Improving MCMC Sampling for Bayesian Compositional Multilevel Models
Flora Le
2023-08-14
1 Introduction
In this vignettes, we present a case study we encountered in the
simulation study for package multilevelcoda. Briefly, we
found that multilevel model with compositional predictors with large
sample size, large between-person heterogeneity and small within-person
heterogeneity (large \(\sigma^2_{u}\)
and small \(\sigma^2_{\varepsilon}\))
produced low bulk effective sample size (ESS). Here, we examined the
diagnostics of the model of interest and explored different methods to
improve the within-chain autocorrelation.
2 Generating Data from Simulation Study
We first generated a dataset consisting of a 3-part behaviour composition with 1200 individuals, 14 observations per individuals, and large random intercept variation (\(\sigma^2_{u} = 1.5\)), coupled with small residual variation (\(\sigma^2_{\varepsilon}: 0.5\)).
set.seed(1)
sampled_cond <- cond[condition == "RElarge_RESsmall" & n_parts == 3 & N == 1200 & K == 14][1]
i <- 1
# condition
N <- sampled_cond[i, N]
K <- sampled_cond[i, K]
rint_sd <- sampled_cond[i, rint_sd]
res_sd <- sampled_cond[i, res_sd]
run <- sampled_cond[i, run]
n_parts <- sampled_cond[i, n_parts]
sbp_n <- sampled_cond[i, sbp]
prefit_n <- sampled_cond[i, prefit]
groundtruth_n <- sampled_cond[i, groundtruth]
parts <- sampled_cond[i, parts]
# inputs
sbp <- meanscovs[[paste(sbp_n)]]
prefit <- get(prefit_n)
groundtruth <- get(groundtruth_n)
parts <- as.vector(strsplit(parts, " ")[[1]])
simd <- with(meanscovs, rbind(
simulateData(
bm = BMeans,
wm = WMeans,
bcov = BCov,
wcov = WCov,
n = N,
k = K,
psi = psi)
))
simd[, Sleep := TST + WAKE]
simd[, PA := MVPA + LPA]
# ILR ---------------------------------------------------
cilr <- compilr(
data = simd,
sbp = sbp,
parts = parts,
idvar = "ID")
tmp <- cbind(cilr$data,
cilr$BetweenILR,
cilr$WithinILR,
cilr$TotalILR)
# random effects ----------------------------------------
redat <- data.table(ID = unique(tmp$ID),
rint = rnorm(
n = length(unique(tmp$ID)),
mean = 0,
sd = rint_sd))
tmp <- merge(tmp, redat, by = "ID")
# outcome -----------------------------------------------
if (n_parts == 3) {
tmp[, sleepy := rnorm(
n = nrow(simd),
mean = groundtruth$b_Intercept + rint +
(groundtruth$b_bilr1 * bilr1) +
(groundtruth$b_bilr2 * bilr2) +
(groundtruth$b_wilr1 * wilr1) +
(groundtruth$b_wilr2 * wilr2),
sd = res_sd)]
}
simd$sleepy <- tmp$sleepy
cilr <- compilr(simd, sbp, parts, total = 1440, idvar = "ID")
dat <- cbind(cilr$data, cilr$BetweenILR, cilr$WithinILR)Here is the dataset dat, along with our variables of
interest.
| ID | sleepy | Sleep | PA | SB | bilr1 | bilr2 | wilr1 | wilr2 |
|---|---|---|---|---|---|---|---|---|
| 1 | 2.91 | 437 | 267.9 | 735 | 0.466 | -1.16 | -0.478 | 0.447 |
| 1 | 1.02 | 665 | 116.2 | 659 | 0.466 | -1.16 | 0.250 | -0.067 |
| 1 | 1.89 | 526 | 209.8 | 704 | 0.466 | -1.16 | -0.210 | 0.304 |
| 1 | 1.88 | 611 | 89.7 | 739 | 0.466 | -1.16 | 0.240 | -0.331 |
| 1 | 2.35 | 488 | 124.9 | 827 | 0.466 | -1.16 | -0.125 | -0.177 |
| 1 | 2.03 | 382 | 201.4 | 856 | 0.466 | -1.16 | -0.534 | 0.137 |
3 Example Models
The model of interest is a multilevel model with 3-part composition (Sleep, Physical Activity, Sedentary Behaviour), expressed as a 2 sets of 2-part between and within- \(ilr\) coordinates predicting sleepiness.
fit <- brmcoda(cilr,
sleepy ~ bilr1 + bilr2 + wilr1 + wilr2 + (1 | ID),
cores = 4,
chains = 4,
iter = 3000,
warmup = 500,
seed = 13,
backend = "cmdstanr"
)A summary of the model indicates that the bulk ESS values for the constant and varying intercept as well as the constant coefficients of the between \(ilr\) coordinates are low.
summary(fit)
#> Family: gaussian
#> Links: mu = identity; sigma = identity
#> Formula: sleepy ~ bilr1 + bilr2 + wilr1 + wilr2 + (1 | ID)
#> Data: tmp (Number of observations: 16800)
#> Draws: 4 chains, each with iter = 3000; warmup = 500; thin = 1;
#> total post-warmup draws = 10000
#>
#> Group-Level Effects:
#> ~ID (Number of levels: 1200)
#> Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
#> sd(Intercept) 1.22 0.03 1.17 1.27 1.01 488 770
#>
#> Population-Level Effects:
#> Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
#> Intercept 2.35 0.15 2.04 2.65 1.02 286 640
#> bilr1 0.19 0.18 -0.18 0.52 1.01 312 520
#> bilr2 0.28 0.14 -0.00 0.56 1.02 280 532
#> wilr1 -0.80 0.03 -0.85 -0.74 1.00 9447 7579
#> wilr2 -0.26 0.02 -0.31 -0.22 1.00 9731 7932
#>
#> Family Specific Parameters:
#> Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
#> sigma 0.71 0.00 0.71 0.72 1.00 17695 7853
#>
#> Draws were sampled using sample(hmc). For each parameter, Bulk_ESS
#> and Tail_ESS are effective sample size measures, and Rhat is the potential
#> scale reduction factor on split chains (at convergence, Rhat = 1).Further inspection of density and trace lots for these parameters show some areas of the samples being drawn from outside of the parameter space, but no strong evidence for non-convergence.
Density and Trace Plot of Parameters with Low ESS
4 Improving within-chain autocorrelation of MCMC sampling
The low bulk ESS for random intercept model has been observed
previously, for example, see
here
and here.
We explored two potential solution to improve MCMC sampling efficiency:
increasing posterior draws and reparameterisation.
4.1 Increased posterior draws
As a first step, we test if increasing iterations and warmups helps.
fit_it <- brmcoda(cilr,
sleepy ~ bilr1 + bilr2 + wilr1 + wilr2 + (1 | ID),
cores = 4,
chains = 4,
iter = 10000,
warmup = 1000,
seed = 13,
backend = "cmdstanr"
)
print(rstan::get_elapsed_time(fit$Model$fit_it))
#> Error in object@mode: no applicable method for `@` applied to an object of class "NULL"summary(fit_it)
#> Family: gaussian
#> Links: mu = identity; sigma = identity
#> Formula: sleepy ~ bilr1 + bilr2 + wilr1 + wilr2 + (1 | ID)
#> Data: tmp (Number of observations: 16800)
#> Draws: 4 chains, each with iter = 10000; warmup = 1000; thin = 1;
#> total post-warmup draws = 36000
#>
#> Group-Level Effects:
#> ~ID (Number of levels: 1200)
#> Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
#> sd(Intercept) 1.22 0.03 1.17 1.27 1.00 1997 3971
#>
#> Population-Level Effects:
#> Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
#> Intercept 2.37 0.15 2.08 2.66 1.00 1130 2114
#> bilr1 0.18 0.18 -0.17 0.54 1.01 938 2477
#> bilr2 0.29 0.14 0.03 0.56 1.00 1182 2510
#> wilr1 -0.80 0.03 -0.85 -0.74 1.00 44344 28415
#> wilr2 -0.26 0.02 -0.31 -0.22 1.00 47155 29351
#>
#> Family Specific Parameters:
#> Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
#> sigma 0.71 0.00 0.71 0.72 1.00 62444 27585
#>
#> Draws were sampled using sample(hmc). For each parameter, Bulk_ESS
#> and Tail_ESS are effective sample size measures, and Rhat is the potential
#> scale reduction factor on split chains (at convergence, Rhat = 1).
Density and Trace Plot after increasing iterations and warmups
It is a good sign that increasing interation and warmups increase the
ESS, supported by trace plots. The ratios of ESS of
b_Intercept, b_bilr1, b_bilr2 and
sd_ID_intercept to b_wilr1,
b_wilr2, and sigma remain somewhat a
concern.
4.2 Centered Parameterisation
By default, brms uses the non-centered parametrization.
However, Betancourt and Girolami (2015)
explains correlation depends on the amount of data, and the efficacy of
the parameterization depends on the relative strength of the data. For
small data sets, the computational implementation of the model using
non-parameterisation is more efficient. When there is enough data,
however, this parameterization is unnecessary and it may be more
efficient to use the centered parameterisation. Reparameterisation is
further discussed in Betancourt’s
case study and Nicenboim,
Schad, and Vasishth’s chapter on complex models and
reparameterisation.
Given that there is a small variation in our large simulated data set, we are going to test if centered parameterisation improves the sampling. We first obtain the Stan code for the example model
make_stancode(sleepy ~ bilr1 + bilr2 + wilr1 + wilr2 + (1 | ID), data = dat)
#> // generated with brms 2.19.0
#> functions {
#> }
#> data {
#> int<lower=1> N; // total number of observations
#> vector[N] Y; // response variable
#> int<lower=1> K; // number of population-level effects
#> matrix[N, K] X; // population-level design matrix
#> // data for group-level effects of ID 1
#> int<lower=1> N_1; // number of grouping levels
#> int<lower=1> M_1; // number of coefficients per level
#> int<lower=1> J_1[N]; // grouping indicator per observation
#> // group-level predictor values
#> vector[N] Z_1_1;
#> int prior_only; // should the likelihood be ignored?
#> }
#> transformed data {
#> int Kc = K - 1;
#> matrix[N, Kc] Xc; // centered version of X without an intercept
#> vector[Kc] means_X; // column means of X before centering
#> for (i in 2:K) {
#> means_X[i - 1] = mean(X[, i]);
#> Xc[, i - 1] = X[, i] - means_X[i - 1];
#> }
#> }
#> parameters {
#> vector[Kc] b; // population-level effects
#> real Intercept; // temporary intercept for centered predictors
#> real<lower=0> sigma; // dispersion parameter
#> vector<lower=0>[M_1] sd_1; // group-level standard deviations
#> vector[N_1] z_1[M_1]; // standardized group-level effects
#> }
#> transformed parameters {
#> vector[N_1] r_1_1; // actual group-level effects
#> real lprior = 0; // prior contributions to the log posterior
#> r_1_1 = (sd_1[1] * (z_1[1]));
#> lprior += student_t_lpdf(Intercept | 3, 2.1, 2.5);
#> lprior += student_t_lpdf(sigma | 3, 0, 2.5)
#> - 1 * student_t_lccdf(0 | 3, 0, 2.5);
#> lprior += student_t_lpdf(sd_1 | 3, 0, 2.5)
#> - 1 * student_t_lccdf(0 | 3, 0, 2.5);
#> }
#> model {
#> // likelihood including constants
#> if (!prior_only) {
#> // initialize linear predictor term
#> vector[N] mu = rep_vector(0.0, N);
#> mu += Intercept;
#> for (n in 1:N) {
#> // add more terms to the linear predictor
#> mu[n] += r_1_1[J_1[n]] * Z_1_1[n];
#> }
#> target += normal_id_glm_lpdf(Y | Xc, mu, b, sigma);
#> }
#> // priors including constants
#> target += lprior;
#> target += std_normal_lpdf(z_1[1]);
#> }
#> generated quantities {
#> // actual population-level intercept
#> real b_Intercept = Intercept - dot_product(means_X, b);
#> }We then can manually edit the generated brms code to center all parameters. The modified Stan code is
m_centered <- cmdstan_model("fit_centered.stan")
print(m_centered)
#> data {
#> int<lower=1> N; // total number of observations
#> vector[N] Y; // response variable
#> int<lower=1> K; // number of population-level effects
#> matrix[N, K] X; // population-level design matrix
#> // data for group-level effects of ID 1
#> int<lower=1> N_1; // number of grouping levels
#> int<lower=1> M_1; // number of coefficients per level
#> int<lower=1> J_1[N]; // grouping indicator per observation
#> // group-level predictor values
#> vector[N] Z_1_1;
#> int prior_only; // should the likelihood be ignored?
#> }
#> transformed data {
#> int Kc = K - 1;
#> matrix[N, Kc] Xc; // centered version of X without an intercept
#> vector[Kc] means_X; // column means of X before centering
#> for (i in 2:K) {
#> means_X[i - 1] = mean(X[, i]);
#> Xc[, i - 1] = X[, i] - means_X[i - 1];
#> }
#> }
#> parameters {
#> vector[Kc] b; // population-level effects
#> real Intercept; // temporary intercept for centered predictors
#> real<lower=0> sigma; // dispersion parameter
#> vector<lower=0>[M_1] sd_1; // group-level standard deviations
#> vector[N_1] z_1[M_1]; // standardized group-level effects
#>
#> // Manually added
#> vector[N_1] r_1_1; // actual group-level effects
#> }
#> //transformed parameters {
#> // vector[N_1] r_1_1; // actual group-level effects
#> // real lprior = 0; // prior contributions to the log posterior
#> // r_1_1 = (sd_1[1] * (z_1[1]));
#> // lprior += student_t_lpdf(Intercept | 3, 2, 2.5);
#> // lprior += student_t_lpdf(sigma | 3, 0, 2.5)
#> // - 1 * student_t_lccdf(0 | 3, 0, 2.5);
#> // lprior += student_t_lpdf(sd_1 | 3, 0, 2.5)
#> // - 1 * student_t_lccdf(0 | 3, 0, 2.5);
#> //}
#> model {
#> // likelihood including constants
#> if (!prior_only) {
#> // initialize linear predictor term
#> vector[N] mu = rep_vector(0.0, N);
#> mu += Intercept;
#> for (n in 1:N) {
#> // add more terms to the linear predictor
#> mu[n] += r_1_1[J_1[n]] * Z_1_1[n];
#>
#> }
#> target += normal_id_glm_lpdf(Y | Xc, mu, b, sigma);
#> }
#> // priors including constants
#> // target += lprior;
#> // target += std_normal_lpdf(z_1[1]);
#>
#> target += student_t_lpdf(Intercept | 3, 2, 2.5);
#> target += student_t_lpdf(sigma | 3, 0, 2.5)
#> - 1 * student_t_lccdf(0 | 3, 0, 2.5);
#> target += student_t_lpdf(sd_1 | 3, 0, 2.5)
#> - 1 * student_t_lccdf(0 | 3, 0, 2.5);
#>
#> // Manually added
#> target += normal_lpdf(r_1_1 | 0, sd_1[1]);
#>
#> // manual soft sum to zero constraint
#> // target += normal_lpdf(sum(r_1_1) | 0, 0.0001 *N_1);
#> }
#> generated quantities {
#> // actual population-level intercept
#> real b_Intercept = Intercept - dot_product(means_X, b);
#> }Now we build data for the model and sample with
cmdstanr
sdat <- make_standata(sleepy ~ bilr1 + bilr2 + wilr1 + wilr2 + (1 | ID), data = dat)
fit_centered <- m_centered$sample(data = sdat,
parallel_chains = 4,
iter_sampling = 2500,
iter_warmup = 500)bm_centered <- brm(sleepy ~ bilr1 + bilr2 + wilr1 + wilr2 + (1 | ID), data = dat, empty = TRUE)
bm_centered$fit <- rstan::read_stan_csv(fit_centered$output_files())
bm_centered <- rename_pars(bm_centered)summary(bm_centered)
#> Family: gaussian
#> Links: mu = identity; sigma = identity
#> Formula: sleepy ~ bilr1 + bilr2 + wilr1 + wilr2 + (1 | ID)
#> Data: dat (Number of observations: 16800)
#> Draws: 4 chains, each with iter = 3000; warmup = 500; thin = 1;
#> total post-warmup draws = 10000
#>
#> Group-Level Effects:
#> ~ID (Number of levels: 1200)
#> Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
#> sd(Intercept) 1.22 0.03 1.17 1.27 1.00 9848 9674
#>
#> Population-Level Effects:
#> Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
#> Intercept 2.36 0.15 2.06 2.66 1.00 10497 5483
#> bilr1 0.19 0.18 -0.17 0.55 1.00 10539 5853
#> bilr2 0.29 0.14 0.01 0.56 1.00 10683 5784
#> wilr1 -0.80 0.03 -0.85 -0.74 1.00 10030 7217
#> wilr2 -0.26 0.02 -0.31 -0.22 1.00 9805 7335
#>
#> Family Specific Parameters:
#> Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
#> sigma 0.71 0.00 0.71 0.72 1.00 8197 6616
#>
#> Draws were sampled using sample(hmc). For each parameter, Bulk_ESS
#> and Tail_ESS are effective sample size measures, and Rhat is the potential
#> scale reduction factor on split chains (at convergence, Rhat = 1).Indeed, the centered parameterisation has improved the ESS for all parameters.
Density and Trace Plot when using centered parameterisation
5 Computational Time
# example model
print(rstan::get_elapsed_time(fit$Model$fit))
#> warmup sample
#> chain:1 16.7 36.6
#> chain:2 14.6 36.6
#> chain:3 19.3 36.3
#> chain:4 15.4 36.7
# increased iter
print(rstan::get_elapsed_time(fit_it$Model$fit))
#> warmup sample
#> chain:1 23.2 128
#> chain:2 20.9 129
#> chain:3 25.3 129
#> chain:4 21.8 129
# centered parameterisation
rstan::get_elapsed_time(bm_centered$fit)
#> warmup sample
#> chain:1 200 1125
#> chain:2 199 1121
#> chain:3 202 1121
#> chain:4 200 1124It took 607.051 to obtain an average-ish of 1000 across the four parameters with initially low ESS. To achieve an ESS of 10 000 across all parameters, simplying increasing iterations should take around 6070.51, whereas centered parameterisation takes around 5292.319.
6 Conclusion
multilevelcoda demonstrates excellent performances
across different conditions of number of individuals, number of
observations, number of compositional parts, and magnitude of modeled
variance (full results of simulation study will be published at a later
date). However, in the case of large data and small sample variation, we
recommend centered parameterisation or increasing iterations to achieve
efficient MCMC sampling. Based on results of the case study, centered
parameterisation would give more reliable results in terms of
ESS-to-draw ratios for all parameters and computational time.