和 \(\tau\) 的先验称为 超先验分布。
- 超参数是多层模型的标志。
stan_program <- "
data {
int<lower=1> N;
int<lower=2> J;
int<lower=1, upper=J> county[N];
vector[N] y;
}
parameters {
vector[J] mu;
real mu_a;
real<lower=0> sigma_y;
real<lower=0> sigma_a;
}
model {
mu_a ~ normal(0, 5);
sigma_a ~ exponential(1);
sigma_y ~ exponential(1);
mu ~ normal(mu_a, sigma_a);
for(i in 1:N) {
y[i] ~ normal(mu[county[i]], sigma_y);
}
}
"
stan_data <- list(
N = nrow(radon),
J = length(unique(radon$county)),
county = as.numeric(radon$county),
y = radon$log_radon
)
fit_partial_pooling <- stan(model_code = stan_program, data = stan_data)
summary(fit_partial_pooling)$summary
## mean se_mean sd 2.5% 25%
## mu[1] 1.1078355 0.0027454493 0.23834601 0.6333122 0.9508426
## mu[2] 0.9450242 0.0012467073 0.10212212 0.7448281 0.8773785
## mu[3] 1.2652001 0.0027689918 0.25439763 0.7598871 1.1002471
## mu[4] 1.2684206 0.0021953750 0.20880167 0.8712869 1.1211045
## mu[5] 1.3235012 0.0025281996 0.24150064 0.8399759 1.1615953
## mu[6] 1.4074491 0.0026596465 0.24914916 0.9185417 1.2415848
## mu[7] 1.7417078 0.0021005395 0.17267697 1.4182688 1.6207701
## mu[8] 1.4700642 0.0024939837 0.24197816 1.0081738 1.3070037
## mu[9] 1.1268923 0.0021501773 0.19106740 0.7543447 0.9974486
## mu[10] 1.2881253 0.0022317497 0.21277997 0.8594222 1.1481606
## mu[11] 1.3860896 0.0022952865 0.22273198 0.9643650 1.2358524
## mu[12] 1.5028724 0.0026767133 0.24320235 1.0375237 1.3372851
## mu[13] 1.2232756 0.0023687785 0.21251024 0.8181867 1.0807665
## mu[14] 1.6613298 0.0019433911 0.17346129 1.3191756 1.5457948
## mu[15] 1.2241526 0.0027789696 0.24108917 0.7381662 1.0724035
## mu[16] 1.1993859 0.0028921285 0.26138496 0.6717348 1.0340207
## mu[17] 1.1203480 0.0028025503 0.24585634 0.6248790 0.9559731
## mu[18] 1.1167916 0.0019972236 0.17954754 0.7561280 0.9992883
## mu[19] 1.3310413 0.0010344161 0.09268103 1.1427419 1.2693385
## mu[20] 1.5046668 0.0029289711 0.24938811 1.0276605 1.3386367
## mu[21] 1.5370235 0.0021610435 0.20241806 1.1508433 1.3992822
## mu[22] 1.0179385 0.0028713891 0.22659123 0.5595025 0.8638776
## mu[23] 1.2832841 0.0029636194 0.28276491 0.7258049 1.1002855
## mu[24] 1.7022146 0.0022809246 0.20165238 1.3231889 1.5643004
## mu[25] 1.6999415 0.0021051267 0.17189976 1.3716837 1.5855441
## mu[26] 1.3233230 0.0008389469 0.07301332 1.1787336 1.2746605
## mu[27] 1.4436584 0.0025523248 0.21631564 1.0270983 1.2961917
## mu[28] 1.1290467 0.0027981257 0.22821873 0.6845372 0.9707580
## mu[29] 1.2649408 0.0026131028 0.25169714 0.7652296 1.0945358
## mu[30] 1.1147893 0.0020343747 0.18420346 0.7479824 0.9923177
## mu[31] 1.6470717 0.0030044569 0.23390177 1.1891265 1.4901458
## mu[32] 1.3149453 0.0026688135 0.24380948 0.8260011 1.1567885
## mu[33] 1.6284786 0.0029143338 0.24652058 1.1528643 1.4616859
## mu[34] 1.2843639 0.0027345359 0.25385377 0.7770673 1.1171266
## mu[35] 0.8981030 0.0030812740 0.21981805 0.4583350 0.7542704
## mu[36] 1.6528116 0.0036981329 0.26835832 1.1472550 1.4708911
## mu[37] 0.8026789 0.0029104334 0.20333472 0.3810473 0.6726981
## mu[38] 1.4159214 0.0025286409 0.23762590 0.9548412 1.2542099
## mu[39] 1.4670920 0.0025477790 0.23028118 1.0173879 1.3117021
## mu[40] 1.6578907 0.0030432670 0.24381499 1.1940901 1.4897338
## mu[41] 1.6476624 0.0026473896 0.20898938 1.2488275 1.5043861
## mu[42] 1.3582240 0.0033584898 0.28556575 0.7860746 1.1731364
## mu[43] 1.2956667 0.0021837916 0.19644247 0.9013475 1.1689808
## mu[44] 1.1731198 0.0023990249 0.20899782 0.7578611 1.0358474
## mu[45] 1.1910305 0.0019748210 0.17358656 0.8476617 1.0738521
## mu[46] 1.3016272 0.0025960697 0.22863654 0.8493190 1.1494103
## mu[47] 1.1711470 0.0030616725 0.27453042 0.6035732 0.9896220
## mu[48] 1.2058997 0.0020400972 0.19682140 0.8202083 1.0707590
## mu[49] 1.5302808 0.0017653463 0.17353190 1.1982175 1.4148151
## mu[50] 1.5087962 0.0031449562 0.29428669 0.9321895 1.3162862
## mu[51] 1.6702379 0.0030468484 0.24361835 1.2085620 1.5021277
## mu[52] 1.5335939 0.0029539824 0.25273123 1.0485275 1.3606160
## mu[53] 1.2546903 0.0029376055 0.24750353 0.7636885 1.0946190
## mu[54] 1.2741097 0.0014988336 0.13969700 1.0030774 1.1794097
## mu[55] 1.3688785 0.0023237886 0.20564361 0.9580721 1.2344322
## mu[56] 1.1536089 0.0029655102 0.26151546 0.6238667 0.9766329
## mu[57] 1.0389823 0.0024205383 0.21869385 0.5921468 0.8888190
## mu[58] 1.4855410 0.0028367425 0.24563658 1.0076856 1.3208399
## mu[59] 1.3677210 0.0026679875 0.23754562 0.9034060 1.2117265
## mu[60] 1.3396296 0.0027861916 0.28478249 0.7751571 1.1509544
## mu[61] 1.1685084 0.0012836093 0.12099807 0.9290895 1.0864858
## mu[62] 1.5744808 0.0026958433 0.22876036 1.1388601 1.4264548
## mu[63] 1.3871896 0.0027158083 0.25815502 0.8952266 1.2125363
## mu[64] 1.6307898 0.0022918320 0.18611776 1.2749660 1.5008903
## mu[65] 1.3435299 0.0027961447 0.26775321 0.8135972 1.1679293
## mu[66] 1.3077750 0.0020990968 0.17310151 0.9735963 1.1875085
## mu[67] 1.5234797 0.0019611773 0.17766022 1.1703005 1.4043789
## mu[68] 1.2223014 0.0020751412 0.20465642 0.8180204 1.0918267
## mu[69] 1.3181098 0.0026330743 0.24057484 0.8462605 1.1546964
## mu[70] 0.8568514 0.0008051756 0.06961936 0.7188560 0.8118551
## mu[71] 1.3984328 0.0014652834 0.13960688 1.1309018 1.3031781
## mu[72] 1.5031999 0.0020917972 0.18895453 1.1257356 1.3758769
## mu[73] 1.4616715 0.0031974708 0.27411601 0.9258221 1.2774809
## mu[74] 1.2254111 0.0025624477 0.23841042 0.7578354 1.0651079
## mu[75] 1.3974931 0.0026269731 0.24700308 0.9116682 1.2399500
## mu[76] 1.5383816 0.0029332661 0.24131978 1.0668479 1.3745173
## mu[77] 1.5500975 0.0024993243 0.21217175 1.1314780 1.4039038
## mu[78] 1.2200589 0.0024576596 0.22796389 0.7654840 1.0661569
## mu[79] 1.0393686 0.0029341686 0.24315110 0.5416823 0.8776455
## mu[80] 1.2971427 0.0012350127 0.10447206 1.0857010 1.2273055
## mu[81] 1.6287420 0.0034583395 0.25604453 1.1421543 1.4544480
## mu[82] 1.4778907 0.0034190386 0.28702685 0.9260767 1.2888288
## mu[83] 1.4476175 0.0019163190 0.17871488 1.1006388 1.3284018
## mu[84] 1.5248573 0.0018659167 0.17722966 1.1809732 1.4035755
## mu[85] 1.3178735 0.0028987471 0.26051106 0.7974437 1.1413241
## mu_a 1.3499440 0.0006989300 0.04579328 1.2622459 1.3190204
## sigma_y 0.7670301 0.0002319138 0.01917573 0.7318580 0.7534510
## sigma_a 0.3035564 0.0013170502 0.04584803 0.2209930 0.2718363
## lp__ -157.8015346 0.3584503656 9.52369007 -176.9315963 -163.9781383
## 50% 75% 97.5% n_eff Rhat
## mu[1] 1.1169384 1.2657447 1.5647613 7536.8335 0.9993684
## mu[2] 0.9452647 1.0130696 1.1483541 6709.8164 0.9993822
## mu[3] 1.2645587 1.4287342 1.7667652 8440.7839 0.9992430
## mu[4] 1.2693453 1.4115832 1.6660971 9045.8729 0.9995937
## mu[5] 1.3203998 1.4881061 1.7960894 9124.6008 0.9993487
## mu[6] 1.4075062 1.5720227 1.8923654 8775.4826 0.9993286
## mu[7] 1.7435297 1.8555346 2.0873844 6757.8285 0.9994379
## mu[8] 1.4677138 1.6295159 1.9430145 9413.8037 1.0001544
## mu[9] 1.1268087 1.2558056 1.5003476 7896.3179 0.9991831
## mu[10] 1.2890894 1.4290256 1.7030453 9090.1391 0.9992794
## mu[11] 1.3860542 1.5345644 1.8208000 9416.5401 0.9990889
## mu[12] 1.4988774 1.6648891 1.9694471 8255.2808 0.9995079
## mu[13] 1.2211518 1.3633913 1.6415599 8048.4239 0.9995119
## mu[14] 1.6608617 1.7798213 2.0036146 7966.8144 1.0004857
## mu[15] 1.2262217 1.3810271 1.6885138 7526.4089 1.0000322
## mu[16] 1.2035107 1.3721524 1.7026946 8168.1933 0.9995079
## mu[17] 1.1226915 1.2906682 1.5885962 7695.8395 0.9997217
## mu[18] 1.1176956 1.2370039 1.4754812 8081.7527 0.9992661
## mu[19] 1.3324428 1.3954516 1.5069882 8027.7010 0.9994481
## mu[20] 1.5015584 1.6627226 2.0131753 7249.7214 0.9994163
## mu[21] 1.5364731 1.6675143 1.9463748 8773.4718 0.9994722
## mu[22] 1.0218569 1.1732538 1.4516953 6227.3331 0.9997092
## mu[23] 1.2835248 1.4716841 1.8434120 9103.4535 0.9992049
## mu[24] 1.6985973 1.8324651 2.1026265 7816.0053 0.9993975
## mu[25] 1.6971506 1.8133396 2.0462826 6667.9771 0.9991344
## mu[26] 1.3235673 1.3716243 1.4695491 7574.1725 0.9992996
## mu[27] 1.4449960 1.5914248 1.8759235 7182.9682 0.9993675
## mu[28] 1.1287409 1.2848984 1.5679803 6652.2435 0.9990872
## mu[29] 1.2656105 1.4334676 1.7583454 9277.7690 0.9996743
## mu[30] 1.1113598 1.2370597 1.4797347 8198.4863 0.9994936
## mu[31] 1.6423134 1.8077425 2.1074174 6060.8716 0.9993707
## mu[32] 1.3141220 1.4721831 1.8019702 8345.7373 0.9997944
## mu[33] 1.6213228 1.7891378 2.1261397 7155.2990 1.0001663
## mu[34] 1.2843425 1.4517979 1.7763304 8617.8708 0.9993220
## mu[35] 0.8997456 1.0470795 1.3225364 5089.3940 1.0010202
## mu[36] 1.6466252 1.8345607 2.1980887 5265.8091 1.0004946
## mu[37] 0.8092612 0.9367922 1.1934323 4880.9878 1.0002064
## mu[38] 1.4199477 1.5771672 1.8770837 8831.0680 0.9995758
## mu[39] 1.4628513 1.6200893 1.9402769 8169.4607 0.9994252
## mu[40] 1.6523191 1.8176921 2.1596890 6418.6055 0.9996533
## mu[41] 1.6420389 1.7854406 2.0709821 6231.7887 0.9993254
## mu[42] 1.3565849 1.5474653 1.9209223 7229.7651 0.9996136
## mu[43] 1.2949444 1.4269853 1.6748497 8091.8602 0.9993775
## mu[44] 1.1753598 1.3097298 1.5950901 7589.5151 0.9993226
## mu[45] 1.1928447 1.3099681 1.5225500 7726.3913 0.9998309
## mu[46] 1.2980083 1.4548427 1.7436223 7756.3712 0.9992997
## mu[47] 1.1802272 1.3529394 1.7077173 8040.1380 0.9994605
## mu[48] 1.2103853 1.3382149 1.5905475 9307.7116 0.9997337
## mu[49] 1.5280431 1.6460546 1.8728475 9662.7077 0.9993806
## mu[50] 1.5072001 1.6995948 2.1038509 8756.1267 0.9993610
## mu[51] 1.6671780 1.8296374 2.1668095 6393.2002 0.9997520
## mu[52] 1.5329855 1.7035805 2.0371242 7319.8468 0.9993024
## mu[53] 1.2550589 1.4174483 1.7428291 7098.6512 0.9996058
## mu[54] 1.2743815 1.3699784 1.5445140 8686.9495 0.9995465
## mu[55] 1.3669686 1.5083417 1.7799605 7831.3603 0.9992907
## mu[56] 1.1498922 1.3344582 1.6556087 7776.7101 0.9998806
## mu[57] 1.0439881 1.1920100 1.4581743 8162.9892 0.9995094
## mu[58] 1.4805049 1.6491533 1.9702421 7498.0140 0.9991861
## mu[59] 1.3694552 1.5169917 1.8427379 7927.3213 0.9995389
## mu[60] 1.3392361 1.5296180 1.9134666 10447.3132 0.9993162
## mu[61] 1.1681516 1.2493052 1.4115024 8885.6901 0.9996095
## mu[62] 1.5728904 1.7226016 2.0395881 7200.6593 0.9994111
## mu[63] 1.3862015 1.5658122 1.8948110 9035.7231 0.9992142
## mu[64] 1.6305440 1.7539010 2.0028515 6594.9281 0.9997190
## mu[65] 1.3462720 1.5139854 1.8828968 9169.5936 0.9992291
## mu[66] 1.3107814 1.4255655 1.6416546 6800.4364 0.9996596
## mu[67] 1.5225261 1.6428089 1.8759448 8206.2865 0.9993290
## mu[68] 1.2271781 1.3581812 1.6104177 9726.4745 0.9994652
## mu[69] 1.3201200 1.4784969 1.7934454 8347.8413 0.9994231
## mu[70] 0.8570901 0.9029779 0.9949332 7476.1662 0.9994037
## mu[71] 1.3983536 1.4920465 1.6721854 9077.5867 0.9992892
## mu[72] 1.5039051 1.6322885 1.8834759 8159.7239 0.9993921
## mu[73] 1.4623304 1.6448563 2.0120321 7349.4631 0.9995899
## mu[74] 1.2280951 1.3871312 1.6915879 8656.4627 0.9993487
## mu[75] 1.3976216 1.5551214 1.8941375 8840.8385 0.9991892
## mu[76] 1.5330760 1.6919286 2.0421158 6768.3518 1.0001668
## mu[77] 1.5479112 1.6972867 1.9623699 7206.5911 0.9993215
## mu[78] 1.2263972 1.3741112 1.6668108 8603.7671 0.9995464
## mu[79] 1.0440576 1.2010567 1.5009966 6867.2418 0.9996874
## mu[80] 1.2987245 1.3672508 1.4999565 7155.7890 0.9995775
## mu[81] 1.6235498 1.8002821 2.1420728 5481.4536 1.0001790
## mu[82] 1.4677632 1.6675195 2.0649620 7047.5319 0.9997944
## mu[83] 1.4447813 1.5663151 1.7939616 8697.3269 0.9991505
## mu[84] 1.5221791 1.6448752 1.8774427 9021.6987 0.9994987
## mu[85] 1.3197250 1.4950314 1.8196886 8076.6573 0.9999474
## mu_a 1.3508595 1.3803184 1.4409502 4292.7544 0.9994196
## sigma_y 0.7664086 0.7795749 0.8056477 6836.7692 0.9991387
## sigma_a 0.3014830 0.3334791 0.4038968 1211.8161 1.0044759
## lp__ -157.7176301 -151.4218874 -139.3209969 705.9151 1.0113196
68.2.5 对比三个模型
对比三个模型的结果
overall_mean <- broom.mixed::tidyMCMC(fit_pooling) %>%
filter(term == "mu") %>%
pull(estimate)
df_no_pooling <- fit_no_pooling %>%
tidybayes::gather_draws(mu[i]) %>%
tidybayes::mean_hdi() %>%
ungroup() %>%
mutate(
type = "no_pooling"
) %>%
select(type, .value) %>%
bind_cols(df_n_county)
df_partial_pooling <- fit_partial_pooling %>%
tidybayes::gather_draws(mu[i]) %>%
tidybayes::mean_hdi() %>%
ungroup() %>%
mutate(
type = "partial_pooling"
) %>%
select(type, .value) %>%
bind_cols(df_n_county)
bind_rows(df_no_pooling, df_partial_pooling) %>%
ggplot(
aes(x = n, y = .value, color = type)
) +
geom_point(size = 3) +
geom_hline(yintercept = overall_mean) +
scale_x_log10()
- 层级模型可以实现不同分组之间的信息交换
- 分组的均值向整体的均值靠拢(收缩)
- 分组的样本量越小,收缩效应越明显
用我们四川火锅记住他们。
68.3 增加预测变量
68.3.1 增加楼层floor作为预测变量
\[ \begin{aligned} y_i &\sim N(\mu_i, \sigma^2) \\ \mu_i &= \alpha_{j[i]} + \beta~\mathtt{floor}_i \\ \alpha_j &\sim \operatorname{normal}(\gamma, \tau) \\ \beta &\sim \operatorname{normal}(0, 2.5)\\ \gamma &\sim \operatorname{normal}(0, 10) \\ \tau &\sim \operatorname{exp}(1) \\ \end{aligned} \] 不同的县有不同的截距,但有共同的\(\beta\),因此被称为变化的截距。
stan_program <- "
data {
int<lower=1> N;
int<lower=2> J;
int<lower=1, upper=J> county[N];
vector[N] x;
vector[N] y;
}
parameters {
vector[J] alpha;
real beta;
real gamma;
real<lower=0> sigma_y;
real<lower=0> sigma_a;
}
model {
vector[N] mu;
for(i in 1:N) {
mu[i] = alpha[county[i]] + beta * x[i];
}
for(i in 1:N) {
y[i] ~ normal(mu[i], sigma_y);
}
alpha ~ normal(gamma, sigma_a);
gamma ~ normal(0, 10);
beta ~ normal(0, 2.5);
sigma_a ~ exponential(1);
sigma_y ~ exponential(1);
}
"
stan_data <- list(
N = nrow(radon),
J = length(unique(radon$county)),
county = as.numeric(radon$county),
x = radon$floor,
y = radon$log_radon
)
fit_intercept_partial <- stan(model_code = stan_program, data = stan_data)
summary(fit_intercept_partial)$summary
## mean se_mean sd 2.5% 25%
## alpha[1] 1.2341771 0.0031082316 0.24279128 0.7535863 1.0754394
## alpha[2] 0.9838383 0.0013466167 0.09865125 0.7908418 0.9167684
## alpha[3] 1.5085429 0.0034374507 0.25619266 0.9961306 1.3383977
## alpha[4] 1.5378967 0.0028340506 0.21438214 1.1136958 1.3911780
## alpha[5] 1.4709565 0.0029103138 0.24210303 1.0065934 1.3005174
## alpha[6] 1.5119591 0.0030517888 0.25427073 1.0139909 1.3468809
## alpha[7] 1.8733998 0.0022014215 0.17130372 1.5316554 1.7586038
## alpha[8] 1.7039983 0.0032788793 0.25284739 1.2071369 1.5381718
## alpha[9] 1.2023261 0.0022997228 0.18394253 0.8388158 1.0751469
## alpha[10] 1.5242719 0.0026003080 0.21449469 1.1001360 1.3799134
## alpha[11] 1.4670822 0.0026075931 0.22330607 1.0280239 1.3163855
## alpha[12] 1.6025584 0.0029308648 0.24476709 1.1317711 1.4417562
## alpha[13] 1.2765172 0.0028579877 0.21790068 0.8410490 1.1337229
## alpha[14] 1.8591817 0.0022201183 0.16763367 1.5285440 1.7444400
## alpha[15] 1.4311016 0.0028113780 0.24098741 0.9523492 1.2740916
## alpha[16] 1.2756993 0.0034279498 0.27830032 0.7167610 1.0972018
## alpha[17] 1.3890714 0.0027877742 0.23158174 0.9237427 1.2365600
## alpha[18] 1.2611156 0.0021819683 0.17694959 0.9168994 1.1391662
## alpha[19] 1.3801588 0.0009696178 0.08505835 1.2127625 1.3228117
## alpha[20] 1.6091828 0.0030922758 0.25092479 1.1232125 1.4475746
## alpha[21] 1.6517031 0.0023524592 0.19566942 1.2617338 1.5197683
## alpha[22] 1.1193281 0.0029632106 0.22683875 0.6735621 0.9670613
## alpha[23] 1.4725805 0.0032971510 0.27828079 0.9182043 1.2838504
## alpha[24] 1.8777719 0.0027049126 0.19735472 1.4896719 1.7439812
## alpha[25] 1.8345686 0.0023673259 0.16730683 1.5101676 1.7180730
## alpha[26] 1.3946586 0.0008255136 0.06910588 1.2590302 1.3491100
## alpha[27] 1.6428501 0.0030118036 0.22510492 1.2067128 1.4942903
## alpha[28] 1.3828605 0.0027588306 0.22365415 0.9419528 1.2324503
## alpha[29] 1.3505131 0.0028552468 0.24871558 0.8740902 1.1860678
## alpha[30] 1.1408361 0.0023311308 0.18439092 0.7681148 1.0171588
## alpha[31] 1.7633962 0.0029185178 0.22889557 1.3298168 1.6080559
## alpha[32] 1.3971255 0.0028141848 0.24158286 0.9325936 1.2310258
## alpha[33] 1.7422072 0.0032999402 0.24452122 1.2725307 1.5781832
## alpha[34] 1.5311530 0.0033744460 0.26016671 1.0207198 1.3611445
## alpha[35] 1.1319218 0.0028079473 0.21084867 0.7244647 0.9877906
## alpha[36] 1.9008329 0.0043153405 0.28263531 1.3610572 1.7064411
## alpha[37] 0.8625804 0.0030590776 0.21023827 0.4369244 0.7219994
## alpha[38] 1.6569077 0.0030318669 0.24108030 1.2002431 1.4857043
## alpha[39] 1.6261119 0.0027138131 0.22575064 1.1867030 1.4689241
## alpha[40] 1.8488500 0.0033966950 0.24285433 1.3778842 1.6865921
## alpha[41] 1.7795452 0.0027253082 0.20128026 1.3882330 1.6403105
## alpha[42] 1.4787880 0.0034758718 0.29072208 0.9175392 1.2807412
## alpha[43] 1.5774642 0.0023847391 0.19102984 1.2076431 1.4477501
## alpha[44] 1.2740258 0.0028472195 0.20414634 0.8542849 1.1401377
## alpha[45] 1.3650549 0.0020124697 0.17020376 1.0283513 1.2486172
## alpha[46] 1.3727369 0.0028812830 0.23335735 0.9153409 1.2181685
## alpha[47] 1.3426364 0.0034298053 0.27578988 0.7999980 1.1528151
## alpha[48] 1.2934942 0.0022244532 0.19280109 0.9209228 1.1599354
## alpha[49] 1.6546323 0.0020332892 0.16897088 1.3281295 1.5416668
## alpha[50] 1.6578680 0.0039922796 0.29688761 1.0913495 1.4600105
## alpha[51] 1.7879346 0.0034298594 0.24878312 1.3071472 1.6167029
## alpha[52] 1.6644739 0.0034329996 0.25073202 1.1779788 1.4973778
## alpha[53] 1.4148599 0.0031017003 0.25713690 0.9051383 1.2438256
## alpha[54] 1.3681743 0.0016740272 0.13748955 1.0954831 1.2757907
## alpha[55] 1.5764009 0.0024571055 0.20113826 1.1754492 1.4440119
## alpha[56] 1.3773281 0.0034421183 0.25994476 0.8590428 1.2127669
## alpha[57] 1.1235537 0.0030015804 0.22121929 0.6770065 0.9758431
## alpha[58] 1.6538300 0.0032544230 0.24621729 1.1705652 1.4938870
## alpha[59] 1.5939537 0.0029828800 0.23446837 1.1423512 1.4358356
## alpha[60] 1.4382794 0.0031221133 0.28077623 0.9034553 1.2531193
## alpha[61] 1.2355359 0.0013403435 0.12171103 0.9926576 1.1537035
## alpha[62] 1.7343024 0.0029892932 0.23237285 1.2800817 1.5785489
## alpha[63] 1.5620336 0.0031823411 0.25704096 1.0516074 1.3913950
## alpha[64] 1.7428082 0.0019217328 0.17700592 1.4008936 1.6219951
## alpha[65] 1.4502243 0.0035983177 0.27056699 0.9181711 1.2726624
## alpha[66] 1.6213711 0.0021408975 0.17006838 1.2830556 1.5062310
## alpha[67] 1.7177277 0.0021980179 0.17305161 1.3676931 1.6021960
## alpha[68] 1.2689898 0.0026072252 0.19839689 0.8757977 1.1329047
## alpha[69] 1.4009745 0.0027297794 0.23085882 0.9461544 1.2488287
## alpha[70] 0.9466394 0.0009331736 0.06513651 0.8212172 0.9021656
## alpha[71] 1.5110153 0.0015913921 0.13197267 1.2477667 1.4213682
## alpha[72] 1.5638988 0.0023107714 0.19143513 1.2012276 1.4354749
## alpha[73] 1.5805028 0.0034554312 0.26686866 1.0473999 1.4034845
## alpha[74] 1.2880860 0.0029882980 0.24310421 0.8114095 1.1293767
## alpha[75] 1.5806238 0.0030193519 0.25382404 1.0932004 1.4067153
## alpha[76] 1.7183588 0.0031008160 0.24135703 1.2580231 1.5530815
## alpha[77] 1.6896485 0.0025164861 0.21303536 1.2717048 1.5382971
## alpha[78] 1.4058183 0.0027182064 0.22098209 0.9767842 1.2584847
## alpha[79] 1.1470895 0.0032866557 0.23842744 0.6739538 0.9873502
## alpha[80] 1.3753752 0.0012124056 0.10013406 1.1820727 1.3090826
## alpha[81] 1.9245395 0.0039837714 0.26872435 1.4257995 1.7370430
## alpha[82] 1.6177690 0.0032993520 0.29837481 1.0349770 1.4219464
## alpha[83] 1.6044163 0.0020684568 0.16991217 1.2739508 1.4860786
## alpha[84] 1.6137412 0.0020883924 0.17303118 1.2828360 1.4995805
## alpha[85] 1.4149168 0.0034331127 0.27202398 0.8813020 1.2331781
## beta -0.6636116 0.0010440044 0.06626257 -0.7936687 -0.7085433
## gamma 1.4933699 0.0008340008 0.04870441 1.3967587 1.4610136
## sigma_y 0.7268850 0.0002371290 0.01817082 0.6932983 0.7141963
## sigma_a 0.3184676 0.0012553504 0.04405155 0.2397795 0.2874022
## lp__ -112.4010460 0.2967100183 8.53067514 -129.5273262 -117.9979628
## 50% 75% 97.5% n_eff Rhat
## alpha[1] 1.2331212 1.4000509 1.7090087 6101.5395 0.9994385
## alpha[2] 0.9842524 1.0502752 1.1778507 5366.8221 0.9993058
## alpha[3] 1.5055571 1.6726123 2.0141333 5554.6973 0.9993901
## alpha[4] 1.5367637 1.6884978 1.9537798 5722.1866 0.9997065
## alpha[5] 1.4653854 1.6381692 1.9410567 6920.2343 0.9997675
## alpha[6] 1.5124288 1.6827284 2.0150175 6941.9870 0.9991148
## alpha[7] 1.8732744 1.9899201 2.2110598 6055.1814 0.9996049
## alpha[8] 1.7041258 1.8682744 2.2075534 5946.5640 0.9993162
## alpha[9] 1.2035594 1.3280979 1.5566922 6397.5449 0.9994920
## alpha[10] 1.5227416 1.6682964 1.9537904 6804.3004 0.9999289
## alpha[11] 1.4664318 1.6193546 1.9144964 7333.6711 0.9993754
## alpha[12] 1.6000753 1.7674605 2.0789784 6974.5225 0.9993098
## alpha[13] 1.2786345 1.4192218 1.7056741 5812.9484 1.0000039
## alpha[14] 1.8592097 1.9712809 2.1915199 5701.2521 1.0000576
## alpha[15] 1.4304302 1.5871045 1.9097590 7347.6799 0.9999325
## alpha[16] 1.2815765 1.4602587 1.8245511 6591.1091 0.9991034
## alpha[17] 1.3923293 1.5454335 1.8396757 6900.7044 0.9991477
## alpha[18] 1.2653554 1.3804229 1.5963723 6576.6119 0.9995124
## alpha[19] 1.3798986 1.4375659 1.5479345 7695.4278 0.9991038
## alpha[20] 1.6088904 1.7746145 2.1087569 6584.6197 0.9997220
## alpha[21] 1.6517081 1.7847972 2.0325520 6918.3368 0.9996741
## alpha[22] 1.1247192 1.2754756 1.5497925 5860.1597 0.9995082
## alpha[23] 1.4750699 1.6618305 2.0123459 7123.4238 0.9994798
## alpha[24] 1.8777284 2.0103108 2.2723030 5323.3934 0.9995632
## alpha[25] 1.8369818 1.9457812 2.1666411 4994.7214 0.9998918
## alpha[26] 1.3937170 1.4422362 1.5311756 7007.7968 0.9991012
## alpha[27] 1.6388155 1.7954945 2.0844332 5586.2025 0.9993411
## alpha[28] 1.3812237 1.5367809 1.8073521 6572.0951 1.0000726
## alpha[29] 1.3527213 1.5119101 1.8307615 7587.8490 0.9995323
## alpha[30] 1.1436843 1.2634944 1.4981343 6256.7063 1.0001648
## alpha[31] 1.7600618 1.9132680 2.2218658 6151.0613 0.9995917
## alpha[32] 1.3974235 1.5653441 1.8519678 7369.3131 0.9994361
## alpha[33] 1.7435082 1.8976810 2.2319158 5490.6149 0.9998154
## alpha[34] 1.5292303 1.6979349 2.0688322 5944.2697 0.9994575
## alpha[35] 1.1332153 1.2828968 1.5329935 5638.5035 1.0002100
## alpha[36] 1.8964360 2.0834845 2.4808808 4289.6585 1.0002155
## alpha[37] 0.8641311 1.0049534 1.2604920 4723.2678 0.9998492
## alpha[38] 1.6587978 1.8259539 2.1147352 6322.7086 0.9992978
## alpha[39] 1.6258314 1.7787346 2.0701179 6919.8725 0.9995221
## alpha[40] 1.8422603 2.0138317 2.3369804 5111.8560 0.9998795
## alpha[41] 1.7824726 1.9161410 2.1717328 5454.7028 0.9999438
## alpha[42] 1.4781008 1.6711749 2.0545345 6995.6578 0.9999712
## alpha[43] 1.5760485 1.7069013 1.9483788 6416.8322 0.9994406
## alpha[44] 1.2774254 1.4098381 1.6706790 5140.9251 0.9999165
## alpha[45] 1.3666067 1.4822228 1.6912166 7152.8583 1.0000599
## alpha[46] 1.3707863 1.5293081 1.8347486 6559.5061 0.9994298
## alpha[47] 1.3411143 1.5283189 1.8754373 6465.7325 0.9992925
## alpha[48] 1.2946463 1.4264631 1.6639195 7512.2914 0.9995128
## alpha[49] 1.6526638 1.7650774 1.9788220 6905.9813 0.9996570
## alpha[50] 1.6519106 1.8535046 2.2590925 5530.2178 0.9995980
## alpha[51] 1.7860080 1.9542965 2.3002270 5261.2528 1.0006851
## alpha[52] 1.6602781 1.8342629 2.1559763 5334.2342 1.0001562
## alpha[53] 1.4190286 1.5872525 1.9117951 6872.7278 0.9994035
## alpha[54] 1.3657728 1.4607707 1.6350977 6745.5036 1.0002918
## alpha[55] 1.5764196 1.7123460 1.9723892 6701.0332 0.9997858
## alpha[56] 1.3825726 1.5401232 1.8894196 5703.0941 0.9993746
## alpha[57] 1.1253622 1.2700161 1.5609764 5431.8281 0.9991605
## alpha[58] 1.6502837 1.8104771 2.1622035 5723.8614 1.0012880
## alpha[59] 1.5889465 1.7519823 2.0573067 6178.6979 0.9994608
## alpha[60] 1.4408127 1.6230764 1.9930203 8087.6688 0.9997020
## alpha[61] 1.2345522 1.3176607 1.4704036 8245.7021 0.9991599
## alpha[62] 1.7368800 1.8870550 2.1897934 6042.7376 0.9999751
## alpha[63] 1.5659990 1.7385489 2.0689474 6523.9591 1.0003280
## alpha[64] 1.7427766 1.8609954 2.0883033 8483.7843 1.0000881
## alpha[65] 1.4492716 1.6289924 1.9784422 5653.9324 0.9995803
## alpha[66] 1.6217200 1.7339206 1.9452554 6310.3787 1.0002613
## alpha[67] 1.7177880 1.8302424 2.0654307 6198.5321 0.9998875
## alpha[68] 1.2730311 1.4031207 1.6496155 5790.4543 0.9994630
## alpha[69] 1.3986444 1.5537886 1.8534438 7152.1701 0.9996164
## alpha[70] 0.9461263 0.9896974 1.0726619 4872.1881 1.0001475
## alpha[71] 1.5125007 1.6012605 1.7622722 6877.2307 1.0002868
## alpha[72] 1.5607890 1.6925710 1.9354295 6863.2418 0.9992212
## alpha[73] 1.5805637 1.7588989 2.1099543 5964.7281 0.9998568
## alpha[74] 1.2893997 1.4500669 1.7551057 6618.1583 0.9993402
## alpha[75] 1.5805516 1.7532071 2.0713911 7067.0483 1.0000471
## alpha[76] 1.7124312 1.8810068 2.2026189 6058.5387 0.9995450
## alpha[77] 1.6888231 1.8393355 2.0949555 7166.6191 0.9992999
## alpha[78] 1.4063669 1.5491743 1.8494659 6609.2060 0.9993618
## alpha[79] 1.1467608 1.3075405 1.6079330 5262.6445 1.0000585
## alpha[80] 1.3764424 1.4423446 1.5690641 6821.3100 0.9994182
## alpha[81] 1.9157492 2.0999410 2.4736201 4550.1447 1.0003923
## alpha[82] 1.6111457 1.8120464 2.2193480 8178.3750 0.9993475
## alpha[83] 1.6041648 1.7175488 1.9383902 6747.7045 0.9994418
## alpha[84] 1.6143557 1.7283508 1.9503691 6864.7468 0.9997444
## alpha[85] 1.4200387 1.5992577 1.9339475 6278.2450 0.9994482
## beta -0.6642821 -0.6188949 -0.5336899 4028.3937 1.0008453
## gamma 1.4935239 1.5253940 1.5905826 3410.3875 1.0001835
## sigma_y 0.7261399 0.7393328 0.7622820 5871.9175 0.9993105
## sigma_a 0.3163937 0.3464567 0.4122320 1231.3808 1.0018482
## lp__ -112.0614001 -106.4044557 -96.4620097 826.6133 1.0016620
68.3.2 截距中增加预测因子
相当于在第二层参数中增加预测因子
\[ \begin{aligned} y_i &\sim N(\mu_i, ~\sigma) \\ \mu_i &= \alpha_{j[i]} + \beta~\mathtt{floor}_i \\ \alpha_j &\sim \operatorname{normal}(\gamma_0 + \gamma_1~u_j, ~\tau) \\ \beta &\sim \operatorname{normal}(0, 1)\\ \gamma_0 &\sim \operatorname{normal}(0, 2.5)\\ \gamma_1 &\sim \operatorname{normal}(0, 2.5)\\ \tau &\sim \operatorname{exp}(1) \\ \end{aligned} \]
stan_program <- "
data {
int<lower=0> N;
vector[N] y;
int<lower=0, upper=1> x[N];
int<lower=2> J;
int<lower=1, upper=J> county[N];
vector[J] u;
}
parameters {
vector[J] alpha;
real beta;
real gamma0;
real gamma1;
real<lower=0> sigma_a;
real<lower=0> sigma_y;
}
model {
vector[N] mu;
for(i in 1:N) {
mu[i] = alpha[county[i]] + x[i] * beta;
}
for(j in 1:J) {
alpha[j] ~ normal(gamma0 + gamma1 * u[j], sigma_a);
}
y ~ normal(mu, sigma_y);
beta ~ normal(0, 1);
gamma0 ~ normal(0, 2.5);
gamma1 ~ normal(0, 2.5);
sigma_a ~ exponential(1);
sigma_y ~ exponential(1);
}
"
stan_data <- list(
N = nrow(radon),
J = length(unique(radon$county)),
county = as.numeric(radon$county),
x = radon$floor,
y = radon$log_radon,
u = unique(radon$log_uranium)
)
fit_intercept_partial_2 <- stan(model_code = stan_program, data = stan_data)
## mean se_mean sd 2.5% 25% 50%
## beta -0.6465663 0.0010213090 0.06806418 -0.7814113 -0.6939567 -0.6456922
## gamma0 1.4926610 0.0008813975 0.04478036 1.4052806 1.4624673 1.4932624
## gamma1 0.5892888 0.0018442793 0.10780197 0.3783845 0.5152805 0.5889562
## sigma_y 0.7247127 0.0001833809 0.01768431 0.6904711 0.7131013 0.7242356
## sigma_a 0.2438118 0.0012433887 0.03934467 0.1721244 0.2160493 0.2422651
## 75% 97.5% n_eff Rhat
## beta -0.5996350 -0.5149011 4441.432 0.9999587
## gamma0 1.5215209 1.5813333 2581.258 0.9995895
## gamma1 0.6621449 0.7973865 3416.646 1.0024602
## sigma_y 0.7362497 0.7595757 9299.681 0.9993431
## sigma_a 0.2690141 0.3250065 1001.286 1.0024075
beta怎么解释? - 负号,说明楼上比楼下氡含量低
68.3.3 变化的截距和斜率
之前模型假定,不管哪个县,所有的房屋一楼和二楼的氡含量的差别是一样的(beta系数是不变的),现在,我们将模型进一步扩展,假定一楼和二楼的氡含量的差别不是固定不变的,而是随县变化的,也就说不同县的房屋,一二楼氡含量差别是不同的。
写出变化的截距和斜率模型的数学表达式
\[ \begin{aligned}[t] y_i &\sim \operatorname{Normal}(\mu_i, \sigma_y) \\ \mu_i &= \alpha_{j[i]} + \beta_{j[i]}~\mathtt{floor}_i \\ \begin{bmatrix} \alpha_j \\ \beta_j \end{bmatrix} & \sim \operatorname{MVNormal} \left( \begin{bmatrix} \gamma_0^{\alpha} + \gamma_1^{\alpha} ~ u_j \\ \gamma_0^{\beta} + \gamma_1^{\beta} ~ u_j \\ \end{bmatrix}, ~\mathbf S \right) \\ \mathbf S & = \begin{bmatrix} \sigma_\alpha & 0 \\ 0 & \sigma_\beta \end{bmatrix} \mathbf R \begin{bmatrix} \sigma_\alpha & 0 \\ 0 & \sigma_\beta \end{bmatrix} \\ & = \begin{bmatrix} \sigma_\alpha & 0 \\ 0 & \sigma_\beta \end{bmatrix} \begin{bmatrix} 1 & \rho \\ \rho & 1 \end{bmatrix} \begin{bmatrix} \sigma_\alpha & 0 \\ 0 & \sigma_\beta \end{bmatrix} \\ \gamma_a & \sim \operatorname{Normal}(0, 4) \\ \gamma_b & \sim \operatorname{Normal}(0, 4) \\ \sigma & \sim \operatorname{Exponential}(1) \\ \sigma_\alpha & \sim \operatorname{Exponential}(1) \\ \sigma_\beta & \sim \operatorname{Exponential}(1) \\ \mathbf R & \sim \operatorname{LKJcorr}(2) \end{aligned} \]
模型表达式中 \(\alpha_j\) 和 \(\beta_j\) 不是直接给先验,而是给的层级先验。
\(\alpha_j\) 和 \(\beta_j\) 也可能存在关联,常见的有,多元正态分布(Multivariate Gaussian Distribution)
\[ \begin{aligned}[t] \begin{bmatrix} \alpha_j \\ \beta_j \end{bmatrix} &\sim \operatorname{MVNormal}\left(\begin{bmatrix}\gamma_{\alpha} \\ \gamma_{\beta} \end{bmatrix}, \mathbf S\right) \\ \end{aligned} \]
68.3.4 协方差矩阵(covariance matrix)
MASS::mvrnorm(n, mu, Sigma)
产生多元高斯分布的随机数,每组随机变量高度相关。
比如,人的身高服从正态分布,人的体重也服从正态分布,同时身高和体重又存在强烈的关联。
-
n
: 随机样本的大小 -
mu
: 多元高斯分布的均值向量 -
Sigma
: 协方差矩阵,主要这里是大写的S (Sigma),提醒我们它是一个矩阵,不是一个数值
a <- 3.5
b <- -1
sigma_a <- 1
sigma_b <- 0.5
rho <- -0.7
mu <- c(a, b)
cov_ab <- sigma_a * sigma_b * rho
sigma <- matrix(c(sigma_a^2, cov_ab,
cov_ab, sigma_b^2), ncol = 2)
sigma
## [,1] [,2]
## [1,] 1.00 -0.35
## [2,] -0.35 0.25
d <- MASS::mvrnorm(1000, mu = mu, Sigma = sigma) %>%
data.frame() %>%
set_names("group_a", "group_b")
head(d)
## group_a group_b
## 1 2.715244 -0.2723168
## 2 3.021387 -0.9497089
## 3 2.995002 -0.8270510
## 4 2.826995 -1.0425705
## 5 1.849620 -1.1043509
## 6 3.606869 -0.6291369
d %>%
ggplot(aes(x = group_a)) +
geom_density(
color = "transparent",
fill = "dodgerblue3",
alpha = 1 / 2
) +
stat_function(
fun = dnorm,
args = list(mean = 3.5, sd = 1),
linetype = 2
)
d %>%
ggplot(aes(x = group_b)) +
geom_density(
color = "transparent",
fill = "dodgerblue3",
alpha = 1 / 2
) +
stat_function(
fun = dnorm,
args = list(mean = -1, sd = 0.5),
linetype = 2
)
d %>%
ggplot(aes(x = group_a, y = group_b)) +
geom_point() +
stat_ellipse(type = "norm", level = 0.95)
68.3.5 回到模型
在stan中要给协方差矩阵指定一个先验,Priors for Covariances
stan_program <- "
data {
int<lower=0> N;
vector[N] y;
int<lower=0, upper=1> x[N];
int<lower=2> J;
int<lower=1, upper=J> county[N];
vector[J] u;
}
parameters {
vector[J] alpha;
vector[J] beta;
vector[2] gamma_a;
vector[2] gamma_b;
real<lower=0> sigma;
vector<lower=0>[2] tau;
corr_matrix[2] Rho;
}
transformed parameters {
vector[2] YY[J];
for (j in 1:J) {
YY[j] = [alpha[j], beta[j]]';
}
}
model {
vector[N] mu;
vector[2] MU[J];
sigma ~ exponential(1);
tau ~ exponential(1);
Rho ~ lkj_corr(2);
gamma_a ~ normal(0, 2);
gamma_b ~ normal(0, 2);
for(i in 1:N) {
mu[i] = alpha[county[i]] + beta[county[i]] * x[i];
}
for(j in 1:J) {
MU[j, 1] = gamma_a[1] + gamma_a[2] * u[j];
MU[j, 2] = gamma_b[1] + gamma_b[2] * u[j];
}
target += multi_normal_lpdf(YY | MU, quad_form_diag(Rho, tau));
y ~ normal(mu, sigma);
}
"
stan_data <- list(
N = nrow(radon),
J = length(unique(radon$county)),
county = as.numeric(radon$county),
x = radon$floor,
y = radon$log_radon,
u = unique(radon$log_uranium)
)
fit_slope_partial <- stan(model_code = stan_program, data = stan_data)
## mean se_mean sd 2.5% 25% 50%
## alpha[1] 0.9519317 0.003353654 0.21468785 0.5331852 0.8065040 0.9537085
## alpha[2] 0.9213506 0.001520614 0.09297252 0.7445764 0.8583941 0.9206529
## alpha[3] 1.4327532 0.003307994 0.22069738 1.0009574 1.2847138 1.4281543
## alpha[4] 1.2307084 0.003544460 0.20498548 0.8505676 1.0908605 1.2258605
## alpha[5] 1.4012124 0.002736854 0.20435606 1.0024084 1.2688080 1.3972693
## alpha[6] 1.7006649 0.003069700 0.21971719 1.2582892 1.5586131 1.7037156
## alpha[7] 1.8682765 0.002742384 0.15871713 1.5601888 1.7593660 1.8687053
## alpha[8] 1.7839273 0.003117164 0.20637049 1.3879590 1.6467750 1.7816495
## alpha[9] 1.1288098 0.002610434 0.16795887 0.8002425 1.0131178 1.1279613
## alpha[10] 1.5784437 0.002905269 0.19507045 1.1998763 1.4443787 1.5778675
## alpha[11] 1.2004181 0.002979086 0.20199305 0.8037032 1.0640679 1.2004750
## alpha[12] 1.7024737 0.002629637 0.20405450 1.3050602 1.5652270 1.6995846
## alpha[13] 1.0251240 0.002792856 0.19031646 0.6502916 0.9002770 1.0220134
## alpha[14] 1.9189194 0.002530061 0.16226987 1.6033615 1.8091695 1.9167418
## alpha[15] 1.4144309 0.002873039 0.21273116 0.9851256 1.2790922 1.4175438
## alpha[16] 1.0645622 0.003452293 0.22491820 0.6074773 0.9165784 1.0695671
## alpha[17] 1.6500046 0.003687618 0.22155579 1.2270210 1.4998423 1.6502365
## alpha[18] 1.0525670 0.002652472 0.16403507 0.7284906 0.9446956 1.0543608
## alpha[19] 1.3883093 0.001178226 0.08812721 1.2140297 1.3289647 1.3891252
## alpha[20] 1.7128467 0.002897388 0.21225267 1.3113888 1.5667382 1.7088012
## alpha[21] 1.6679869 0.002397118 0.17307648 1.3370031 1.5491531 1.6641465
## alpha[22] 1.3355807 0.003620887 0.20240956 0.9227366 1.1981750 1.3406356
## alpha[23] 1.7222808 0.003323876 0.23273001 1.2505402 1.5689641 1.7238414
## alpha[24] 1.8808092 0.003070899 0.18119698 1.5388245 1.7551287 1.8747088
## alpha[25] 1.8091645 0.002458036 0.15724280 1.5000183 1.7033750 1.8112576
## alpha[26] 1.3961411 0.001031404 0.07040717 1.2613581 1.3485908 1.3952360
## alpha[27] 1.8327158 0.003011070 0.19462635 1.4566815 1.7053590 1.8325327
## alpha[28] 1.1850781 0.003468087 0.21114908 0.7779572 1.0469643 1.1836832
## alpha[29] 1.0077382 0.003212421 0.22125759 0.5874808 0.8543263 1.0051908
## alpha[30] 0.9952942 0.002399774 0.16835525 0.6678215 0.8873337 0.9959623
## alpha[31] 1.8205261 0.002781380 0.20301114 1.4312303 1.6789819 1.8195778
## alpha[32] 1.3967406 0.002575971 0.20249457 0.9969899 1.2593385 1.3977493
## alpha[33] 1.7235691 0.003052286 0.20784215 1.3270668 1.5836191 1.7175826
## alpha[34] 1.5132556 0.003088099 0.22089601 1.0886321 1.3639818 1.5097109
## alpha[35] 0.7610688 0.003932438 0.20830745 0.3344719 0.6268445 0.7653319
## alpha[36] 1.9272049 0.004115546 0.24632230 1.4731471 1.7584748 1.9148040
## alpha[37] 0.7339738 0.002942820 0.17681842 0.3788598 0.6172423 0.7415420
## alpha[38] 1.2148739 0.003601584 0.21835942 0.7955514 1.0695808 1.2060466
## alpha[39] 1.6692999 0.003096599 0.19752040 1.2867499 1.5378543 1.6671692
## alpha[40] 1.9687914 0.003428570 0.21495861 1.5664819 1.8215939 1.9646679
## alpha[41] 1.8564128 0.002541474 0.17409119 1.5230290 1.7383874 1.8558836
## alpha[42] 1.5626027 0.003064477 0.23234993 1.0999282 1.4148451 1.5643962
## alpha[43] 1.7218205 0.005637101 0.19738626 1.3623538 1.5860971 1.7138082
## alpha[44] 1.3402833 0.002684746 0.18593309 0.9727795 1.2170085 1.3440359
## alpha[45] 1.4665663 0.002667160 0.16459785 1.1346443 1.3578059 1.4701169
## alpha[46] 1.4120261 0.002584964 0.19135524 1.0338819 1.2883924 1.4126593
## alpha[47] 1.2880182 0.003443675 0.22215974 0.8603349 1.1410978 1.2876299
## alpha[48] 1.2921851 0.002494741 0.17408660 0.9443450 1.1756492 1.2934839
## alpha[49] 1.7241471 0.002540136 0.15529069 1.4228538 1.6195810 1.7256671
## alpha[50] 1.8372352 0.003503701 0.23995656 1.3705600 1.6760922 1.8367269
## alpha[51] 1.8196693 0.003053724 0.20682477 1.4158252 1.6791080 1.8211672
## alpha[52] 1.8066083 0.002935125 0.21162286 1.3927384 1.6670596 1.8043193
## alpha[53] 1.5838625 0.003112178 0.21673020 1.1495365 1.4398343 1.5864813
## alpha[54] 1.4548553 0.001812282 0.13323220 1.1920199 1.3660671 1.4545263
## alpha[55] 1.4454911 0.002540107 0.18112401 1.0892453 1.3268496 1.4419640
## alpha[56] 1.3896750 0.003539458 0.22305591 0.9640651 1.2420396 1.3896368
## alpha[57] 1.1440631 0.002955152 0.19234800 0.7568794 1.0178469 1.1499908
## alpha[58] 1.8436091 0.003135100 0.20717642 1.4370710 1.7049651 1.8426400
## alpha[59] 1.7141568 0.003097634 0.20868751 1.3159720 1.5775359 1.7139989
## alpha[60] 1.6148653 0.003085026 0.22414441 1.1787419 1.4616501 1.6156019
## alpha[61] 1.1671926 0.001641145 0.11759923 0.9359935 1.0874309 1.1668885
## alpha[62] 1.8149507 0.003161927 0.20197701 1.4022872 1.6832495 1.8184751
## alpha[63] 1.7245416 0.003250193 0.21580664 1.2802520 1.5873794 1.7290552
## alpha[64] 1.7613902 0.002060697 0.16079979 1.4564064 1.6516886 1.7609193
## alpha[65] 1.7564018 0.003207661 0.22860426 1.2983539 1.6044767 1.7609477
## alpha[66] 1.4796253 0.002579439 0.16782088 1.1479370 1.3684441 1.4768058
## alpha[67] 1.4581322 0.003242883 0.17002073 1.1272241 1.3427318 1.4572282
## alpha[68] 1.3400376 0.002421674 0.17783606 0.9895438 1.2179056 1.3431086
## alpha[69] 1.1052663 0.002971051 0.21661274 0.6966168 0.9597750 1.0988825
## alpha[70] 0.9757155 0.001295334 0.07382497 0.8323926 0.9279980 0.9758341
## alpha[71] 1.5438858 0.001776335 0.12623121 1.2972565 1.4611862 1.5432967
## alpha[72] 1.6409851 0.002083285 0.16596574 1.3114261 1.5311118 1.6395149
## alpha[73] 1.8131499 0.003428889 0.22175330 1.3758959 1.6631683 1.8116511
## alpha[74] 1.4937826 0.003430274 0.20909286 1.0709270 1.3552306 1.4975707
## alpha[75] 1.5079348 0.003050620 0.20656003 1.1005149 1.3706130 1.5079433
## alpha[76] 1.9041916 0.003575203 0.20310389 1.5150809 1.7700360 1.9037997
## alpha[77] 1.7143489 0.002710865 0.18135671 1.3635311 1.5901476 1.7103102
## alpha[78] 1.0962731 0.003262629 0.20613290 0.6967964 0.9581975 1.0895785
## alpha[79] 1.3554009 0.003470343 0.20696385 0.9375743 1.2182870 1.3630556
## alpha[80] 1.3720130 0.001415465 0.09791249 1.1794778 1.3063741 1.3724494
## alpha[81] 1.8586083 0.004328592 0.23689050 1.4039156 1.7031916 1.8514036
## alpha[82] 1.7071957 0.003915923 0.22770527 1.2747605 1.5537967 1.7019673
## alpha[83] 1.7865354 0.002999872 0.16399051 1.4751124 1.6762375 1.7840108
## alpha[84] 1.5660924 0.002129781 0.15587944 1.2766856 1.4580465 1.5620844
## alpha[85] 1.6351601 0.002929707 0.22301609 1.1830134 1.4888412 1.6417432
## 75% 97.5% n_eff Rhat
## alpha[1] 1.1003015 1.360342 4098.060 0.9997172
## alpha[2] 0.9842125 1.106770 3738.276 0.9999053
## alpha[3] 1.5774113 1.870244 4451.076 1.0004986
## alpha[4] 1.3646823 1.664353 3344.613 1.0008200
## alpha[5] 1.5341600 1.807129 5575.346 1.0001793
## alpha[6] 1.8486566 2.129637 5123.140 0.9997305
## alpha[7] 1.9733099 2.176808 3349.586 1.0002705
## alpha[8] 1.9222286 2.194028 4383.044 0.9995557
## alpha[9] 1.2470978 1.460232 4139.810 1.0004175
## alpha[10] 1.7076384 1.982730 4508.274 1.0004635
## alpha[11] 1.3345524 1.608036 4597.340 1.0001867
## alpha[12] 1.8396405 2.089860 6021.444 1.0000293
## alpha[13] 1.1501953 1.410773 4643.608 1.0004185
## alpha[14] 2.0256995 2.235397 4113.521 1.0002933
## alpha[15] 1.5569713 1.823673 5482.508 0.9993977
## alpha[16] 1.2148245 1.495768 4244.571 0.9999769
## alpha[17] 1.8011947 2.086203 3609.727 1.0027579
## alpha[18] 1.1652075 1.355866 3824.474 1.0001771
## alpha[19] 1.4470976 1.556684 5594.522 0.9998284
## alpha[20] 1.8573444 2.136092 5366.521 0.9995045
## alpha[21] 1.7797779 2.018619 5213.116 1.0009420
## alpha[22] 1.4736449 1.717472 3124.871 1.0005473
## alpha[23] 1.8762236 2.165036 4902.473 0.9991197
## alpha[24] 2.0004046 2.253822 3481.535 1.0008662
## alpha[25] 1.9153384 2.113069 4092.278 1.0013684
## alpha[26] 1.4430359 1.535565 4659.893 1.0016308
## alpha[27] 1.9587928 2.211046 4177.933 0.9996019
## alpha[28] 1.3224614 1.612108 3706.793 1.0001495
## alpha[29] 1.1510521 1.453448 4743.857 0.9998455
## alpha[30] 1.1081339 1.331021 4921.671 0.9995717
## alpha[31] 1.9580927 2.225042 5327.446 1.0001730
## alpha[32] 1.5360485 1.796799 6179.377 0.9993134
## alpha[33] 1.8552389 2.149746 4636.785 1.0002771
## alpha[34] 1.6583388 1.959657 5116.740 1.0001739
## alpha[35] 0.9017612 1.151721 2805.988 1.0011116
## alpha[36] 2.0850028 2.437771 3582.222 1.0000540
## alpha[37] 0.8509241 1.075149 3610.169 1.0009169
## alpha[38] 1.3575566 1.665925 3675.842 1.0003793
## alpha[39] 1.7995900 2.062082 4068.684 0.9991789
## alpha[40] 2.1065268 2.403715 3930.826 0.9999035
## alpha[41] 1.9720484 2.209405 4692.263 0.9996486
## alpha[42] 1.7133396 2.021255 5748.735 1.0001209
## alpha[43] 1.8458754 2.137788 1226.090 1.0024276
## alpha[44] 1.4623153 1.700079 4796.308 1.0015063
## alpha[45] 1.5759521 1.788291 3808.466 1.0008279
## alpha[46] 1.5371528 1.774580 5479.888 0.9993782
## alpha[47] 1.4331916 1.727210 4161.851 1.0004470
## alpha[48] 1.4104354 1.625416 4869.450 0.9996094
## alpha[49] 1.8243277 2.034269 3737.462 0.9999708
## alpha[50] 1.9939146 2.332710 4690.413 0.9993716
## alpha[51] 1.9575679 2.236414 4587.178 0.9992191
## alpha[52] 1.9483241 2.235419 5198.428 0.9999447
## alpha[53] 1.7300816 2.011675 4849.647 0.9995000
## alpha[54] 1.5445952 1.720587 5404.640 0.9997539
## alpha[55] 1.5685743 1.804953 5084.498 1.0008839
## alpha[56] 1.5361079 1.833575 3971.495 0.9999883
## alpha[57] 1.2717847 1.516737 4236.584 1.0000696
## alpha[58] 1.9828200 2.262707 4366.947 0.9999132
## alpha[59] 1.8491338 2.139414 4538.712 1.0007966
## alpha[60] 1.7680276 2.053259 5278.837 0.9998648
## alpha[61] 1.2482358 1.392249 5134.701 0.9995918
## alpha[62] 1.9478640 2.201454 4080.376 1.0012090
## alpha[63] 1.8673545 2.140254 4408.707 0.9991195
## alpha[64] 1.8677874 2.075037 6088.952 0.9995091
## alpha[65] 1.9062534 2.202483 5079.158 0.9998632
## alpha[66] 1.5928385 1.818104 4232.934 0.9999891
## alpha[67] 1.5698649 1.796656 2748.788 1.0013054
## alpha[68] 1.4598779 1.699556 5392.724 0.9997470
## alpha[69] 1.2501871 1.543298 5315.544 0.9999522
## alpha[70] 1.0259323 1.120187 3248.201 1.0005520
## alpha[71] 1.6280134 1.794891 5049.912 0.9993192
## alpha[72] 1.7494469 1.966321 6346.575 0.9993569
## alpha[73] 1.9656252 2.245212 4182.475 0.9994124
## alpha[74] 1.6356340 1.886567 3715.530 0.9998144
## alpha[75] 1.6430099 1.921257 4584.756 0.9998729
## alpha[76] 2.0315432 2.317663 3227.269 0.9997366
## alpha[77] 1.8349298 2.077528 4475.601 0.9994004
## alpha[78] 1.2299953 1.505411 3991.713 0.9996280
## alpha[79] 1.4977606 1.740977 3556.674 0.9997123
## alpha[80] 1.4367455 1.560968 4784.957 0.9994886
## alpha[81] 2.0130290 2.341839 2995.036 1.0009345
## alpha[82] 1.8554026 2.161381 3381.255 1.0001061
## alpha[83] 1.8940519 2.120908 2988.354 1.0031464
## alpha[84] 1.6732702 1.875764 5356.830 0.9999885
## alpha[85] 1.7842689 2.059926 5794.608 0.9997485
## mean se_mean sd 2.5% 25% 50%
## beta[1] -0.3030413 0.006720692 0.3126060 -0.9140722 -0.4940543 -0.3120472
## beta[2] -0.4980363 0.013611312 0.2875532 -1.1154770 -0.6758638 -0.4788113
## beta[3] -0.5883236 0.004386772 0.2570363 -1.1051184 -0.7430379 -0.5832167
## beta[4] -0.4021724 0.005355640 0.2620278 -0.9336048 -0.5662986 -0.3977372
## beta[5] -0.5384383 0.004604968 0.2819605 -1.1030659 -0.7014163 -0.5466950
## beta[6] -0.8159568 0.006641223 0.3104158 -1.4482857 -1.0009603 -0.8184723
## beta[7] -0.5221451 0.016142625 0.2988984 -1.0335387 -0.7274081 -0.5583546
## beta[8] -0.7383255 0.004946105 0.2601781 -1.2586115 -0.9005770 -0.7428251
## beta[9] -0.3465328 0.010091581 0.2899161 -0.8649025 -0.5381192 -0.3764453
## beta[10] -0.7616415 0.005179511 0.2506588 -1.2855322 -0.9095501 -0.7535025
## beta[11] -0.3825179 0.007206771 0.3333535 -1.0368771 -0.5805990 -0.3826561
## beta[12] -0.7709969 0.005734165 0.3072965 -1.4009586 -0.9445013 -0.7687419
## beta[13] -0.3238779 0.007647073 0.3419432 -1.0156753 -0.5246843 -0.3302553
## beta[14] -0.8396599 0.005003122 0.2413719 -1.3390137 -0.9858599 -0.8323510
## beta[15] -0.6153940 0.003896563 0.2581678 -1.1452472 -0.7656193 -0.6170301
## beta[16] -0.4134817 0.006224523 0.3172008 -1.0712782 -0.5944658 -0.4131444
## beta[17] -0.9476459 0.010790606 0.2703088 -1.5379111 -1.1068621 -0.9220920
## beta[18] -0.2735102 0.006228926 0.2622621 -0.7684646 -0.4425521 -0.2792567
## beta[19] -0.7250979 0.004798187 0.2183951 -1.1830943 -0.8609326 -0.7115620
## beta[20] -0.7677315 0.005523243 0.3135258 -1.3925103 -0.9493436 -0.7683732
## beta[21] -0.6705168 0.005356064 0.2803788 -1.1910896 -0.8429943 -0.6863102
## beta[22] -0.7887503 0.005963844 0.3000488 -1.4071099 -0.9654353 -0.7939567
## beta[23] -0.8218970 0.006076073 0.2899649 -1.3943003 -0.9987064 -0.8200531
## beta[24] -0.7261831 0.005195648 0.2632440 -1.2415006 -0.8938480 -0.7339934
## beta[25] -0.5077868 0.014834273 0.2909818 -0.9979299 -0.7123235 -0.5430667
## beta[26] -0.7107391 0.004193608 0.1825726 -1.0980543 -0.8231486 -0.7033650
## beta[27] -0.8146455 0.006288994 0.2838293 -1.3640494 -0.9988645 -0.8227987
## beta[28] -0.4527634 0.004803400 0.2541751 -0.9637185 -0.6048339 -0.4548247
## beta[29] -0.3161594 0.007449517 0.3459894 -1.0248022 -0.5268567 -0.3148994
## beta[30] -0.3460725 0.007119331 0.3274406 -1.0044489 -0.5531515 -0.3504414
## beta[31] -0.7940265 0.005755141 0.3145944 -1.4416483 -0.9734705 -0.7906300
## beta[32] -0.6169924 0.004928547 0.3027245 -1.2148826 -0.7964968 -0.6187311
## beta[33] -0.7074118 0.005834255 0.3170576 -1.3670052 -0.8801510 -0.7017518
## beta[34] -0.6677593 0.003994489 0.2571996 -1.2000831 -0.8205331 -0.6639891
## beta[35] -0.2237865 0.006621125 0.2749385 -0.7457459 -0.4041935 -0.2296301
## beta[36] -0.6269031 0.010623309 0.3159487 -1.1982220 -0.8334872 -0.6558471
## beta[37] -0.3236230 0.006688778 0.3120225 -0.9284546 -0.5225015 -0.3288592
## beta[38] -0.2554516 0.006687635 0.2834597 -0.8066718 -0.4372508 -0.2666320
## beta[39] -0.6684845 0.005954442 0.2894112 -1.2060467 -0.8512198 -0.6806734
## beta[40] -0.8353723 0.006262632 0.2931388 -1.4295812 -1.0169266 -0.8345612
## beta[41] -0.7123072 0.009499444 0.2927183 -1.2511662 -0.9045290 -0.7329133
## beta[42] -0.7082503 0.005149082 0.3084356 -1.3360289 -0.8805794 -0.7119100
## beta[43] -1.0457201 0.019253643 0.2830325 -1.6795114 -1.2158100 -1.0142153
## beta[44] -0.7993056 0.009716235 0.2911761 -1.4366340 -0.9674106 -0.7695560
## beta[45] -0.8350362 0.008276736 0.2382125 -1.3318285 -0.9835211 -0.8190592
## beta[46] -0.6478682 0.005726239 0.3058810 -1.2729869 -0.8257849 -0.6457934
## beta[47] -0.7226032 0.010815687 0.3015054 -1.4028746 -0.8933732 -0.6905117
## beta[48] -0.5405341 0.005304329 0.2697159 -1.0411369 -0.7014851 -0.5533612
## beta[49] -0.9255142 0.010683095 0.2756646 -1.5527112 -1.0872977 -0.9002967
## beta[50] -0.8397182 0.006834991 0.3205389 -1.5114593 -1.0246911 -0.8374269
## beta[51] -0.7676813 0.006103267 0.3120952 -1.3926615 -0.9483248 -0.7697585
## beta[52] -0.8276903 0.006507587 0.3240390 -1.4636376 -1.0127452 -0.8296644
## beta[53] -0.7967042 0.004906244 0.2775771 -1.3754808 -0.9585333 -0.7941758
## beta[54] -0.9094890 0.009996546 0.2545942 -1.4750582 -1.0600454 -0.8833766
## beta[55] -0.4764234 0.005330174 0.2554075 -0.9556680 -0.6394411 -0.4934966
## beta[56] -0.7441212 0.007248965 0.2711338 -1.3444772 -0.8948435 -0.7214473
## beta[57] -0.5961015 0.005121362 0.2819350 -1.1599240 -0.7625197 -0.5946360
## beta[58] -0.7908568 0.006168628 0.2959496 -1.3647272 -0.9762762 -0.8057767
## beta[59] -0.8291218 0.005433591 0.2636252 -1.3783127 -0.9861114 -0.8216350
## beta[60] -0.7680102 0.006238209 0.3172696 -1.4156311 -0.9512974 -0.7730598
## beta[61] -0.2776078 0.010094660 0.2601798 -0.7410178 -0.4537275 -0.3028755
## beta[62] -0.5346912 0.018491146 0.3402042 -1.0770543 -0.7727514 -0.5864294
## beta[63] -0.6972739 0.007888426 0.2979608 -1.2505420 -0.8890613 -0.7200443
## beta[64] -0.7539639 0.004801536 0.2739794 -1.3376980 -0.9132354 -0.7534081
## beta[65] -0.8756623 0.006653224 0.3264299 -1.5263907 -1.0786572 -0.8796967
## beta[66] -0.5001943 0.004542816 0.2092116 -0.9083963 -0.6330584 -0.5054298
## beta[67] -0.2184017 0.014313564 0.2700924 -0.7020678 -0.4078772 -0.2328044
## beta[68] -0.6628048 0.005359459 0.3055929 -1.2822973 -0.8409814 -0.6674304
## beta[69] -0.3437809 0.007418044 0.3418222 -1.0210306 -0.5482539 -0.3465923
## beta[70] -0.6431331 0.005234519 0.1742594 -0.9747187 -0.7625968 -0.6479965
## beta[71] -0.7822887 0.005616177 0.2302088 -1.2384447 -0.9258432 -0.7716008
## beta[72] -0.7619719 0.005343562 0.3103291 -1.4043833 -0.9420532 -0.7643868
## beta[73] -0.8621089 0.006679232 0.3213920 -1.5205982 -1.0466283 -0.8588473
## beta[74] -0.7663318 0.006283631 0.3270461 -1.4077400 -0.9572036 -0.7795805
## beta[75] -0.5015279 0.009723381 0.2900317 -1.0129398 -0.6809862 -0.5267676
## beta[76] -0.8673323 0.006176914 0.2806875 -1.4506170 -1.0399824 -0.8632977
## beta[77] -0.8321239 0.007475239 0.2865095 -1.4472824 -1.0019863 -0.8098555
## beta[78] -0.4128055 0.005950462 0.2870089 -1.0145391 -0.5893904 -0.4094492
## beta[79] -0.8000064 0.006271739 0.2909240 -1.4269699 -0.9685357 -0.7880590
## beta[80] -0.7321706 0.008382554 0.2294750 -1.2319209 -0.8683308 -0.7145151
## beta[81] -0.5395559 0.012883785 0.2925521 -1.0537814 -0.7280829 -0.5745713
## beta[82] -0.7612997 0.005331392 0.3101680 -1.4154587 -0.9372896 -0.7647274
## beta[83] -1.1431209 0.019381658 0.2975025 -1.7912397 -1.3268295 -1.1059823
## beta[84] -0.6276947 0.004805438 0.2762994 -1.2091163 -0.7860138 -0.6223734
## beta[85] -0.8019686 0.005568033 0.3136438 -1.4594796 -0.9897656 -0.8005862
## 75% 97.5% n_eff Rhat
## beta[1] -0.11912508 0.351010525 2163.5485 1.0013650
## beta[2] -0.30057741 0.020061064 446.3098 1.0121605
## beta[3] -0.44049841 -0.056782556 3433.1964 1.0008918
## beta[4] -0.23563538 0.111292978 2393.7120 1.0016570
## beta[5] -0.38490682 0.050641274 3749.0677 1.0002738
## beta[6] -0.63737414 -0.172413312 2184.6993 1.0013362
## beta[7] -0.33603076 0.151569000 342.8459 1.0114602
## beta[8] -0.58673725 -0.184164834 2767.0371 1.0003627
## beta[9] -0.18262043 0.299801499 825.3273 1.0059825
## beta[10] -0.60663571 -0.279568685 2342.0079 1.0019851
## beta[11] -0.18275615 0.297173151 2139.5798 1.0004640
## beta[12] -0.59968545 -0.135266685 2871.9365 0.9997418
## beta[13] -0.11946617 0.365191674 1999.4823 1.0002025
## beta[14] -0.67726377 -0.384023318 2327.5072 1.0018073
## beta[15] -0.46432270 -0.082495188 4389.7617 0.9996521
## beta[16] -0.23054840 0.243602840 2596.9079 1.0006262
## beta[17] -0.77406859 -0.451257594 627.5218 1.0053595
## beta[18] -0.10981851 0.281144450 1772.7388 1.0038056
## beta[19] -0.58449687 -0.310124910 2071.7220 1.0021092
## beta[20] -0.58230348 -0.147836169 3222.2425 1.0004023
## beta[21] -0.51076361 -0.051826722 2740.3044 1.0011318
## beta[22] -0.61266700 -0.175721500 2531.2284 0.9996334
## beta[23] -0.64808959 -0.236589536 2277.4297 1.0002088
## beta[24] -0.56496472 -0.163902252 2567.0684 1.0019776
## beta[25] -0.33320594 0.159170303 384.7682 1.0103853
## beta[26] -0.58554008 -0.374363418 1895.3767 1.0026909
## beta[27] -0.63974931 -0.220641578 2036.8173 1.0014487
## beta[28] -0.29247046 0.059357974 2800.0674 1.0013073
## beta[29] -0.10752871 0.376074618 2157.0956 1.0008307
## beta[30] -0.14767650 0.319960280 2115.3715 1.0010405
## beta[31] -0.61514988 -0.159609501 2988.0653 0.9996404
## beta[32] -0.45128138 0.018368911 3772.7428 1.0000598
## beta[33] -0.53454404 -0.054186815 2953.2870 1.0005211
## beta[34] -0.51097082 -0.149539318 4145.8927 0.9997209
## beta[35] -0.04500423 0.339145023 1724.2793 1.0021092
## beta[36] -0.44835118 0.100508416 884.5321 1.0061291
## beta[37] -0.13432195 0.324575336 2176.0975 1.0023461
## beta[38] -0.07915815 0.323632742 1796.5426 1.0023679
## beta[39] -0.50502242 -0.028683366 2362.3736 1.0013306
## beta[40] -0.65260323 -0.235814358 2190.9514 1.0024495
## beta[41] -0.54035886 -0.061512200 949.5180 1.0037489
## beta[42] -0.53019667 -0.076892808 3588.1408 0.9998515
## beta[43] -0.84115116 -0.586293013 216.0960 1.0191532
## beta[44] -0.61303106 -0.264492616 898.0806 1.0072343
## beta[45] -0.67746588 -0.384095675 828.3446 1.0048936
## beta[46] -0.47542431 -0.004782087 2853.4217 0.9997329
## beta[47] -0.52580992 -0.192019385 777.1092 1.0082638
## beta[48] -0.38910950 0.037311342 2585.5465 1.0004794
## beta[49] -0.73582274 -0.439279750 665.8368 1.0075882
## beta[50] -0.65654604 -0.178681877 2199.3059 1.0003204
## beta[51] -0.59155104 -0.132422338 2614.8658 1.0003848
## beta[52] -0.64477686 -0.125665892 2479.4453 1.0022709
## beta[53] -0.63662980 -0.213984233 3200.8763 1.0006439
## beta[54] -0.74335792 -0.444620177 648.6299 1.0066013
## beta[55] -0.32377037 0.062042204 2296.0673 1.0025998
## beta[56] -0.56696770 -0.274699262 1398.9925 1.0059094
## beta[57] -0.43000226 -0.027116399 3030.5878 1.0011166
## beta[58] -0.61209113 -0.173199482 2301.7507 1.0018637
## beta[59] -0.65987369 -0.307318874 2353.9636 1.0019660
## beta[60] -0.59231317 -0.076258232 2586.6458 1.0014369
## beta[61] -0.11610993 0.287447347 664.2995 1.0068440
## beta[62] -0.34472779 0.250036811 338.4946 1.0143482
## beta[63] -0.52679463 -0.053542807 1426.7159 1.0028267
## beta[64] -0.58936705 -0.187893514 3255.9335 1.0011178
## beta[65] -0.68282182 -0.211397920 2407.2187 0.9995663
## beta[66] -0.37645533 -0.065107681 2120.9054 1.0008409
## beta[67] -0.04859708 0.347004994 356.0648 1.0103891
## beta[68] -0.48811163 -0.021088296 3251.2076 0.9999984
## beta[69] -0.13453942 0.361258163 2123.3511 1.0008679
## beta[70] -0.52590149 -0.301162946 1108.2522 1.0038789
## beta[71] -0.63806985 -0.351301224 1680.2044 1.0036527
## beta[72] -0.58017919 -0.141279399 3372.7432 0.9999572
## beta[73] -0.67438725 -0.198143092 2315.3520 1.0007570
## beta[74] -0.58318193 -0.066150815 2708.9234 0.9995692
## beta[75] -0.34942255 0.141285715 889.7260 1.0053339
## beta[76] -0.68292910 -0.318243818 2064.9189 1.0002657
## beta[77] -0.64833029 -0.322031875 1469.0205 1.0034849
## beta[78] -0.22931408 0.155018694 2326.4269 1.0015869
## beta[79] -0.62224852 -0.234310472 2151.7074 1.0025285
## beta[80] -0.57606428 -0.320129129 749.4076 1.0081500
## beta[81] -0.37831808 0.121988216 515.6079 1.0099209
## beta[82] -0.57827585 -0.128848354 3384.6424 0.9996730
## beta[83] -0.93217154 -0.642387360 235.6130 1.0156904
## beta[84] -0.46318686 -0.072329950 3305.9310 0.9996514
## beta[85] -0.61145908 -0.163757628 3172.9996 1.0002582
## mean se_mean sd 2.5% 25% 50% 75%
## sigma 0.7169821 0.000294179 0.01753899 0.6834949 0.7047496 0.7169355 0.7282565
## 97.5% n_eff Rhat
## sigma 0.7519314 3554.561 1.001291
## mean se_mean sd 2.5% 25% 50%
## gamma_a[1] 1.4919256 0.001165342 0.04291215 1.4090995 1.4622673 1.4913780
## gamma_a[2] 0.6850737 0.002850866 0.11604210 0.4546167 0.6078951 0.6841795
## gamma_b[1] -0.6453958 0.002988174 0.08067028 -0.8086556 -0.6971753 -0.6447844
## gamma_b[2] -0.4466330 0.007599022 0.22231790 -0.8852762 -0.5939574 -0.4434967
## 75% 97.5% n_eff Rhat
## gamma_a[1] 1.5207872 1.577032940 1355.9807 1.002477
## gamma_a[2] 0.7672601 0.912117610 1656.8291 1.001838
## gamma_b[1] -0.5919461 -0.488345518 728.8116 1.005086
## gamma_b[2] -0.3008174 -0.002157828 855.9206 1.002744