bkmr and bkmrhatbkmr 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.
bkmrhat packagekmbaryes function from bkmr, or use multiple parallel chains kmbayes_parallel from bkmrhatkmbayes_diagnose function (uses functions from the rstan package) OR convert the BKMR fit(s) to mcmc (one chain) or mcmc.list (multiple chains) objects from the coda package using as.mcmc or as.mcmc.list from the bkmrhat package. The coda package has a whole host of inference and diagnostic procedures (but may lag behind some of the diagnostics functions from rstan).coda functions or combine chains from a kmbayes_parallel fit using kmbayes_combine. Final posterior inferences can be made on the combined object, which enables use of bkmr package functions for visual summaries of independent and joint effects of exposures in the bkmr model.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
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
## 4.259 0.264 4.531
# 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
## 0.042 0.003 2.108
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 1951 1652
## beta2 0.0 0.1 0.3 0.1 0.1 1.00 3204 2578
## lambda 4.5 9.7 24.0 11.4 6.5 1.01 359 510
## r1 0.0 0.0 0.1 0.1 0.2 1.02 133 66
## r2 0.0 0.0 0.2 0.1 0.2 1.02 116 71
## r3 0.0 0.0 0.1 0.0 0.2 1.02 87 92
## r4 0.0 0.0 0.1 0.0 0.1 1.03 119 78
## r5 0.0 0.0 0.5 0.1 0.2 1.07 49 44
## sigsq.eps 0.2 0.3 0.5 0.3 0.1 1.01 655 431
##
## 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)
