Introduction to bkmr and bkmrhat
bkmr is a package to implement Bayesian kernel machine regression (BKMR) using Markov chain Monte Carlo (MCMC). Notably, bkmr is missing some key features in Bayesian inference and MCMC diagnostics: 1) no facility for running multiple chains in parallel 2) no inference across multiple chains 3) limited posterior summary of parameters 4) limited diagnostics. The bkmrhat package is a lightweight set of function that fills in each of those gaps by enabling post-processing of bkmr output in other packages and building a small framework for parallel processing.
How to use the bkmrhat package
- Fit a BKMR model for a single chain using the
kmbaryesfunction frombkmr, or use multiple parallel chainskmbayes_parallelfrombkmrhat - Perform diagnostics on single or multiple chains using the
kmbayes_diagnosefunction (uses functions from therstanpackage) OR convert the BKMR fit(s) tomcmc(one chain) ormcmc.list(multiple chains) objects from thecodapackage usingas.mcmcoras.mcmc.listfrom thebkmrhatpackage. Thecodapackage has a whole host of inference and diagnostic procedures (but may lag behind some of the diagnostics functions fromrstan). - Perform posterior summaries using
codafunctions or combine chains from akmbayes_parallelfit usingkmbayes_combine. Final posterior inferences can be made on the combined object, which enables use ofbkmrpackage functions for visual summaries of independent and joint effects of exposures in thebkmrmodel.
First, simulate some data from the bkmr function
library("bkmr")
library("bkmrhat")
library("coda")
set.seed(111)
dat <- bkmr::SimData(n = 50, M = 5, ind=1:3, Zgen="realistic")
y <- dat$y
Z <- dat$Z
X <- cbind(dat$X, rnorm(50))
head(cbind(y,Z,X))
## y z1 z2 z3 z4 z5
## [1,] 4.1379128 -0.06359282 -0.02996246 -0.14190647 -0.44089352 -0.1878732
## [2,] 12.0843607 -0.07308834 0.32021690 1.08838691 0.29448354 -1.4609837
## [3,] 7.8859254 0.59604857 0.20602329 0.46218114 -0.03387906 -0.7615902
## [4,] 1.1609768 1.46504863 2.48389356 1.39869461 1.49678590 0.2837234
## [5,] 0.4989372 -0.37549639 0.01159884 1.17891641 -0.05286516 -0.1680664
## [6,] 5.0731242 -0.36904566 -0.49744932 -0.03330522 0.30843805 0.6814844
##
## [1,] 1.0569172 -1.0503824
## [2,] 4.8158570 0.3251424
## [3,] 2.6683461 -2.1048716
## [4,] -0.7492096 -0.9551027
## [5,] -0.5428339 -0.5306399
## [6,] 1.6493251 0.8274405
Example 1: single vs multi-chains
There is some overhead in parallel processing when using the future package, so the payoff when using parallel processing may vary by the problem. Here it is about a 2-4x speedup, but you can see more benefit at higher iterations. Note that this may not yield as many usable iterations as a single large chain if a substantial burnin period is needed, but it will enable useful convergence diagnostics. Note that the future package can implement sequential processing, which effectively turns the kmbayes_parallel into a loop, but still has all other advantages of multiple chains.
# enable parallel processing (up to 4 simultaneous processes here)
future::plan(strategy = future::multisession)
# single run of 4000 observations from bkmr package
set.seed(111)
system.time(kmfit <- suppressMessages(kmbayes(y = y, Z = Z, X = X, iter = 4000, verbose = FALSE, varsel = FALSE)))
## user system elapsed
## 4.714 0.450 5.211
# 4 runs of 1000 observations from bkmrhat package
set.seed(111)
system.time(kmfit5 <- suppressMessages(kmbayes_parallel(nchains=4, y = y, Z = Z, X = X, iter = 1000, verbose = FALSE, varsel = FALSE)))
## Chain 1
## Chain 2
## Chain 3
## Chain 4
## user system elapsed
## 0.359 0.048 9.585
Example 2: Diagnostics
The diagnostics from the rstan package come from the monitor function (see the help files for that function in the rstan pacakge)
# Using rstan functions (set burnin/warmup to zero for comparability with coda numbers given later
# posterior summaries should be performed after excluding warmup/burnin)
singlediag = kmbayes_diagnose(kmfit, warmup=0, digits_summary=2)
## Single chain
## Inference for the input samples (1 chains: each with iter = 4000; warmup = 0):
##
## Q5 Q50 Q95 Mean SD Rhat Bulk_ESS Tail_ESS
## beta1 1.9 2.0 2.1 2.0 0.0 1.00 2820 3194
## beta2 0.0 0.1 0.3 0.1 0.1 1.00 3739 3535
## lambda 3.9 10.0 22.3 11.2 5.9 1.00 346 222
## r1 0.0 0.0 0.1 0.0 0.1 1.01 129 173
## r2 0.0 0.0 0.1 0.0 0.1 1.00 182 181
## r3 0.0 0.0 0.0 0.0 0.0 1.01 158 112
## r4 0.0 0.0 0.1 0.0 0.1 1.03 176 135
## r5 0.0 0.0 0.0 0.0 0.1 1.00 107 114
## sigsq.eps 0.2 0.3 0.5 0.4 0.1 1.00 1262 1563
##
## For each parameter, Bulk_ESS and Tail_ESS are crude measures of
## effective sample size for bulk and tail quantities respectively (an ESS > 100
## per chain is considered good), and Rhat is the potential scale reduction
## factor on rank normalized split chains (at convergence, Rhat <= 1.05).
# Using rstan functions (multiple chains enable R-hat)
multidiag = kmbayes_diagnose(kmfit5, warmup=0, digits_summary=2)
## Parallel chains
## Inference for the input samples (4 chains: each with iter = 1000; warmup = 0):
##
## Q5 Q50 Q95 Mean SD Rhat Bulk_ESS Tail_ESS
## beta1 1.9 2.0 2.1 2.0 0.0 1.00 2334 1512
## beta2 0.0 0.1 0.3 0.1 0.1 1.00 3228 3643
## lambda 4.1 9.9 23.7 11.4 6.1 1.01 384 235
## r1 0.0 0.0 0.6 0.1 0.2 1.11 32 28
## r2 0.0 0.0 0.1 0.0 0.1 1.02 184 138
## r3 0.0 0.0 0.1 0.0 0.1 1.03 101 87
## r4 0.0 0.0 0.1 0.0 0.1 1.03 142 89
## r5 0.0 0.0 0.1 0.0 0.2 1.09 37 53
## sigsq.eps 0.2 0.3 0.5 0.3 0.1 1.00 899 1148
##
## For each parameter, Bulk_ESS and Tail_ESS are crude measures of
## effective sample size for bulk and tail quantities respectively (an ESS > 100
## per chain is considered good), and Rhat is the potential scale reduction
## factor on rank normalized split chains (at convergence, Rhat <= 1.05).
# using coda functions, not using any burnin (for demonstration only)
kmfitcoda = as.mcmc(kmfit, iterstart = 1)
kmfit5coda = as.mcmc.list(kmfit5, iterstart = 1)
# single chain trace plot
traceplot(kmfitcoda)
The trace plots look typical, and fine, but trace plots don’t give a full picture of convergence. Note that there is apparent quick convergence for a couple of parameters demonstrated by movement away from the starting value and concentration of the rest of the samples within a narrow band.
Seeing visual evidence that different chains are sampling from the same marginal distributions is reassuring about the stability of the results.
# multiple chain trace plot
traceplot(kmfit5coda)
Now examine “cross correlation”, which can help identify highly correlated parameters in the posterior, which can be problematic for MCMC sampling. Here there is a block {r3,r4,r5} which appear to be highly correlated. All other things equal, having highly correlated parameters in the posterior means that more samples are needed than would be needed with uncorrelated parameters.
# multiple cross-correlation plot (combines all samples)
crosscorr(kmfit5coda)
## beta1 beta2 lambda r1 r2
## beta1 1.000000000 0.06063846 0.009619123 0.11863388 0.16213875
## beta2 0.060638461 1.00000000 -0.085205911 -0.08886474 -0.08044259
## lambda 0.009619123 -0.08520591 1.000000000 -0.03205822 -0.01564349
## r1 0.118633883 -0.08886474 -0.032058222 1.00000000 0.53623405
## r2 0.162138749 -0.08044259 -0.015643492 0.53623405 1.00000000
## r3 0.137314004 -0.05691727 -0.016089644 0.56704044 0.82325738
## r4 0.208176153 -0.07359633 -0.037404909 0.68444590 0.79050903
## r5 0.092149921 -0.05934214 -0.022712957 0.28395735 0.53975111
## sigsq.eps -0.029864448 0.04941527 -0.302048386 -0.13153370 -0.14208652
## r3 r4 r5 sigsq.eps
## beta1 0.13731400 0.20817615 0.09214992 -0.02986445
## beta2 -0.05691727 -0.07359633 -0.05934214 0.04941527
## lambda -0.01608964 -0.03740491 -0.02271296 -0.30204839
## r1 0.56704044 0.68444590 0.28395735 -0.13153370
## r2 0.82325738 0.79050903 0.53975111 -0.14208652
## r3 1.00000000 0.74555066 0.46137436 -0.12123327
## r4 0.74555066 1.00000000 0.51099917 -0.11946345
## r5 0.46137436 0.51099917 1.00000000 -0.02434892
## sigsq.eps -0.12123327 -0.11946345 -0.02434892 1.00000000
crosscorr.plot(kmfit5coda)
Now examine “autocorrelation” to identify parameters that have high correlation between subsequent iterations of the MCMC sampler, which can lead to inefficient MCMC sampling. All other things equal, having highly autocorrelated parameters in the posterior means that more samples are needed than would be needed with low-autocorrelation parameters.
# multiple chain trace plot
#autocorr(kmfit5coda) # lots of output
autocorr.plot(kmfit5coda)
Graphical tools can be limited, and are sometimes difficult to use effectively with scale parameters (of which bkmr has many). Additionally, no single diagnostic is perfect, leading many authors to advocate the use of multiple, complementary diagnostics. Thus, more formal diagnostics are helpful.
Gelman’s r-hat diagnostic gives an interpretable diagnostic: the expected reduction in the standard error of the posterior means if you could run the chains to an infinite size. These give some idea about when is a fine idea to stop sampling. There are rules of thumb about using r-hat to stop sampling that are available from several authors (for example you can consult the help files for rstan and coda).
Effective sample size is also useful - it estimates the amount of information in your chain, expressed in terms of the number of independent posterior samples it would take to match that information (e.g. if we could just sample from the posterior directly).
# Gelman's r-hat using coda estimator (will differ from rstan implementation)
gelman.diag(kmfit5coda)
## Potential scale reduction factors:
##
## Point est. Upper C.I.
## beta1 1.00 1.01
## beta2 1.00 1.00
## lambda 1.01 1.03
## r1 1.07 1.15
## r2 1.06 1.12
## r3 1.09 1.16
## r4 1.03 1.08
## r5 1.03 1.05
## sigsq.eps 1.00 1.00
##
## Multivariate psrf
##
## 1.04
# effective sample size
effectiveSize(kmfitcoda)
## beta1 beta2 lambda r1 r2 r3 r4
## 2411.61996 2865.78264 431.49111 87.11091 260.29419 328.45388 181.61903
## r5 sigsq.eps
## 123.06100 1719.70470
effectiveSize(kmfit5coda)
## beta1 beta2 lambda r1 r2 r3 r4 r5
## 2585.9305 3682.2160 429.5677 100.1964 189.3471 120.6366 167.5633 181.6534
## sigsq.eps
## 1393.7627
Example 3: Posterior summaries
Posterior kernel marginal densities, 1 chain
# posterior kernel marginal densities using `mcmc` and `mcmc` objects
densplot(kmfitcoda)
Posterior kernel marginal densities, multiple chains combined. Look for multiple modes that may indicate non-convergence of some chains
# posterior kernel marginal densities using `mcmc` and `mcmc` objects
densplot(kmfit5coda)
Other diagnostics from the coda package are available here.
Finally, the chains from the original kmbayes_parallel fit can be combined into a single chain (see the help files for how to deal with burn-in, the default in bkmr is to use the first half of the chain, which is respected here). The kmbayes_combine function smartly first combines the burn-in iterations and then combines the iterations after burnin, such that the burn-in rules of subsequent functions within the bkmr package are respected. Note that unlike the as.mcmc.list function, this function combines all iterations into a single chain, so trace plots will not be good diagnotistics in this combined object, and it should be used once one is assured that all chains have converged and the burn-in is acceptable.
With this combined set of samples, you can follow any of the post-processing functions from the bkmr functions, which are described here: https://jenfb.github.io/bkmr/overview.html. For example, see below the estimation of the posterior mean difference along a series of quantiles of all exposures in Z.
# posterior summaries using `mcmc` and `mcmc` objects
summary(kmfitcoda)
##
## Iterations = 1:4000
## Thinning interval = 1
## Number of chains = 1
## Sample size per chain = 4000
##
## 1. Empirical mean and standard deviation for each variable,
## plus standard error of the mean:
##
## Mean SD Naive SE Time-series SE
## beta1 1.98349 0.04392 0.0006945 0.0008944
## beta2 0.11999 0.08532 0.0013490 0.0015937
## lambda 11.18052 5.90540 0.0933726 0.2842911
## r1 0.03010 0.06385 0.0010095 0.0068408
## r2 0.03656 0.05690 0.0008997 0.0035270
## r3 0.02131 0.04065 0.0006427 0.0022429
## r4 0.02855 0.06641 0.0010500 0.0049278
## r5 0.02904 0.09543 0.0015089 0.0086023
## sigsq.eps 0.35283 0.08228 0.0013009 0.0019840
##
## 2. Quantiles for each variable:
##
## 2.5% 25% 50% 75% 97.5%
## beta1 1.90086 1.95474 1.98287 2.01177 2.07218
## beta2 -0.04513 0.06141 0.12001 0.17716 0.28713
## lambda 3.23941 7.01812 10.00979 14.12075 25.93332
## r1 0.01022 0.01232 0.01807 0.02767 0.09818
## r2 0.01018 0.01433 0.02172 0.04049 0.12353
## r3 0.01015 0.01180 0.01488 0.02198 0.05655
## r4 0.01040 0.01299 0.01670 0.02582 0.08533
## r5 0.01025 0.01219 0.01532 0.01951 0.07021
## sigsq.eps 0.22855 0.29302 0.34057 0.39833 0.54838
summary(kmfit5coda)
##
## Iterations = 1:1000
## Thinning interval = 1
## Number of chains = 4
## Sample size per chain = 1000
##
## 1. Empirical mean and standard deviation for each variable,
## plus standard error of the mean:
##
## Mean SD Naive SE Time-series SE
## beta1 1.98298 0.04612 0.0007293 0.0009553
## beta2 0.11772 0.08773 0.0013871 0.0014504
## lambda 11.40130 6.12844 0.0968991 0.2986803
## r1 0.08451 0.23939 0.0037851 0.0552047
## r2 0.04603 0.10130 0.0016017 0.0076008
## r3 0.04565 0.13176 0.0020833 0.0122476
## r4 0.04559 0.13140 0.0020776 0.0131784
## r5 0.04645 0.15955 0.0025227 0.0254687
## sigsq.eps 0.34965 0.08363 0.0013224 0.0023475
##
## 2. Quantiles for each variable:
##
## 2.5% 25% 50% 75% 97.5%
## beta1 1.89623 1.95258 1.98244 2.01235 2.0742
## beta2 -0.05300 0.05883 0.11761 0.17403 0.2905
## lambda 3.67891 7.12616 9.92129 14.40938 27.3759
## r1 0.01028 0.01329 0.02010 0.03206 1.0863
## r2 0.01046 0.01444 0.02281 0.04011 0.1913
## r3 0.01057 0.01260 0.01542 0.02247 0.4567
## r4 0.01017 0.01199 0.01658 0.02697 0.5292
## r5 0.01022 0.01208 0.01451 0.02095 0.6181
## sigsq.eps 0.22134 0.29111 0.33894 0.39542 0.5413
# highest posterior density intervals using `mcmc` and `mcmc` objects
HPDinterval(kmfitcoda)
## lower upper
## beta1 1.90086937 2.07231225
## beta2 -0.04051413 0.29012992
## lambda 2.68866745 22.62238656
## r1 0.01002094 0.06754473
## r2 0.01002597 0.09683241
## r3 0.01003504 0.04392585
## r4 0.01000511 0.06510500
## r5 0.01005733 0.04755863
## sigsq.eps 0.21411781 0.52229778
## attr(,"Probability")
## [1] 0.95
HPDinterval(kmfit5coda)
## [[1]]
## lower upper
## beta1 1.89901711 2.06836515
## beta2 -0.05823444 0.27896900
## lambda 3.49272910 24.19963619
## r1 0.01005653 0.24866062
## r2 0.01060567 0.10852257
## r3 0.01007904 0.11839418
## r4 0.01000154 0.09244707
## r5 0.01000340 0.03816433
## sigsq.eps 0.21686487 0.51812163
## attr(,"Probability")
## [1] 0.95
##
## [[2]]
## lower upper
## beta1 1.89943022 2.0753674
## beta2 -0.05153992 0.2882669
## lambda 3.43119031 23.1473823
## r1 0.01057793 1.2923038
## r2 0.01081145 0.1262868
## r3 0.01083266 0.1137877
## r4 0.01015433 0.3104132
## r5 0.01068435 0.0511815
## sigsq.eps 0.21825793 0.5414359
## attr(,"Probability")
## [1] 0.95
##
## [[3]]
## lower upper
## beta1 1.89832712 2.07555916
## beta2 -0.04869572 0.28362661
## lambda 3.48106754 23.29163500
## r1 0.01010213 0.08172045
## r2 0.01018210 0.12211655
## r3 0.01021310 0.14258896
## r4 0.01047863 0.06584484
## r5 0.01005888 0.03750793
## sigsq.eps 0.21408996 0.50909708
## attr(,"Probability")
## [1] 0.95
##
## [[4]]
## lower upper
## beta1 1.89141784 2.07007189
## beta2 -0.04577330 0.30009627
## lambda 3.90931112 24.33540839
## r1 0.01009055 0.07067172
## r2 0.01005591 0.08111086
## r3 0.01039775 0.07175424
## r4 0.01007359 0.06401300
## r5 0.01004367 0.98114725
## sigsq.eps 0.21926000 0.54019887
## attr(,"Probability")
## [1] 0.95
# combine multiple chains into a single chain
fitkmccomb = kmbayes_combine(kmfit5)
# For example:
summary(fitkmccomb)
## Fitted object of class 'bkmrfit'
## Iterations: 4000
## Outcome family: gaussian
## Model fit on: 2025-11-19 10:57:04.138257
## Running time: 1.18362 secs
##
## Acceptance rates for Metropolis-Hastings algorithm:
## param rate
## 1 lambda 0.4258565
## 2 r1 0.2245561
## 3 r2 0.2633158
## 4 r3 0.1647912
## 5 r4 0.1845461
## 6 r5 0.1312828
##
## Parameter estimates (based on iterations 2001-4000):
## param mean sd q_2.5 q_97.5
## 1 beta1 1.98139 0.04338 1.89747 2.06594
## 2 beta2 0.12214 0.08642 -0.04635 0.28978
## 3 sigsq.eps 0.35259 0.08181 0.22589 0.53979
## 4 r1 0.02356 0.01873 0.01028 0.07269
## 5 r2 0.03062 0.02677 0.01099 0.11466
## 6 r3 0.02344 0.02086 0.01103 0.09542
## 7 r4 0.02128 0.01811 0.01015 0.07353
## 8 r5 0.01655 0.01042 0.01000 0.04604
## 9 lambda 11.36624 5.98372 3.77510 26.86412
## NULL
mean.difference <- suppressWarnings(OverallRiskSummaries(fit = fitkmccomb, y = y, Z = Z, X = X,
qs = seq(0.25, 0.75, by = 0.05),
q.fixed = 0.5, method = "exact"))
mean.difference
## quantile est sd
## 1 0.25 -0.7171653 0.11869197
## 2 0.30 -0.5786133 0.09642004
## 3 0.35 -0.3865093 0.07915782
## 4 0.40 -0.2705121 0.04634351
## 5 0.45 -0.1507966 0.02668551
## 6 0.50 0.0000000 0.00000000
## 7 0.55 0.2162210 0.04196557
## 8 0.60 0.3341817 0.05137196
## 9 0.65 0.5160848 0.08286888
## 10 0.70 0.8825217 0.14407776
## 11 0.75 0.9750881 0.15359942
with(mean.difference, {
plot(quantile, est, pch=19, ylim=c(min(est - 1.96*sd), max(est + 1.96*sd)),
axes=FALSE, ylab= "Mean difference", xlab = "Joint quantile")
segments(x0=quantile, x1=quantile, y0 = est - 1.96*sd, y1 = est + 1.96*sd)
abline(h=0)
axis(1)
axis(2)
box(bty='l')
})
Example 4: diagnostics and inference when variable selection is used (Bayesian model averaging over the scale parameters of the kernel function)
These results parallel previous session and are given here without comment, other than to note that no fixed effects (X variables) are included, and that it is useful to check the posterior inclusion probabilities to ensure they are similar across chains.
set.seed(111)
system.time(kmfitbma.list <- suppressWarnings(kmbayes_parallel(nchains=4, y = y, Z = Z, X = X, iter = 1000, verbose = FALSE, varsel = TRUE)))
## Chain 1
## Iteration: 100 (10% completed; 0.04698 secs elapsed)
## Iteration: 200 (20% completed; 0.09187 secs elapsed)
## Iteration: 300 (30% completed; 0.13672 secs elapsed)
## Iteration: 400 (40% completed; 0.18241 secs elapsed)
## Iteration: 500 (50% completed; 0.23537 secs elapsed)
## Iteration: 600 (60% completed; 0.28523 secs elapsed)
## Iteration: 700 (70% completed; 0.33103 secs elapsed)
## Iteration: 800 (80% completed; 0.37859 secs elapsed)
## Iteration: 900 (90% completed; 0.42699 secs elapsed)
## Iteration: 1000 (100% completed; 0.47529 secs elapsed)
## Chain 2
## Iteration: 100 (10% completed; 0.0522 secs elapsed)
## Iteration: 200 (20% completed; 0.22663 secs elapsed)
## Iteration: 300 (30% completed; 0.27335 secs elapsed)
## Iteration: 400 (40% completed; 0.31918 secs elapsed)
## Iteration: 500 (50% completed; 0.36388 secs elapsed)
## Iteration: 600 (60% completed; 0.41142 secs elapsed)
## Iteration: 700 (70% completed; 0.45521 secs elapsed)
## Iteration: 800 (80% completed; 0.50126 secs elapsed)
## Iteration: 900 (90% completed; 0.54405 secs elapsed)
## Iteration: 1000 (100% completed; 0.58985 secs elapsed)
## Chain 3
## Iteration: 100 (10% completed; 0.05363 secs elapsed)
## Iteration: 200 (20% completed; 0.10388 secs elapsed)
## Iteration: 300 (30% completed; 0.15206 secs elapsed)
## Iteration: 400 (40% completed; 0.19918 secs elapsed)
## Iteration: 500 (50% completed; 0.24638 secs elapsed)
## Iteration: 600 (60% completed; 0.29031 secs elapsed)
## Iteration: 700 (70% completed; 0.33389 secs elapsed)
## Iteration: 800 (80% completed; 0.38077 secs elapsed)
## Iteration: 900 (90% completed; 0.42459 secs elapsed)
## Iteration: 1000 (100% completed; 0.47101 secs elapsed)
## Chain 4
## Iteration: 100 (10% completed; 0.05046 secs elapsed)
## Iteration: 200 (20% completed; 0.09532 secs elapsed)
## Iteration: 300 (30% completed; 0.14506 secs elapsed)
## Iteration: 400 (40% completed; 0.1892 secs elapsed)
## Iteration: 500 (50% completed; 0.23772 secs elapsed)
## Iteration: 600 (60% completed; 0.28319 secs elapsed)
## Iteration: 700 (70% completed; 0.34597 secs elapsed)
## Iteration: 800 (80% completed; 0.38828 secs elapsed)
## Iteration: 900 (90% completed; 0.43416 secs elapsed)
## Iteration: 1000 (100% completed; 0.4771 secs elapsed)
## user system elapsed
## 0.345 0.053 0.866
bmadiag = kmbayes_diagnose(kmfitbma.list)
## Parallel chains
## Inference for the input samples (4 chains: each with iter = 1000; warmup = 500):
##
## Q5 Q50 Q95 Mean SD Rhat Bulk_ESS Tail_ESS
## beta1 1.9 2.0 2.0 2.0 0.0 1.01 508 1735
## beta2 0.0 0.1 0.3 0.1 0.1 1.00 1634 1800
## lambda 5.3 11.8 31.3 14.2 8.6 1.03 105 111
## r1 0.0 0.0 0.1 0.0 0.0 1.13 35 54
## r2 0.0 0.0 0.1 0.0 0.0 1.18 20 40
## r3 0.0 0.0 0.0 0.0 0.0 1.02 130 100
## r4 0.0 0.0 0.1 0.0 0.0 1.06 53 33
## r5 0.0 0.0 0.0 0.0 0.0 1.03 140 78
## sigsq.eps 0.3 0.4 0.5 0.4 0.1 1.00 627 1494
##
## For each parameter, Bulk_ESS and Tail_ESS are crude measures of
## effective sample size for bulk and tail quantities respectively (an ESS > 100
## per chain is considered good), and Rhat is the potential scale reduction
## factor on rank normalized split chains (at convergence, Rhat <= 1.05).
# posterior exclusion probability of each chain
lapply(kmfitbma.list, function(x) t(ExtractPIPs(x)))
## [[1]]
## [,1] [,2] [,3] [,4] [,5]
## variable "z1" "z2" "z3" "z4" "z5"
## PIP "0.992" "0.764" "0.312" "0.732" "0.548"
##
## [[2]]
## [,1] [,2] [,3] [,4] [,5]
## variable "z1" "z2" "z3" "z4" "z5"
## PIP "0.698" "0.946" "0.436" "0.588" "0.552"
##
## [[3]]
## [,1] [,2] [,3] [,4] [,5]
## variable "z1" "z2" "z3" "z4" "z5"
## PIP "0.938" "0.590" "0.240" "0.766" "0.330"
##
## [[4]]
## [,1] [,2] [,3] [,4] [,5]
## variable "z1" "z2" "z3" "z4" "z5"
## PIP "0.730" "0.882" "0.342" "0.750" "0.492"
kmfitbma.comb = kmbayes_combine(kmfitbma.list)
summary(kmfitbma.comb)
## Fitted object of class 'bkmrfit'
## Iterations: 4000
## Outcome family: gaussian
## Model fit on: 2025-11-19 10:57:13.078278
## Running time: 0.47544 secs
##
## Acceptance rates for Metropolis-Hastings algorithm:
## param rate
## 1 lambda 0.4851213
## 2 r/delta (overall) 0.3308327
## 3 r/delta (move 1) 0.4426478
## 4 r/delta (move 2) 0.2216163
##
## Parameter estimates (based on iterations 2001-4000):
## param mean sd q_2.5 q_97.5
## 1 beta1 1.97474 0.04443 1.88866 2.06270
## 2 beta2 0.11984 0.08669 -0.05194 0.28878
## 3 sigsq.eps 0.37670 0.08844 0.24078 0.58243
## 4 r1 0.02127 0.01734 0.00000 0.06506
## 5 r2 0.02408 0.03050 0.00000 0.10093
## 6 r3 0.00541 0.00986 0.00000 0.02558
## 7 r4 0.02036 0.02397 0.00000 0.08360
## 8 r5 0.00869 0.01187 0.00000 0.03734
## 9 lambda 14.16323 8.61424 4.79547 39.76603
##
## Posterior inclusion probabilities:
## variable PIP
## 1 z1 0.8395
## 2 z2 0.7955
## 3 z3 0.3325
## 4 z4 0.7090
## 5 z5 0.4805
## NULL
ExtractPIPs(kmfitbma.comb) # posterior inclusion probabilities
## variable PIP
## 1 z1 0.8395
## 2 z2 0.7955
## 3 z3 0.3325
## 4 z4 0.7090
## 5 z5 0.4805
mean.difference2 <- suppressWarnings(OverallRiskSummaries(fit = kmfitbma.comb, y = y, Z = Z, X = X, qs = seq(0.25, 0.75, by = 0.05),
q.fixed = 0.5, method = "exact"))
mean.difference2
## quantile est sd
## 1 0.25 -0.6146968 0.12440724
## 2 0.30 -0.4915129 0.10102886
## 3 0.35 -0.3181185 0.08864045
## 4 0.40 -0.2356821 0.04858855
## 5 0.45 -0.1399487 0.02775610
## 6 0.50 0.0000000 0.00000000
## 7 0.55 0.1969654 0.04878284
## 8 0.60 0.2991031 0.06074668
## 9 0.65 0.4678363 0.09732989
## 10 0.70 0.8152414 0.17315362
## 11 0.75 0.8922859 0.17206280
with(mean.difference2, {
plot(quantile, est, pch=19, ylim=c(min(est - 1.96*sd), max(est + 1.96*sd)),
axes=FALSE, ylab= "Mean difference", xlab = "Joint quantile")
segments(x0=quantile, x1=quantile, y0 = est - 1.96*sd, y1 = est + 1.96*sd)
abline(h=0)
axis(1)
axis(2)
box(bty='l')
})
Example 5: Parallel posterior summaries as diagnostics
bkmrhat also has ported versions of the native posterior summarization functions to compare how these summaries vary across parallel chains. Note that these should serve as diagnostics, and final posterior inference should be done on the combined chain. The easiest of these functions to demonstrate is the OverallRiskSummaries_parallel function, which simply runs OverallRiskSummaries (from the bkmr package) on each chain and combines the results. Notably, this function fixes the y-axis at zero for the median, so it under-represents overall predictive variation across chains, but captures variation in effect estimates across the chains. Ideally, that variation is negligible - e.g. if you see differences between chains that would result in different interpretations, you should re-fit the model with more iterations. In this example, the results are reasonably consistent across chains, but one might want to run more iterations if, say, the differences seen across the upper error bounds are of such a magnitude as to be practically meaningful.
set.seed(111)
system.time(kmfitbma.list <- suppressWarnings(kmbayes_parallel(nchains=4, y = y, Z = Z, X = X, iter = 1000, verbose = FALSE, varsel = TRUE)))
## Chain 1
## Iteration: 100 (10% completed; 0.04437 secs elapsed)
## Iteration: 200 (20% completed; 0.08926 secs elapsed)
## Iteration: 300 (30% completed; 0.13463 secs elapsed)
## Iteration: 400 (40% completed; 0.18461 secs elapsed)
## Iteration: 500 (50% completed; 0.22756 secs elapsed)
## Iteration: 600 (60% completed; 0.27209 secs elapsed)
## Iteration: 700 (70% completed; 0.42183 secs elapsed)
## Iteration: 800 (80% completed; 0.47734 secs elapsed)
## Iteration: 900 (90% completed; 0.52534 secs elapsed)
## Iteration: 1000 (100% completed; 0.57117 secs elapsed)
## Chain 2
## Iteration: 100 (10% completed; 0.0453 secs elapsed)
## Iteration: 200 (20% completed; 0.09534 secs elapsed)
## Iteration: 300 (30% completed; 0.14516 secs elapsed)
## Iteration: 400 (40% completed; 0.19311 secs elapsed)
## Iteration: 500 (50% completed; 0.23682 secs elapsed)
## Iteration: 600 (60% completed; 0.40315 secs elapsed)
## Iteration: 700 (70% completed; 0.44757 secs elapsed)
## Iteration: 800 (80% completed; 0.49265 secs elapsed)
## Iteration: 900 (90% completed; 0.54188 secs elapsed)
## Iteration: 1000 (100% completed; 0.58522 secs elapsed)
## Chain 3
## Iteration: 100 (10% completed; 0.04957 secs elapsed)
## Iteration: 200 (20% completed; 0.09726 secs elapsed)
## Iteration: 300 (30% completed; 0.14525 secs elapsed)
## Iteration: 400 (40% completed; 0.32493 secs elapsed)
## Iteration: 500 (50% completed; 0.36995 secs elapsed)
## Iteration: 600 (60% completed; 0.41537 secs elapsed)
## Iteration: 700 (70% completed; 0.46255 secs elapsed)
## Iteration: 800 (80% completed; 0.50596 secs elapsed)
## Iteration: 900 (90% completed; 0.54883 secs elapsed)
## Iteration: 1000 (100% completed; 0.59305 secs elapsed)
## Chain 4
## Iteration: 100 (10% completed; 0.17464 secs elapsed)
## Iteration: 200 (20% completed; 0.22162 secs elapsed)
## Iteration: 300 (30% completed; 0.26513 secs elapsed)
## Iteration: 400 (40% completed; 0.3083 secs elapsed)
## Iteration: 500 (50% completed; 0.4671 secs elapsed)
## Iteration: 600 (60% completed; 0.51094 secs elapsed)
## Iteration: 700 (70% completed; 0.55433 secs elapsed)
## Iteration: 800 (80% completed; 0.59765 secs elapsed)
## Iteration: 900 (90% completed; 0.65502 secs elapsed)
## Iteration: 1000 (100% completed; 0.69891 secs elapsed)
## user system elapsed
## 0.349 0.051 1.092
meandifference_par = OverallRiskSummaries_parallel(kmfitbma.list, y = y, Z = Z, X = X ,qs = seq(0.25, 0.75, by = 0.05), q.fixed = 0.5, method = "exact")
## Chain 1
## Chain 2
## Chain 3
## Chain 4
head(meandifference_par)
## quantile est sd chain
## 1 0.25 -0.6191983 0.12051657 1
## 2 0.30 -0.5019924 0.09747662 1
## 3 0.35 -0.3067245 0.08058220 1
## 4 0.40 -0.2311694 0.04357326 1
## 5 0.45 -0.1454666 0.02463054 1
## 6 0.50 0.0000000 0.00000000 1
nchains = length(unique(meandifference_par$chain))
with(meandifference_par, {
plot.new()
plot.window(ylim=c(min(est - 1.96*sd), max(est + 1.96*sd)),
xlim=c(min(quantile), max(quantile)),
ylab= "Mean difference", xlab = "Joint quantile")
for(cch in seq_len(nchains)){
width = diff(quantile)[1]
jit = runif(1, -width/5, width/5)
points(jit+quantile[chain==cch], est[chain==cch], pch=19, col=cch)
segments(x0=jit+quantile[chain==cch], x1=jit+quantile[chain==cch], y0 = est[chain==cch] - 1.96*sd[chain==cch], y1 = est[chain==cch] + 1.96*sd[chain==cch], col=cch)
}
abline(h=0)
axis(1)
axis(2)
box(bty='l')
legend("bottom", col=1:nchains, pch=19, lty=1, legend=paste("chain", 1:nchains), bty="n")
})
regfuns_par = PredictorResponseUnivar_parallel(kmfitbma.list, y = y, Z = Z, X = X ,qs = seq(0.25, 0.75, by = 0.05), q.fixed = 0.5, method = "exact")
## Chain 1
## Chain 2
## Chain 3
## Chain 4
head(regfuns_par)
## variable z est se chain
## 1 z1 -2.186199 -1.435648 0.5915233 1
## 2 z1 -2.082261 -1.379339 0.5677579 1
## 3 z1 -1.978323 -1.321802 0.5442158 1
## 4 z1 -1.874385 -1.263092 0.5208719 1
## 5 z1 -1.770446 -1.203272 0.4977049 1
## 6 z1 -1.666508 -1.142409 0.4746991 1
nchains = length(unique(meandifference_par$chain))
# single variable
with(regfuns_par[regfuns_par$variable=="z1",], {
plot.new()
plot.window(ylim=c(min(est - 1.96*se), max(est + 1.96*se)),
xlim=c(min(z), max(z)),
ylab= "Predicted Y", xlab = "Z")
pc = c("#000000", "#E69F00", "#56B4E9", "#009E73", "#F0E442", "#0072B2", "#D55E00", "#CC79A7", "#999999")
pc2 = c("#0000001A", "#E69F001A", "#56B4E91A", "#009E731A", "#F0E4421A", "#0072B21A", "#D55E001A", "#CC79A71A", "#9999991A")
for(cch in seq_len(nchains)){
ribbonX = c(z[chain==cch], rev(z[chain==cch]))
ribbonY = c(est[chain==cch] + 1.96*se[chain==cch], rev(est[chain==cch] - 1.96*se[chain==cch]))
polygon(x=ribbonX, y = ribbonY, col=pc2[cch], border=NA)
lines(z[chain==cch], est[chain==cch], pch=19, col=pc[cch])
}
axis(1)
axis(2)
box(bty='l')
legend("bottom", col=1:nchains, pch=19, lty=1, legend=paste("chain", 1:nchains), bty="n")
})
Example 6: Continuing a fit
Sometimes you just need to run more samples in an existing chain. For example, you run a bkmr fit for 3 days, only to find you don’t have enough samples. A “continued” fit just means that you can start off at the last iteration you were at and just keep building on an existing set of results by lengthening the Markov chain. Unfortunately, due to how the kmbayes function accepts starting values (for the official install version), you can’t quite do this exactly in many cases (The function will relay a message and possible solutions, if any. bkmr package authors are aware of this issue). The kmbayes_continue function continues a bkmr fit as well as the bkmr package will allow. The r parameters from the fit must all be initialized at the same value, so kmbayes_continue starts a new MCMC fit at the final values of all parameters from the prior bkmr fit, but sets all of the r parameters to the mean at the last iteration from the previous fit. Additionally, if h.hat parameters are estimated, these are fixed to be above zero to meet similar constraints, either by fixing them at their posterior mean or setting to a small positive value. One should inspect trace plots to see whether this will cause issues (e.g. if the traceplots demonstrate different patterns in the samples before and after the continuation). Here’s an example with a quick check of diagnostics of the first part of the chain, and the combined chain (which could be used for inference or extended again, if necessary). We caution users that this function creates 2 distinct, if very similar Markov chains, and to use appropriate caution if traceplots differ before and after each continuation. Nonetheless, in many cases one can act as though all samples are from the same Markov chain.
Note that if you install the developmental version of the bkmr package you can continue fits from exactly where they left off, so you get a true, single Markov chain. You can install that via the commented code below
# install dev version of bkmr to allow true continued fits.
#install.packages("devtools")
#devtools::install_github("jenfb/bkmr")
set.seed(111)
# run 100 initial iterations for a model with only 2 exposures
Z2 = Z[,1:2]
kmfitbma.start <- suppressWarnings(kmbayes(y = y, Z = Z2, X = X, iter = 500, verbose = FALSE, varsel = FALSE))
## Iteration: 50 (10% completed; 0.02397 secs elapsed)
## Iteration: 100 (20% completed; 0.05547 secs elapsed)
## Iteration: 150 (30% completed; 0.08043 secs elapsed)
## Iteration: 200 (40% completed; 0.10795 secs elapsed)
## Iteration: 250 (50% completed; 0.13285 secs elapsed)
## Iteration: 300 (60% completed; 0.16364 secs elapsed)
## Iteration: 350 (70% completed; 0.18805 secs elapsed)
## Iteration: 400 (80% completed; 0.21684 secs elapsed)
## Iteration: 450 (90% completed; 0.24108 secs elapsed)
## Iteration: 500 (100% completed; 0.27007 secs elapsed)
kmbayes_diag(kmfitbma.start)
## Single chain
## Inference for the input samples (1 chains: each with iter = 500; warmup = 250):
##
## Q5 Q50 Q95 Mean SD Rhat Bulk_ESS Tail_ESS
## beta1 1.9 2.0 2.0 2.0 0.0 1.02 184 238
## beta2 0.0 0.1 0.2 0.1 0.1 1.00 259 197
## lambda 6.4 15.0 33.2 17.0 8.2 1.03 32 38
## r1 0.0 0.0 0.0 0.0 0.0 1.20 5 16
## r2 0.0 0.0 0.1 0.0 0.0 1.01 26 28
## sigsq.eps 0.3 0.4 0.5 0.4 0.1 1.00 195 157
##
## For each parameter, Bulk_ESS and Tail_ESS are crude measures of
## effective sample size for bulk and tail quantities respectively (an ESS > 100
## per chain is considered good), and Rhat is the potential scale reduction
## factor on rank normalized split chains (at convergence, Rhat <= 1.05).
## mean se_mean sd 2.5% 25%
## beta1 1.96002469 0.002996717 0.04071499 1.88732666 1.93049212
## beta2 0.09934846 0.005601868 0.09100445 -0.06425935 0.03866907
## lambda 17.03096186 1.480162990 8.16066439 5.99261263 10.62650587
## r1 0.02445179 0.006963033 0.01503557 0.01009566 0.01650405
## r2 0.02931652 0.003839555 0.02294888 0.01050284 0.01300914
## sigsq.eps 0.38177766 0.006062528 0.08483668 0.25337749 0.32965181
## 50% 75% 97.5% n_eff Rhat valid Q5
## beta1 1.96168526 1.98761915 2.04042002 192 1.0174270 1 1.89496869
## beta2 0.09494353 0.15892943 0.26004214 263 0.9979147 1 -0.04566104
## lambda 14.97716478 22.17297877 35.85230190 32 1.0322926 1 6.39370841
## r1 0.02078562 0.03098429 0.04827171 12 1.1969882 1 0.01009566
## r2 0.02369039 0.03766762 0.09155390 38 1.0072685 1 0.01213386
## sigsq.eps 0.36834066 0.42589940 0.61832629 187 0.9989252 1 0.26493273
## Q50 Q95 MCSE_Q2.5 MCSE_Q25 MCSE_Q50
## beta1 1.96168526 2.02239974 0.0061439984 0.003896226 0.004052288
## beta2 0.09494353 0.24851228 0.0125216874 0.009226277 0.006555085
## lambda 14.97716478 33.24311222 1.2308679562 0.873763048 1.767460797
## r1 0.02078562 0.04809447 0.0006957382 0.002532962 0.004430201
## r2 0.02369039 0.07473096 0.0011501569 0.001816037 0.004755380
## sigsq.eps 0.36834066 0.53371298 0.0112954790 0.004960890 0.004978161
## MCSE_Q75 MCSE_Q97.5 MCSE_SD Bulk_ESS Tail_ESS
## beta1 0.001302716 0.010539232 0.002122356 184 238
## beta2 0.008043386 0.008643991 0.003965505 259 197
## lambda 3.451601827 1.237140963 1.056843473 32 38
## r1 0.006131920 0.001462048 0.005269340 5 16
## r2 0.002075801 0.013178244 0.002737455 26 28
## sigsq.eps 0.006630407 0.039435542 0.004293256 195 157
# run 2000 additional iterations
moreiterations = kmbayes_continue(kmfitbma.start, iter=2000)
## Validating control.params...
## Validating starting.values...
## Iteration: 201 (10% completed; 0.12364 secs elapsed)
## Acceptance rates for Metropolis-Hastings algorithm:
## param rate
## 1 lambda 0.430
## 2 r1 0.245
## 3 r2 0.235
## Iteration: 401 (20% completed; 0.24282 secs elapsed)
## Acceptance rates for Metropolis-Hastings algorithm:
## param rate
## 1 lambda 0.4675
## 2 r1 0.1850
## 3 r2 0.2500
## Iteration: 601 (30% completed; 0.36783 secs elapsed)
## Acceptance rates for Metropolis-Hastings algorithm:
## param rate
## 1 lambda 0.4966667
## 2 r1 0.1633333
## 3 r2 0.2333333
## Iteration: 801 (40% completed; 0.4825 secs elapsed)
## Acceptance rates for Metropolis-Hastings algorithm:
## param rate
## 1 lambda 0.49625
## 2 r1 0.18500
## 3 r2 0.25250
## Iteration: 1001 (50% completed; 0.59947 secs elapsed)
## Acceptance rates for Metropolis-Hastings algorithm:
## param rate
## 1 lambda 0.489
## 2 r1 0.192
## 3 r2 0.243
## Iteration: 1201 (60% completed; 0.72636 secs elapsed)
## Acceptance rates for Metropolis-Hastings algorithm:
## param rate
## 1 lambda 0.4908333
## 2 r1 0.1900000
## 3 r2 0.2416667
## Iteration: 1401 (70% completed; 0.84681 secs elapsed)
## Acceptance rates for Metropolis-Hastings algorithm:
## param rate
## 1 lambda 0.4921429
## 2 r1 0.1928571
## 3 r2 0.2457143
## Iteration: 1601 (80% completed; 0.9701 secs elapsed)
## Acceptance rates for Metropolis-Hastings algorithm:
## param rate
## 1 lambda 0.490625
## 2 r1 0.199375
## 3 r2 0.246250
## Iteration: 1801 (90% completed; 1.08822 secs elapsed)
## Acceptance rates for Metropolis-Hastings algorithm:
## param rate
## 1 lambda 0.4888889
## 2 r1 0.1983333
## 3 r2 0.2516667
## Iteration: 2001 (100% completed; 1.20897 secs elapsed)
## Acceptance rates for Metropolis-Hastings algorithm:
## param rate
## 1 lambda 0.4940
## 2 r1 0.2005
## 3 r2 0.2485
kmbayes_diag(moreiterations)
## Single chain
## Inference for the input samples (1 chains: each with iter = 2500; warmup = 1250):
##
## Q5 Q50 Q95 Mean SD Rhat Bulk_ESS Tail_ESS
## beta1 1.9 2.0 2.0 2.0 0.0 1 1226 1188
## beta2 0.0 0.1 0.2 0.1 0.1 1 1342 1153
## lambda 4.8 11.6 27.8 13.5 7.2 1 132 118
## r1 0.0 0.0 0.1 0.0 0.0 1 36 60
## r2 0.0 0.0 0.1 0.0 0.0 1 63 74
## sigsq.eps 0.3 0.4 0.5 0.4 0.1 1 687 907
##
## For each parameter, Bulk_ESS and Tail_ESS are crude measures of
## effective sample size for bulk and tail quantities respectively (an ESS > 100
## per chain is considered good), and Rhat is the potential scale reduction
## factor on rank normalized split chains (at convergence, Rhat <= 1.05).
## mean se_mean sd 2.5% 25% 50%
## beta1 1.95902186 0.001313130 0.04611042 1.86406579 1.92824507 1.95995430
## beta2 0.10364082 0.002472674 0.09078254 -0.08028629 0.04483808 0.10217365
## lambda 13.45858048 0.575042840 7.18975272 4.63158214 8.06723496 11.62070182
## r1 0.02558935 0.002505863 0.01684645 0.01051616 0.01389383 0.02000374
## r2 0.03152169 0.002840685 0.02568259 0.01042071 0.01378317 0.02173483
## sigsq.eps 0.39671746 0.003232309 0.08256885 0.26948029 0.33468348 0.38833815
## 75% 97.5% n_eff Rhat valid Q5 Q50
## beta1 1.99013424 2.05059125 1236 1.000076 1 1.88233558 1.95995430
## beta2 0.16648882 0.27919590 1353 1.001507 1 -0.04502216 0.10217365
## lambda 17.07843757 32.36093597 156 1.000211 1 4.81793593 11.62070182
## r1 0.03034907 0.07050527 45 1.004470 1 0.01097881 0.02000374
## r2 0.03773703 0.10918759 80 1.001841 1 0.01069795 0.02173483
## sigsq.eps 0.44446826 0.59030414 650 1.000453 1 0.28159002 0.38833815
## Q95 MCSE_Q2.5 MCSE_Q25 MCSE_Q50 MCSE_Q75
## beta1 2.03392196 0.0034878092 0.0018218271 0.001705603 0.001924244
## beta2 0.24728033 0.0063559462 0.0036553685 0.002849542 0.003241571
## lambda 27.83773681 0.2173104821 0.6064432977 0.646607402 0.583887847
## r1 0.06248365 0.0004568228 0.0017076967 0.002442041 0.003929030
## r2 0.07985574 0.0002483599 0.0009401994 0.002913607 0.004161742
## sigsq.eps 0.54458362 0.0028952538 0.0027590593 0.003419585 0.004708623
## MCSE_Q97.5 MCSE_SD Bulk_ESS Tail_ESS
## beta1 0.004277831 0.0009293335 1226 1188
## beta2 0.009449827 0.0017774270 1342 1153
## lambda 1.708604468 0.4073777680 132 118
## r1 0.005777434 0.0017834748 36 60
## r2 0.011995755 0.0020158792 63 74
## sigsq.eps 0.007480563 0.0023104526 687 907
TracePlot(moreiterations, par="beta")
TracePlot(moreiterations, par="r")
# move to sequential processing (end of vignette)
future::plan(strategy = future::sequential)
Acknowledgments
Thanks to Haotian “Howie” Wu for invaluable feedback on early versions of the package.