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 
##  15.016  27.562  10.995

# 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 
##   0.713   0.034  15.309

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  2.0  2.0 0.0  1.00     2665     2969
## beta2     0.0  0.1  0.3  0.1 0.1  1.00     3041     3693
## lambda    4.3 11.1 27.6 13.1 8.2  1.01      259      196
## r1        0.0  0.0  0.1  0.0 0.1  1.00      170      162
## r2        0.0  0.0  0.1  0.0 0.1  1.00      310      244
## r3        0.0  0.0  0.0  0.0 0.0  1.00      155      128
## r4        0.0  0.0  0.1  0.0 0.1  1.00      170      155
## r5        0.0  0.0  0.0  0.0 0.1  1.01       90      131
## sigsq.eps 0.2  0.3  0.5  0.4 0.1  1.00     1326     1631
## 
## 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  2.0  2.0 0.0  1.00     2367     1711
## beta2     0.0  0.1  0.3  0.1 0.1  1.00     2009     1954
## lambda    4.3 10.1 24.4 11.8 7.0  1.02      207      102
## r1        0.0  0.0  0.1  0.1 0.2  1.02      128       67
## r2        0.0  0.0  0.2  0.1 0.2  1.03      149       56
## r3        0.0  0.0  0.1  0.0 0.2  1.04       78       64
## r4        0.0  0.0  0.1  0.0 0.1  1.02       98      145
## r5        0.0  0.0  0.2  0.1 0.2  1.03       99       53
## sigsq.eps 0.2  0.3  0.5  0.4 0.1  1.01      439      407
## 
## 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      1.00000000  0.07733726  0.020816066  0.15791108  0.14489557
## beta2      0.07733726  1.00000000 -0.059673636 -0.11479475 -0.12891028
## lambda     0.02081607 -0.05967364  1.000000000  0.04963909  0.04178173
## r1         0.15791108 -0.11479475  0.049639091  1.00000000  0.79825315
## r2         0.14489557 -0.12891028  0.041781725  0.79825315  1.00000000
## r3         0.15952865 -0.08849151  0.006925921  0.50841621  0.68184307
## r4         0.23091705 -0.06415646 -0.002732878  0.67347750  0.71342585
## r5         0.15971059 -0.10964670  0.043259501  0.75711064  0.93661034
## sigsq.eps -0.03541153  0.07444138 -0.327970696 -0.23171096 -0.23704523
##                     r3           r4         r5   sigsq.eps
## beta1      0.159528646  0.230917049  0.1597106 -0.03541153
## beta2     -0.088491506 -0.064156458 -0.1096467  0.07444138
## lambda     0.006925921 -0.002732878  0.0432595 -0.32797070
## r1         0.508416212  0.673477505  0.7571106 -0.23171096
## r2         0.681843070  0.713425851  0.9366103 -0.23704523
## r3         1.000000000  0.689998376  0.7737409 -0.10128083
## r4         0.689998376  1.000000000  0.7600625 -0.13636196
## r5         0.773740905  0.760062544  1.0000000 -0.21093636
## sigsq.eps -0.101280830 -0.136361955 -0.2109364  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       1.00
## beta2           1.00       1.00
## lambda          1.05       1.13
## r1              1.02       1.03
## r2              1.05       1.09
## r3              1.07       1.15
## r4              1.07       1.13
## r5              1.05       1.11
## sigsq.eps       1.00       1.01
## 
## Multivariate psrf
## 
## 1.07

# effective sample size
effectiveSize(kmfitcoda)

##     beta1     beta2    lambda        r1        r2        r3        r4        r5 
## 2892.7570 3198.3261  202.6782  118.1804  284.6426  295.7300  146.9502  115.5348 
## sigsq.eps 
## 1242.4771

effectiveSize(kmfit5coda)

##     beta1     beta2    lambda        r1        r2        r3        r4        r5 
## 2143.5876 2880.0101  318.0519  123.0036  119.2547  136.5004  193.8485  105.6125 
## sigsq.eps 
##  850.5044

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      1.97357 0.04598 0.0007270      0.0008549
## beta2      0.13518 0.08710 0.0013771      0.0015401
## lambda    13.08699 8.23403 0.1301915      0.5783746
## r1         0.02829 0.06333 0.0010013      0.0058256
## r2         0.03552 0.05200 0.0008222      0.0030823
## r3         0.02140 0.04096 0.0006476      0.0023818
## r4         0.02931 0.06623 0.0010472      0.0054636
## r5         0.03055 0.10618 0.0016788      0.0098783
## sigsq.eps  0.35970 0.08495 0.0013432      0.0024101
## 
## 2. Quantiles for each variable:
## 
##               2.5%     25%      50%      75%    97.5%
## beta1      1.88651 1.94226  1.97310  2.00380  2.06407
## beta2     -0.03357 0.07694  0.13345  0.19279  0.30592
## lambda     3.95746 7.68835 11.09318 16.25233 33.16143
## r1         0.01031 0.01229  0.01676  0.02568  0.08740
## r2         0.01056 0.01474  0.02290  0.04175  0.10979
## r3         0.01025 0.01194  0.01437  0.02179  0.06415
## r4         0.01021 0.01286  0.01763  0.02860  0.07571
## r5         0.01014 0.01198  0.01451  0.02031  0.06762
## sigsq.eps  0.23236 0.29752  0.34779  0.40793  0.55917

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      1.97220 0.04757 0.0007521       0.001134
## beta2      0.12948 0.09183 0.0014520       0.001740
## lambda    11.76360 7.02139 0.1110180       0.410121
## r1         0.05423 0.16378 0.0025897       0.018419
## r2         0.07722 0.18602 0.0029412       0.021687
## r3         0.04928 0.16250 0.0025694       0.016215
## r4         0.04291 0.11688 0.0018480       0.008879
## r5         0.05352 0.16240 0.0025678       0.018949
## sigsq.eps  0.36171 0.09018 0.0014258       0.003302
## 
## 2. Quantiles for each variable:
## 
##               2.5%     25%      50%      75%   97.5%
## beta1      1.88251 1.94143  1.97146  2.00155  2.0671
## beta2     -0.05827 0.07128  0.13124  0.19054  0.3044
## lambda     3.25664 7.11622 10.11615 14.53353 30.3804
## r1         0.01010 0.01223  0.01707  0.02712  0.7265
## r2         0.01084 0.01617  0.02846  0.05229  0.8885
## r3         0.01035 0.01198  0.01473  0.02104  0.6824
## r4         0.01025 0.01270  0.01763  0.02984  0.2784
## r5         0.01025 0.01159  0.01487  0.02220  0.7232
## sigsq.eps  0.21768 0.29936  0.34981  0.41255  0.5662

# highest posterior density intervals using `mcmc` and `mcmc` objects
HPDinterval(kmfitcoda)

##                 lower       upper
## beta1      1.88272469  2.05913505
## beta2     -0.03379651  0.30462773
## lambda     2.62734881 27.66498240
## r1         0.01003506  0.05828581
## r2         0.01000265  0.08925073
## r3         0.01009384  0.04319179
## r4         0.01001438  0.06122670
## r5         0.01003380  0.03968790
## sigsq.eps  0.21094543  0.52732628
## attr(,"Probability")
## [1] 0.95

HPDinterval(kmfit5coda)

## [[1]]
##                 lower       upper
## beta1      1.87881466  2.05640734
## beta2     -0.06790542  0.29408811
## lambda     4.03387403 25.18297025
## r1         0.01049732  0.45345607
## r2         0.01013077  0.20479069
## r3         0.01007166  0.07531484
## r4         0.01027842  0.08344169
## r5         0.01044085  0.35237108
## sigsq.eps  0.20873868  0.55829021
## attr(,"Probability")
## [1] 0.95
## 
## [[2]]
##                 lower       upper
## beta1      1.89301316  2.06171257
## beta2     -0.03168293  0.33174439
## lambda     2.16371834 22.42148597
## r1         0.01010240  0.07291425
## r2         0.01031875  0.14093974
## r3         0.01014711  0.06031589
## r4         0.01007560  0.12774120
## r5         0.01031048  0.04156094
## sigsq.eps  0.22080250  0.56493912
## attr(,"Probability")
## [1] 0.95
## 
## [[3]]
##                 lower       upper
## beta1      1.86689907  2.06291839
## beta2     -0.04088964  0.31660729
## lambda     3.15093841 25.77385790
## r1         0.01006494  0.05443499
## r2         0.01024398  0.44847169
## r3         0.01005822  0.91002697
## r4         0.01010409  0.09978123
## r5         0.01006096  0.44352639
## sigsq.eps  0.18776118  0.52538413
## attr(,"Probability")
## [1] 0.95
## 
## [[4]]
##                 lower       upper
## beta1      1.87431092  2.06061875
## beta2     -0.04925949  0.30091574
## lambda     3.25664492 24.56205553
## r1         0.01002125  0.60943465
## r2         0.01067705  0.80759889
## r3         0.01009028  0.06928882
## r4         0.01013029  0.09261106
## r5         0.01009554  0.53367948
## sigsq.eps  0.20121944  0.52602621
## 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: 2025-03-09 04:12:30.708982 
## Running time:  13.39785 secs 
## 
## Acceptance rates for Metropolis-Hastings algorithm:
##    param      rate
## 1 lambda 0.4161040
## 2     r1 0.1897974
## 3     r2 0.3438360
## 4     r3 0.1472868
## 5     r4 0.1865466
## 6     r5 0.1370343
## 
## Parameter estimates (based on iterations 2001-4000):
##       param     mean      sd    q_2.5   q_97.5
## 1     beta1  1.97170 0.04505  1.88241  2.06101
## 2     beta2  0.13308 0.08748 -0.04784  0.30174
## 3 sigsq.eps  0.36394 0.08443  0.22955  0.55831
## 4        r1  0.02266 0.01794  0.01041  0.06867
## 5        r2  0.03716 0.03347  0.01032  0.13517
## 6        r3  0.01822 0.01304  0.01019  0.05190
## 7        r4  0.02410 0.02436  0.01025  0.08196
## 8        r5  0.01802 0.01052  0.01025  0.04156
## 9    lambda 11.84660 6.34991  3.82256 28.37833

## 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 -0.43409703 0.09329187
## 2      0.30 -0.37823674 0.06917024
## 3      0.35 -0.18115927 0.04230407
## 4      0.40 -0.14181918 0.03513813
## 5      0.45 -0.06350457 0.03092162
## 6      0.50  0.00000000 0.00000000
## 7      0.55  0.17379452 0.05137174
## 8      0.60  0.31492642 0.07284092
## 9      0.65  0.58316326 0.11592165
## 10     0.70  0.71234910 0.13873256
## 11     0.75  0.85574860 0.17996696

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 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; 0.20548 secs elapsed)

## Iteration: 200 (20% completed; 0.76547 secs elapsed)

## Iteration: 300 (30% completed; 1.3945 secs elapsed)

## Iteration: 400 (40% completed; 1.99192 secs elapsed)

## Iteration: 500 (50% completed; 2.65605 secs elapsed)

## Iteration: 600 (60% completed; 3.2037 secs elapsed)

## Iteration: 700 (70% completed; 3.80628 secs elapsed)

## Iteration: 800 (80% completed; 4.37069 secs elapsed)

## Iteration: 900 (90% completed; 5.00516 secs elapsed)

## Iteration: 1000 (100% completed; 5.56893 secs elapsed)

## Chain 2

## Iteration: 100 (10% completed; 0.3113 secs elapsed)

## Iteration: 200 (20% completed; 0.85779 secs elapsed)

## Iteration: 300 (30% completed; 1.53719 secs elapsed)

## Iteration: 400 (40% completed; 2.13433 secs elapsed)

## Iteration: 500 (50% completed; 2.6291 secs elapsed)

## Iteration: 600 (60% completed; 3.17072 secs elapsed)

## Iteration: 700 (70% completed; 3.74548 secs elapsed)

## Iteration: 800 (80% completed; 4.76075 secs elapsed)

## Iteration: 900 (90% completed; 5.23622 secs elapsed)

## Iteration: 1000 (100% completed; 5.90344 secs elapsed)

## Chain 3

## Iteration: 100 (10% completed; 0.53021 secs elapsed)

## Iteration: 200 (20% completed; 1.17362 secs elapsed)

## Iteration: 300 (30% completed; 1.87651 secs elapsed)

## Iteration: 400 (40% completed; 2.46672 secs elapsed)

## Iteration: 500 (50% completed; 3.00859 secs elapsed)

## Iteration: 600 (60% completed; 3.6063 secs elapsed)

## Iteration: 700 (70% completed; 4.16999 secs elapsed)

## Iteration: 800 (80% completed; 4.73244 secs elapsed)

## Iteration: 900 (90% completed; 5.32029 secs elapsed)

## Iteration: 1000 (100% completed; 6.02845 secs elapsed)

## Chain 4

## Iteration: 100 (10% completed; 0.60657 secs elapsed)

## Iteration: 200 (20% completed; 1.61362 secs elapsed)

## Iteration: 300 (30% completed; 2.27458 secs elapsed)

## Iteration: 400 (40% completed; 2.81039 secs elapsed)

## Iteration: 500 (50% completed; 3.39985 secs elapsed)

## Iteration: 600 (60% completed; 3.96603 secs elapsed)

## Iteration: 700 (70% completed; 4.68908 secs elapsed)

## Iteration: 800 (80% completed; 5.25211 secs elapsed)

## Iteration: 900 (90% completed; 5.61278 secs elapsed)

## Iteration: 1000 (100% completed; 5.86945 secs elapsed)

##    user  system elapsed 
##   0.383   0.011   6.425

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 -0.43399519 0.09677667     1
## 2     0.30 -0.36352052 0.07236802     1
## 3     0.35 -0.17642660 0.04594553     1
## 4     0.40 -0.14811018 0.03930450     1
## 5     0.45 -0.04764707 0.03047827     1
## 6     0.50  0.00000000 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 -2.159373 -0.3297703 0.5129312     1
## 2       z1 -2.048986 -0.3141977 0.4904068     1
## 3       z1 -1.938600 -0.2985819 0.4680000     1
## 4       z1 -1.828214 -0.2829365 0.4457449     1
## 5       z1 -1.717827 -0.2672755 0.4236794     1
## 6       z1 -1.607441 -0.2516131 0.4018457     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; 0.05753 secs elapsed)

## Iteration: 100 (20% completed; 0.12966 secs elapsed)

## Iteration: 150 (30% completed; 0.18762 secs elapsed)

## Iteration: 200 (40% completed; 0.25323 secs elapsed)

## Iteration: 250 (50% completed; 0.31158 secs elapsed)

## Iteration: 300 (60% completed; 0.37598 secs elapsed)

## Iteration: 350 (70% completed; 0.44495 secs elapsed)

## Iteration: 400 (80% completed; 0.53054 secs elapsed)

## Iteration: 450 (90% completed; 0.58863 secs elapsed)

## Iteration: 500 (100% completed; 0.65308 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  1.9  2.0  1.9 0.0  1.00      279      217
## beta2     -0.1  0.1  0.3  0.1 0.1  1.00      238      206
## lambda     4.6 13.0 31.3 14.9 7.8  1.08       13       16
## r1         0.0  0.0  0.0  0.0 0.0  1.17        5        7
## r2         0.0  0.0  0.1  0.1 0.0  1.05       10       16
## sigsq.eps  0.3  0.4  0.6  0.4 0.1  1.00       87      219
## 
## 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      1.94302414 0.002744458 0.04643461  1.84782583 1.91330142  1.94655854
## beta2      0.10483405 0.006191034 0.09620251 -0.07303313 0.04304370  0.10467199
## lambda    14.90060019 1.869438290 7.81070475  4.60230691 8.65519030 13.04053754
## r1         0.01476940 0.002347690 0.00449794  0.01078574 0.01235942  0.01320025
## r2         0.05291092 0.009623519 0.04332210  0.01010751 0.01924294  0.03815578
## sigsq.eps  0.41328122 0.010095801 0.09214694  0.27259916 0.35051490  0.39841682
##                   75%       97.5% n_eff     Rhat valid          Q5         Q50
## beta1      1.97193285  2.02693739   282 1.004931     1  1.86400517  1.94655854
## beta2      0.16804699  0.27700828   236 0.997853     1 -0.05155040  0.10467199
## lambda    18.65686378 34.19469565    18 1.084344     1  4.60230691 13.04053754
## r1         0.01703200  0.02877183     9 1.167807     1  0.01078574  0.01320025
## r2         0.07782864  0.15639005    21 1.052302     1  0.01010751  0.03815578
## sigsq.eps  0.46665521  0.62932750    94 1.002317     1  0.28064363  0.39841682
##                   Q95    MCSE_Q2.5     MCSE_Q25    MCSE_Q50    MCSE_Q75
## beta1      2.01504706 0.0098237593 0.0048432127 0.003083228 0.004034983
## beta2      0.26110337 0.0166696373 0.0093358891 0.007387626 0.010829577
## lambda    31.32637623 1.1971720562 2.0194134613 2.532679950 1.535664875
## r1         0.02153674 0.0002641772 0.0009430751 0.001429907 0.004168245
## r2         0.14102786 0.0028040064 0.0071858659 0.011789241 0.013548261
## sigsq.eps  0.60258149 0.0068027724 0.0052486067 0.005123756 0.008923788
##            MCSE_Q97.5     MCSE_SD Bulk_ESS Tail_ESS
## beta1     0.006818301 0.001946731      279      217
## beta2     0.008917280 0.005057796      238      206
## lambda    1.781705411 1.344630129       13       16
## r1        0.004563786 0.001813399        5        7
## r2        0.010363171 0.006905270       10       16
## sigsq.eps 0.013662390 0.007248003       87      219

# run 2000 additional iterations
moreiterations = kmbayes_continue(kmfitbma.start, iter=2000)

## Validating control.params...

## Validating starting.values...

## Iteration: 201 (10% completed; 0.26671 secs elapsed)

## Acceptance rates for Metropolis-Hastings algorithm:
##    param  rate
## 1 lambda 0.425
## 2     r1 0.140
## 3     r2 0.475

## Iteration: 401 (20% completed; 0.53244 secs elapsed)

## Acceptance rates for Metropolis-Hastings algorithm:
##    param   rate
## 1 lambda 0.4500
## 2     r1 0.1475
## 3     r2 0.3975

## Iteration: 601 (30% completed; 0.84683 secs elapsed)

## Acceptance rates for Metropolis-Hastings algorithm:
##    param      rate
## 1 lambda 0.4750000
## 2     r1 0.1400000
## 3     r2 0.4083333

## Iteration: 801 (40% completed; 1.11445 secs elapsed)

## Acceptance rates for Metropolis-Hastings algorithm:
##    param    rate
## 1 lambda 0.48250
## 2     r1 0.13875
## 3     r2 0.40125

## Iteration: 1001 (50% completed; 1.38025 secs elapsed)

## Acceptance rates for Metropolis-Hastings algorithm:
##    param  rate
## 1 lambda 0.457
## 2     r1 0.137
## 3     r2 0.412

## Iteration: 1201 (60% completed; 1.64621 secs elapsed)

## Acceptance rates for Metropolis-Hastings algorithm:
##    param      rate
## 1 lambda 0.4683333
## 2     r1 0.1483333
## 3     r2 0.3975000

## Iteration: 1401 (70% completed; 1.95021 secs elapsed)

## Acceptance rates for Metropolis-Hastings algorithm:
##    param      rate
## 1 lambda 0.4885714
## 2     r1 0.1500000
## 3     r2 0.4028571

## Iteration: 1601 (80% completed; 2.21696 secs elapsed)

## Acceptance rates for Metropolis-Hastings algorithm:
##    param     rate
## 1 lambda 0.484375
## 2     r1 0.155625
## 3     r2 0.408750

## Iteration: 1801 (90% completed; 2.6581 secs elapsed)

## Acceptance rates for Metropolis-Hastings algorithm:
##    param      rate
## 1 lambda 0.4911111
## 2     r1 0.1627778
## 3     r2 0.4061111

## Iteration: 2001 (100% completed; 2.95569 secs elapsed)

## Acceptance rates for Metropolis-Hastings algorithm:
##    param   rate
## 1 lambda 0.4935
## 2     r1 0.1590
## 3     r2 0.4055

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  1.9  2.0  1.9 0.0  1.00     1195     1239
## beta2     0.0  0.1  0.3  0.1 0.1  1.00      861     1219
## lambda    4.9 12.1 30.5 14.3 8.5  1.00      126       93
## r1        0.0  0.0  0.1  0.0 0.0  1.03       77       78
## r2        0.0  0.0  0.2  0.1 0.0  1.00      112      100
## sigsq.eps 0.3  0.4  0.6  0.4 0.1  1.00      514      890
## 
## 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      1.94451464 0.001354599 0.04679152  1.84957428 1.91337229  1.94425420
## beta2      0.10591205 0.003098008 0.09190734 -0.07244492 0.04307369  0.10417774
## lambda    14.34371537 0.741761549 8.52169987  4.05070504 7.75788463 12.05628895
## r1         0.02298157 0.001565890 0.01454607  0.01069964 0.01263807  0.01684080
## r2         0.05795423 0.004171398 0.04711284  0.01155283 0.02140563  0.04287896
## sigsq.eps  0.40958211 0.004090066 0.09282983  0.26471559 0.34187272  0.39924364
##                   75%       97.5% n_eff      Rhat valid          Q5         Q50
## beta1      1.97704630  2.03445739  1186 0.9998051     1  1.86691892  1.94425420
## beta2      0.16721410  0.29665914   874 0.9996921     1 -0.04406332  0.10417774
## lambda    18.43684053 35.00798030   130 1.0031465     1  4.88809530 12.05628895
## r1         0.02691796  0.06024277    87 1.0285881     1  0.01105215  0.01684080
## r2         0.08141999  0.18286772   128 1.0017863     1  0.01242185  0.04287896
## sigsq.eps  0.46162498  0.62034160   515 0.9992769     1  0.28188526  0.39924364
##                   Q95    MCSE_Q2.5     MCSE_Q25     MCSE_Q50    MCSE_Q75
## beta1      2.02154219 0.0053480030 0.0018254915 0.0017327069 0.001809085
## beta2      0.25197027 0.0060693244 0.0032074359 0.0037835712 0.004042631
## lambda    30.54061347 0.6391431563 0.5063700931 0.7650849805 1.072408138
## r1         0.05704602 0.0002311747 0.0003811785 0.0007469754 0.002080573
## r2         0.16096251 0.0004350924 0.0020817839 0.0040577461 0.005804157
## sigsq.eps  0.58415925 0.0046659948 0.0037588012 0.0037328009 0.005981460
##            MCSE_Q97.5      MCSE_SD Bulk_ESS Tail_ESS
## beta1     0.003404986 0.0009580805     1195     1239
## beta2     0.010762886 0.0021913487      861     1219
## lambda    2.300751507 0.5314745343      126       93
## r1        0.001167341 0.0011110153       77       78
## r2        0.005557337 0.0029563949      112      100
## sigsq.eps 0.004102877 0.0028937522      514      890

TracePlot(moreiterations, par="beta")
TracePlot(moreiterations, par="r")

Acknowledgments

Thanks to Haotian “Howie” Wu for invaluable feedback on early versions of the package.