Simulation 3: Two-dimensional Non-linear Regression
Dayi Li
2025-04-19
Last updated: 2025-05-09
Checks: 7 0
Knit directory: BOSS_website/
This reproducible R Markdown analysis was created with workflowr (version 1.7.1). The Checks tab describes the reproducibility checks that were applied when the results were created. The Past versions tab lists the development history.
Great! Since the R Markdown file has been committed to the Git repository, you know the exact version of the code that produced these results.
Great job! The global environment was empty. Objects defined in the global environment can affect the analysis in your R Markdown file in unknown ways. For reproduciblity it’s best to always run the code in an empty environment.
The command set.seed(20250415) was run prior to running
the code in the R Markdown file. Setting a seed ensures that any results
that rely on randomness, e.g. subsampling or permutations, are
reproducible.
Great job! Recording the operating system, R version, and package versions is critical for reproducibility.
Nice! There were no cached chunks for this analysis, so you can be confident that you successfully produced the results during this run.
Great job! Using relative paths to the files within your workflowr project makes it easier to run your code on other machines.
Great! You are using Git for version control. Tracking code development and connecting the code version to the results is critical for reproducibility.
The results in this page were generated with repository version c72ffd4. See the Past versions tab to see a history of the changes made to the R Markdown and HTML files.
Note that you need to be careful to ensure that all relevant files for
the analysis have been committed to Git prior to generating the results
(you can use wflow_publish or
wflow_git_commit). workflowr only checks the R Markdown
file, but you know if there are other scripts or data files that it
depends on. Below is the status of the Git repository when the results
were generated:
Ignored files:
Ignored: .DS_Store
Ignored: .Rhistory
Ignored: .Rproj.user/
Ignored: analysis/.DS_Store
Ignored: data/.DS_Store
Ignored: data/dimension/
Ignored: data/sim3/
Ignored: output/.DS_Store
Unstaged changes:
Modified: output/sim3/MCMC_small_n_sim3.rda
Note that any generated files, e.g. HTML, png, CSS, etc., are not included in this status report because it is ok for generated content to have uncommitted changes.
These are the previous versions of the repository in which changes were
made to the R Markdown (analysis/sim3.Rmd) and HTML
(docs/sim3.html) files. If you’ve configured a remote Git
repository (see ?wflow_git_remote), click on the hyperlinks
in the table below to view the files as they were in that past version.
| File | Version | Author | Date | Message |
|---|---|---|---|---|
| Rmd | c72ffd4 | david.li | 2025-05-09 | wflow_publish("analysis/sim3.Rmd") |
| html | 1881d2e | david.li | 2025-05-08 | Build site. |
| Rmd | 3270116 | david.li | 2025-05-08 | wflow_publish("analysis/sim3.Rmd") |
| html | 43c115a | david.li | 2025-05-07 | Build site. |
| Rmd | a114090 | david.li | 2025-05-07 | wflow_publish("analysis/sim3.Rmd") |
| Rmd | 304b28d | Ziang Zhang | 2025-05-05 | update |
| html | a07691d | david.li | 2025-04-20 | Build site. |
| html | 3478135 | david.li | 2025-04-20 | Build site. |
| Rmd | 3f8195f | david.li | 2025-04-20 | wflow_publish("analysis/sim3.Rmd") |
| html | 6cea47c | david.li | 2025-04-20 | Build site. |
| Rmd | d9facf4 | david.li | 2025-04-20 | wflow_publish("analysis/sim3.Rmd") |
| html | 657dd74 | david.li | 2025-04-20 | Build site. |
| Rmd | 1026e5b | david.li | 2025-04-20 | wflow_publish("analysis/sim3.Rmd") |
| html | 3528574 | david.li | 2025-04-20 | Build site. |
| html | 13358d2 | david.li | 2025-04-20 | Build site. |
| Rmd | 78186b1 | david.li | 2025-04-20 | Simulation 3 |
| html | 78186b1 | david.li | 2025-04-20 | Simulation 3 |
| html | cc32863 | david.li | 2025-04-19 | Build site. |
| Rmd | 2011696 | david.li | 2025-04-19 | wflow_publish("analysis/sim3.Rmd") |
Data
library(tidyverse)
library(tikzDevice)
library(rstan)
library(INLA)
library(inlabru)
library(modeest)
function_path <- "./code"
output_path <- "./output/sim3"
data_path <- "./data/sim3"
source(paste0(function_path, "/00_BOSS.R"))Consider the following non-linear regression model:
\[\begin{align*} y_i \mid \log(\rho_i) &\overset{ind}{\sim}\mathcal{N}(\log\rho(r_i), \sigma^2), \\ \log\rho(r_i) &= \log\rho_0 - \gamma\log\left\{1 + (r_i/R)^\beta\right\}. \end{align*}\]
We simulate \(n = 200\) data points based on the above model with \(\rho_0 = 10\), \(R = 2\), \(\beta = 2\), \(\gamma = -2.5\), and \(\sigma = 0.5\). The inferential goal is the nuisance parameters \(R\) and \(\beta\).
r <- seq(0, 20, length.out = 200)
beta <- 10
a <- 2
b <- 2
c <- -2.5
set.seed(1234)
Ir <- beta*(1 + (r/a)^b)^c
lr <- log(Ir) + rnorm(length(r), 0, 0.5)
data <- data.frame(r, lr)
ggplot(data, aes(r, lr)) + geom_point() + ylab('y')
| Version | Author | Date |
|---|---|---|
| cc32863 | david.li | 2025-04-19 |
inlabru
We first run inlabru to to fit the model. We set the
following priors for the parameters:
\[\begin{align*} \rho_0 \sim \mathcal{N}(0, 1000), \ & R \sim \mathrm{Unif}(0.1, 5), \\ \beta \sim \mathrm{Unif}(0.1, 4), \ \gamma \sim \mathcal{N}(0, & 1000), \ \sigma^2 \sim \mathrm{Inv-Gamma}(1, 10^{-5}). \end{align*}\]
a_fun <- function(u){
qunif(pnorm(u), 0.1, 5)
}
b_fun <- function(u){
qunif(pnorm(u), 0.1, 4)
}
cmp <- ~ a(1, model="linear", mean.linear=0, prec.linear=1) +
b(1, model="linear", mean.linear=0, prec.linear=1) +
c(1) + Intercept(1)
form <- lr ~ Intercept + c*log(1 + (r/a_fun(a))^b_fun(b))
fit <- bru(cmp, formula = form, data = data, family = 'gaussian')BOSS
Now let’s run BOSS.
BOSS-modal
We first specify the (unnormalized) log-posterior for \((R,\beta)\). Note that for this specific problem, the unnormalized log-posterior has a closed-form expression:
# specify the objective function for BOSS: unnormalized log posterior of (R, beta)
eval_func <- function(par, x = r, y = lr){
a <- par[1]
b <- par[2]
n <- length(r)
X <- matrix(cbind(rep(1, n), log(1 + (r/a)^b)), ncol = 2)
Vb <- solve(t(X) %*% X + diag(1/1000, 2))
P <- diag(n) - X %*% Vb %*% t(X)
mlik <- log(det(Vb))/2 - log(1000) + lgamma((n+1)/2) - (n+1)/2*log(1e-5 + t(y) %*% P %*% y/2) -
n/2*log(pi) -5*log(10)
return(mlik)
}BOSS-modal
Next, we run the BOSS algorithm where the stopping criteria is based on the convergence of the posterior mode. Specifically, we check the modal convergence every BO iteration, and consider the convergence statistics of the average \(5\) nearest neighbor distance around the current mode.
set.seed(123)
res_opt_modal <- BOSS(eval_func, criterion = 'modal', update_step = 5, max_iter = 100, D = 2,
lower = rep(0.1, 2), upper = c(5, 4),
noise_var = 1e-6,
modal_iter_check = 1, modal_check_warmup = 20, modal_k.nn = 5,
modal_eps = 0.01,
initial_design = 5, delta = 0.01^2,
optim.n = 5, optim.max.iter = 100)
save(res_opt_modal, file = paste0(output_path, "/BOSS_modal_sim3.rda"))The above BOSS algorithm with modal convergence
criterion converged in \(55\)
iterations.
Let’s check the convergence diagnostic plot:
load(paste0(output_path, "/BOSS_modal_sim3.rda"))
plot(res_opt_modal$modal_result$modal_max_diff ~ res_opt_modal$modal_result$i, type = 'o', xlab = 'Iteration', ylab = 'Modal Stats')
| Version | Author | Date |
|---|---|---|
| 43c115a | david.li | 2025-05-07 |
MCMC
Lastly, we implement the MCMC-based method using stan to
obtain the oracle.
set.seed(1234)
MCMC_fit <- stan(
file = "code/nlreg.stan", # Stan program
data = list(x = r, y = lr, N = length(r)), # named list of data
chains = 4, # number of Markov chains
warmup = 1000, # number of warmup iterations per chain
iter = 20000, # total number of iterations per chain
cores = 4, # number of cores (could use one per chain)
algorithm = 'NUTS')
# thin the samples fo plotting
MCMC_samp <- as.data.frame(MCMC_fit)
#MCMC_samp_thin <- MCMC_samp[seq(1, 76000, by = 8),]
save(MCMC_samp, file = paste0(output_path, "/MCMC_sim3.rda"))Results Comparison
Marginal Posterior Distributions
We first compare the marginal posterior distributions of \(R\) and \(\beta\) from inlabru, BOSS,
and MCMC.
# inlabru marginal samples
set.seed(1234)
inla.samples.a <- a_fun(inla.rmarginal(49500, fit$marginals.fixed$a))
inla.samples.b <- b_fun(inla.rmarginal(49500, fit$marginals.fixed$b))
# BOSS-modal marginal samples
data_to_smooth <- list()
unique_data <- unique(data.frame(x = res_opt_modal$result$x, y = res_opt_modal$result$y))
data_to_smooth$x <- as.matrix(dplyr::select(unique_data, -y))
data_to_smooth$y <- (unique_data$y - mean(unique_data$y))
square_exp_cov <- square_exp_cov_generator_nd(length_scale = res_opt_modal$length_scale, signal_var = res_opt_modal$signal_var)
surrogate <- function(xvalue, data_to_smooth, cov){
predict_gp(data_to_smooth, x_pred = xvalue, choice_cov = cov, noise_var = 1e-6)$mean
}
ff <- list()
ff$fn <- function(x) as.numeric(surrogate(x, data_to_smooth = data_to_smooth, cov = square_exp_cov))
x.1 <- (seq(from = 0.1, to = 5, length.out = 300) - 0.1)/4.9
x.2 <- (seq(from = 0.1, to = 4, length.out = 300) - 0.1)/3.9
x_vals <- expand.grid(x.1, x.2)
names(x_vals) <- c('x.1','x.2')
x_original <- t(t(x_vals)*(c(5, 4) - c(0.1, 0.1)) + c(0.1, 0.1))
fn_vals <- apply(x_vals, 1, function(x) ff$fn(x = matrix(x, ncol = 2))) + mean(unique_data$y)
# normalize
lognormal_const <- log(sum(exp(fn_vals))*0.0098*0.0078*25/9)
post_x_modal <- data.frame(x_original, pos = exp(fn_vals - lognormal_const))
dx <- unique(post_x_modal$x.1)[2] - unique(post_x_modal$x.1)[1]
dy <- unique(post_x_modal$x.2)[2] - unique(post_x_modal$x.2)[1]
set.seed(123456)
sample_idx <- rmultinom(1:90000, size = 49500, prob = post_x_modal$pos)
sample_x_modal <- data.frame(post_x_modal, n = sample_idx)
samples_BOSS_modal <- data.frame(do.call(rbind, apply(sample_x_modal, 1, function(x) cbind(runif(x[4], x[1], x[1]+dx), runif(x[4], x[2], x[2] + dy)))))
# MCMC marginal samples
load(paste0(output_path, "/MCMC_sim3.rda"))
# Combine all samples together
R_marginal <- data.frame(R = c(inla.samples.a, samples_BOSS_modal[,1], MCMC_samp$a),
method = rep(c('inlabru', 'BOSS', 'MCMC'),
c(length(inla.samples.a),
length(samples_BOSS_modal[,1]),
length(MCMC_samp$a))))
beta_marginal <- data.frame(beta = c(inla.samples.b, samples_BOSS_modal[,2], MCMC_samp$b),
method = rep(c('inlabru', 'BOSS', 'MCMC'),
c(length(inla.samples.b),
length(samples_BOSS_modal[,2]),
length(MCMC_samp$b))))Plot the marginal posterior densities
ggplot(R_marginal, aes(R)) + geom_density(aes(color = method)) + theme_minimal() + xlab('$R$')
ggplot(beta_marginal, aes(beta)) + geom_density(aes(color = method)) + theme_minimal() + xlab('$\\beta$')
Joint Posterior Distribution
We now compare the results of the posterior distributions from
inlabru, modal-based BOSS, and AGHQ-based BOSS, and
MCMC.
inlabru joint posterior distribution:
# get joint posterior of (R, beta) from inlabru
joint_samp <- inla.posterior.sample(10000, fit, selection = list(a = 1, b = 1), seed = 12345)
joint_samp <- do.call('rbind', lapply(joint_samp, function(x) matrix(x$latent, ncol = 2)))
inla.joint.samps <- data.frame(a = a_fun(joint_samp[,1]), b = b_fun(joint_samp[,2]))
# plot joint posterior of (R, beta) from inlabru
ggplot(inla.joint.samps, aes(a, b)) + stat_density_2d(
geom = "raster",
aes(fill = after_stat(density)), n = 300,
contour = FALSE) +
geom_point(data = data.frame(a = a_fun(fit$summary.fixed$mode[1]), b = b_fun(fit$summary.fixed$mode[2])), color = 'red', shape = 1, size =0.5) +
geom_point(data = data.frame(a = 2, b = 2), color = 'orange', size =0.5) +
coord_fixed() + scale_fill_viridis_c(name = 'Density') + theme_minimal() + xlab('$R$') + ylab('$\\beta$') + xlim(c(0.1, 5)) + ylim(c(0.1, 4))
BOSS joint posterior distribution:
# plot joint posterior of (R, beta) from BOSS
ggplot(post_x_modal, aes(x.1,x.2)) + geom_raster(aes(fill = (pos))) +
geom_point(data = data.frame(x.1 = post_x_modal$x.1[which.max(post_x_modal$pos)], x.2 = post_x_modal$x.2[which.max(post_x_modal$pos)]), color = 'red', shape = 1, size =0.5) +
geom_point(data = data.frame(x.1 = 2, x.2 = 2), color = 'orange', size =0.5) + coord_fixed() + scale_fill_viridis_c(name = 'Density') + theme_minimal() + xlab('$R$') + ylab('$\\beta$')
MCMC joint posterior distribution:
ggplot(MCMC_samp, aes(a, b)) + stat_density_2d(
geom = "raster",
aes(fill = after_stat(density)), n = 300,
contour = FALSE) +
geom_point(data = data.frame(a = post_x_modal$x.1[which.max(post_x_modal$pos)], b = post_x_modal$x.2[which.max(post_x_modal$pos)]), color = 'red', shape = 1, size =0.5) +
geom_point(data = data.frame(a = 2, b = 2), color = 'orange', size =0.5) + coord_fixed() + scale_fill_viridis_c(name = 'Density') + theme_minimal() + xlab('$R$') + ylab('$\\beta$') + xlim(c(0.1, 5)) + ylim(c(0.1, 4))
From the above results, it is clear that BOSS is much better at
depicting the joint posterior distribution than inlabru.
The joint distribution from inlabru is simply the product
of the marginal distribution, which completely ignores the more complex
structures in the joint posterior.
Does Starting Design Points Matter?
We here check if BOSS is robust towards the starting design points. We run BOSS for \(20\) times with different initial design points and run until the modal statistics have reached below \(\epsilon =0.01\).
res_list <- vector('list', 20)
for (i in 1:20) {
set.seed(i)
res_list[[i]] <- BOSS(eval_func, criterion = 'modal', update_step = 5, max_iter = 100, D = 2,
lower = rep(0.1, 2), upper = c(5, 4),
noise_var = 1e-6,
modal_iter_check = 1, modal_check_warmup = 20, modal_k.nn = 5,
modal_eps = 0.01,
initial_design = 5, delta = 0.01^2,
optim.n = 5, optim.max.iter = 100)
}
save(res_list, file = paste0(output_path, "/BOSS_robust_sim3.rda"))load(paste0(output_path, "/BOSS_robust_sim3.rda"))
sample_list <- vector('list', 20)
for(i in 1:20){
# BOSS-modal marginal samples
data_to_smooth <- list()
unique_data <- unique(data.frame(x = res_list[[i]]$result$x, y = res_list[[i]]$result$y))
data_to_smooth$x <- as.matrix(dplyr::select(unique_data, -y))
data_to_smooth$y <- (unique_data$y - mean(unique_data$y))
square_exp_cov <- square_exp_cov_generator_nd(length_scale = res_list[[i]]$length_scale, signal_var = res_list[[i]]$signal_var)
surrogate <- function(xvalue, data_to_smooth, cov){
predict_gp(data_to_smooth, x_pred = xvalue, choice_cov = cov, noise_var = 1e-6)$mean
}
ff <- list()
ff$fn <- function(x) as.numeric(surrogate(x, data_to_smooth = data_to_smooth, cov = square_exp_cov))
x.1 <- (seq(from = 0.1, to = 5, length.out = 300) - 0.1)/4.9
x.2 <- (seq(from = 0.1, to = 4, length.out = 300) - 0.1)/3.9
x_vals <- expand.grid(x.1, x.2)
names(x_vals) <- c('x.1','x.2')
x_original <- t(t(x_vals)*(c(5, 4) - c(0.1, 0.1)) + c(0.1, 0.1))
fn_vals <- apply(x_vals, 1, function(x) ff$fn(x = matrix(x, ncol = 2))) + mean(unique_data$y)
# normalize
lognormal_const <- log(sum(exp(fn_vals))*0.0098*0.0078*25/9)
post_x_modal <- data.frame(x_original, pos = exp(fn_vals - lognormal_const))
dx <- unique(post_x_modal$x.1)[2] - unique(post_x_modal$x.1)[1]
dy <- unique(post_x_modal$x.2)[2] - unique(post_x_modal$x.2)[1]
set.seed(123456)
sample_idx <- rmultinom(1:90000, size = 49500, prob = post_x_modal$pos)
sample_x_modal <- data.frame(post_x_modal, n = sample_idx)
sample_list[[i]] <- data.frame(do.call(rbind, apply(sample_x_modal, 1, function(x) cbind(runif(x[4], x[1], x[1]+dx), runif(x[4], x[2], x[2] + dy)))))
}
save(sample_list, file = paste0(output_path, "/BOSS_sample_robust_sim3.rda"))load(paste0(output_path, "/BOSS_sample_robust_sim3.rda"))
load(paste0(output_path, "/MCMC_sim3.rda"))
p_R <- ggplot(MCMC_samp, aes(x = a)) + theme_minimal() + xlab('$R$')
for (i in 1:20) {
p_R <- p_R + geom_density(data = sample_list[[i]], aes(x = X1), color = 'red', alpha = 0.1)
}
p_R <- p_R + geom_density(color = 'violet')
p_R
| Version | Author | Date |
|---|---|---|
| 43c115a | david.li | 2025-05-07 |
p_b <- ggplot(MCMC_samp, aes(x = b)) + theme_minimal() + xlab('$beta$')
for (i in 1:20) {
p_b <- p_b + geom_density(data = sample_list[[i]], aes(x = X2), color = 'red', alpha = 0.1)
}
p_b <- p_b + geom_density(color = 'violet')
p_b
| Version | Author | Date |
|---|---|---|
| 43c115a | david.li | 2025-05-07 |
In the above, violet density comes from MCMC while red ones come from BOSS. We can see that BOSS is highly robust to the initial design points selection.
What Happens with Smaller Sample Size?
We now test the performance of BOSS when the sample size is small. This is interesting since small sample size typically translates to a posterior distribution that may have highly complex and distorted geometry that deviates significantly from a normal distribution.
Here we use the same simulation example but with \(n = 50\).
r <- seq(0, 20, length.out = 50)
beta <- 10
a <- 2
b <- 2
c <- -2.5
set.seed(1234)
Ir <- beta*(1 + (r/a)^b)^c
lr <- log(Ir) + rnorm(length(r), 0, 0.5)We run BOSS for \(20\) times with different initial design points while turning off the stopping criteria. We want to check what the BOSS posterior looks like.
res_list <- vector('list', 20)
for (i in 1:20) {
set.seed(i)
res_list[[i]] <- BOSS(eval_func, criterion = 'modal', update_step = 5, max_iter = 100, D = 2,
lower = rep(0.1, 2), upper = c(5, 4),
noise_var = 1e-6,
modal_iter_check = 1, modal_check_warmup = 20, modal_k.nn = 5,
modal_eps = 0,
initial_design = 5, delta = 0.01^2,
optim.n = 5, optim.max.iter = 100)
}
save(res_list, file = paste0(output_path, "/BOSS_small_n_sim3.rda"))set.seed(1234)
MCMC_fit <- stan(
file = "code/nlreg.stan", # Stan program
data = list(x = r, y = lr, N = length(r)), # named list of data
chains = 4, # number of Markov chains
warmup = 1000, # number of warmup iterations per chain
iter = 40000, # total number of iterations per chain
cores = 4, # number of cores (could use one per chain)
algorithm = 'NUTS')
# thin the samples fo plotting
MCMC_samp <- as.data.frame(MCMC_fit)
#MCMC_samp_thin <- MCMC_samp[seq(1, 76000, by = 8),]
save(MCMC_samp, file = paste0(output_path, "/MCMC_small_n_sim3.rda"))load(paste0(output_path, "/BOSS_small_n_sim3.rda"))
sample_list <- vector('list', 20)
for(i in 1:20){
# BOSS-modal marginal samples
data_to_smooth <- list()
unique_data <- unique(data.frame(x = res_list[[i]]$result$x, y = res_list[[i]]$result$y))
data_to_smooth$x <- as.matrix(dplyr::select(unique_data, -y))
data_to_smooth$y <- (unique_data$y - mean(unique_data$y))
square_exp_cov <- square_exp_cov_generator_nd(length_scale = res_list[[i]]$length_scale, signal_var = res_list[[i]]$signal_var)
surrogate <- function(xvalue, data_to_smooth, cov){
predict_gp(data_to_smooth, x_pred = xvalue, choice_cov = cov, noise_var = 1e-6)$mean
}
ff <- list()
ff$fn <- function(x) as.numeric(surrogate(x, data_to_smooth = data_to_smooth, cov = square_exp_cov))
x.1 <- (seq(from = 0.1, to = 5, length.out = 300) - 0.1)/4.9
x.2 <- (seq(from = 0.1, to = 4, length.out = 300) - 0.1)/3.9
x_vals <- expand.grid(x.1, x.2)
names(x_vals) <- c('x.1','x.2')
x_original <- t(t(x_vals)*(c(5, 4) - c(0.1, 0.1)) + c(0.1, 0.1))
fn_vals <- apply(x_vals, 1, function(x) ff$fn(x = matrix(x, ncol = 2))) + mean(unique_data$y)
# normalize
lognormal_const <- log(sum(exp(fn_vals))*0.0098*0.0078*25/9)
post_x_modal <- data.frame(x_original, pos = exp(fn_vals - lognormal_const))
dx <- unique(post_x_modal$x.1)[2] - unique(post_x_modal$x.1)[1]
dy <- unique(post_x_modal$x.2)[2] - unique(post_x_modal$x.2)[1]
set.seed(123456)
sample_idx <- rmultinom(1:90000, size = 49500, prob = post_x_modal$pos)
sample_x_modal <- data.frame(post_x_modal, n = sample_idx)
sample_list[[i]] <- data.frame(do.call(rbind, apply(sample_x_modal, 1, function(x) cbind(runif(x[4], x[1], x[1]+dx), runif(x[4], x[2], x[2] + dy)))))
}
save(sample_list, file = paste0(output_path, "/BOSS_sample_small_n_sim3.rda"))load(paste0(output_path, "/BOSS_sample_small_n_sim3.rda"))
load(paste0(output_path, "/MCMC_small_n_sim3.rda"))
plot(MCMC_samp[seq(1, 156000, by = 10),c('a','b')], asp = 1)
p_R <- ggplot(MCMC_samp, aes(x = a)) + theme_minimal() + xlab('$R$')
for (i in 1:20) {
p_R <- p_R + geom_density(data = sample_list[[i]], aes(x = X1), color = 'red', alpha = 0.1)
}
p_R <- p_R + geom_density(color = 'violet')
p_R
p_b <- ggplot(MCMC_samp, aes(x = b)) + theme_minimal() + xlab('$beta$')
for (i in 1:20) {
p_b <- p_b + geom_density(data = sample_list[[i]], aes(x = X2), color = 'red', alpha = 0.1)
}
p_b <- p_b + geom_density(color = 'violet')
p_b
Compared to the case when \(n = 200\), having a smaller sample size causes BOSS to have inferior accuracy. Specifically, surrogate posterior from BOSS (red densities) is lighter-tailed compared to the true posterior (violet densities). The reason here is that BvM theorem does not apply when sample size is small. Under such scenarios, the tail region of the posterior is likely underestimated since the design points used for interpolation for the tail regions are ostly near the boundaries of the search domain and of very low values.
sessionInfo()R version 4.4.1 (2024-06-14)
Platform: aarch64-apple-darwin20
Running under: macOS 15.0
Matrix products: default
BLAS: /Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/lib/libRlapack.dylib; LAPACK version 3.12.0
locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
time zone: America/Toronto
tzcode source: internal
attached base packages:
[1] stats graphics grDevices utils datasets methods base
other attached packages:
[1] modeest_2.4.0 inlabru_2.11.1 fmesher_0.1.7
[4] INLA_24.06.27 sp_2.1-4 Matrix_1.7-0
[7] rstan_2.32.6 StanHeaders_2.32.10 tikzDevice_0.12.6
[10] lubridate_1.9.3 forcats_1.0.0 stringr_1.5.1
[13] dplyr_1.1.4 purrr_1.0.2 readr_2.1.5
[16] tidyr_1.3.1 tibble_3.2.1 ggplot2_3.5.1
[19] tidyverse_2.0.0 workflowr_1.7.1
loaded via a namespace (and not attached):
[1] mnormt_2.1.1 DBI_1.2.3 gridExtra_2.3
[4] inline_0.3.19 rlang_1.1.4 magrittr_2.0.3
[7] clue_0.3-65 git2r_0.33.0 matrixStats_1.4.1
[10] e1071_1.7-16 compiler_4.4.1 getPass_0.2-4
[13] loo_2.8.0 callr_3.7.6 vctrs_0.6.5
[16] rmutil_1.1.10 pkgconfig_2.0.3 fastmap_1.2.0
[19] labeling_0.4.3 utf8_1.2.4 promises_1.3.0
[22] rmarkdown_2.28 tzdb_0.4.0 ps_1.8.0
[25] MatrixModels_0.5-3 xfun_0.47 cachem_1.1.0
[28] jsonlite_1.8.9 highr_0.11 later_1.3.2
[31] parallel_4.4.1 cluster_2.1.6 R6_2.5.1
[34] bslib_0.8.0 stringi_1.8.4 rpart_4.1.23
[37] numDeriv_2016.8-1.1 jquerylib_0.1.4 Rcpp_1.0.13
[40] knitr_1.48 filehash_2.4-6 httpuv_1.6.15
[43] splines_4.4.1 timechange_0.3.0 tidyselect_1.2.1
[46] rstudioapi_0.16.0 yaml_2.3.10 timeDate_4041.110
[49] codetools_0.2-20 processx_3.8.4 pkgbuild_1.4.4
[52] lattice_0.22-6 plyr_1.8.9 withr_3.0.1
[55] evaluate_1.0.0 stable_1.1.6 sf_1.0-19
[58] units_0.8-5 proxy_0.4-27 RcppParallel_5.1.10
[61] pillar_1.9.0 whisker_0.4.1 KernSmooth_2.23-24
[64] stats4_4.4.1 sn_2.1.1 generics_0.1.3
[67] rprojroot_2.0.4 hms_1.1.3 munsell_0.5.1
[70] scales_1.3.0 timeSeries_4041.111 class_7.3-22
[73] glue_1.7.0 statip_0.2.3 tools_4.4.1
[76] spatial_7.3-17 fBasics_4041.97 fs_1.6.4
[79] grid_4.4.1 QuickJSR_1.6.0 colorspace_2.1-1
[82] cli_3.6.3 fansi_1.0.6 viridisLite_0.4.2
[85] gtable_0.3.5 stabledist_0.7-2 sass_0.4.9
[88] digest_0.6.37 classInt_0.4-10 farver_2.1.2
[91] htmltools_0.5.8.1 lifecycle_1.0.4 httr_1.4.7
[94] MASS_7.3-61