Explore Complex Posterior Functions

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Last updated: 2025-04-24

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Introduction

In this example, we assess the accuracy of the proposed BOSS algorithm for approximating (un-normalized) posterior functions with different complexities.

We consider a conditioning parameter \(\alpha\) with support \(\Omega = [0,10]\), and assume its unnormalized log posteriors are defined respectively as \(f(\alpha) = \alpha \sin(\alpha)\), \(\log(\alpha+1)\sin(2\alpha) - \alpha \cos(2\alpha)\), and \(\log(\alpha + 1) (\sin(4\alpha) + \cos(2\alpha))\) for simple, medium, and hard settings.

In the simple setting, the log posterior has two local modes and its corresponding posterior is close to uni-modal. In the medium scenario, the log posterior has three local modes, with the corresponding posterior that is close to bi-modal. In the hard scenario, the log posterior has seven local modes and the posterior is close to tri-modal. The proposed BOSS algorithm is then applied with different numbers of BO iterations \(B\) with five initial values that are equally placed from 0 to 10.

library(npreg)

Package 'npreg' version 1.1.0
Type 'citation("npreg")' to cite this package.

library(ggplot2)
library(aghq)

set.seed(123)
noise_var = 1e-6
function_path <- "./code"
output_path <- "./output/simA1"
data_path <- "./data/simA1"
source(paste0(function_path, "/00_BOSS.R"))

surrogate <- function(xvalue, data_to_smooth){
  predict(ss(x = as.numeric(data_to_smooth$x), y = data_to_smooth$y, df = length(unique(as.numeric(data_to_smooth$x))), m = 2, all.knots = TRUE), x = xvalue)$y
}
lower = 0
upper = 10

integrate_aghq <- function(f, k = 100, startingvalue = 0){
  ff <- list(fn = f, gr = function(x) numDeriv::grad(f, x), he = function(x) numDeriv::hessian(f, x))
  aghq(ff = ff, k = k, startingvalue = startingvalue)$normalized_posterior$lognormconst
}

#### Compute the KL distance:
Compute_KL <- function(x, logpx, logqx){
  dx <- diff(x)
  left <- c(0,dx)
  right <- c(dx,0)
  0.5 * sum(left * (logpx - logqx) * exp(logpx)) + 0.5 * sum(right * (logpx - logqx) * exp(logpx))
}

#### Compute the KS distance:
Compute_KS <- function(x, qx, px){
  dx <- c(diff(x),0)
  max(abs(cumsum(qx * dx) - cumsum(px * dx)))
}

Easy Example:

Define the log posterior function:

log_prior <- function(x){
  1
}
log_likelihood <- function(x){
  x*sin(x)
}
eval_once <- function(x){
  log_prior(x) + log_likelihood(x)
}
eval_once_mapped <- function(y){
  eval_once(pnorm(y) * (upper - lower) + lower) + dnorm(y, log = T) + log(upper - lower)
}
x <- seq(0.01,9.99, by = 0.01)
y <- qnorm((x - lower)/(upper - lower))
true_log_norm_constant <- integrate_aghq(f = function(y) eval_once_mapped(y))
true_log_post_mapped <- function(y) {eval_once_mapped(y) - true_log_norm_constant}
plot((true_log_post_mapped(y)) ~ y, type = "l", cex.lab = 1.5, cex.axis = 1.5, 
     xlab = "y", ylab = "log density", lwd = 2, col = "blue")

Past versions of unnamed-chunk-2-1.png

Version	Author	Date
d35a475	Ziang Zhang	2025-04-21

true_log_post <- function(x) {true_log_post_mapped(qnorm((x - lower)/(upper - lower))) - dnorm(qnorm((x - lower)/(upper - lower)), log = T) - log(upper - lower)}
integrate(function(x) exp(true_log_post(x)), lower = 0, upper = 10)

1.000275 with absolute error < 9.7e-05

Let’s run BOSS with the above log posterior function:

objective_func <- eval_once
eval_num <- seq(5, 100, by = 5)

result_ad <- BOSS(
  func = eval_once, initial_design = 5,
  update_step = 5, max_iter = (max(eval_num) - 5),
  opt.lengthscale.grid = 100, opt.grid = 1000,
  delta = 0.01, noise_var = noise_var,
  lower = lower, upper = upper,
  verbose = 0,
  modal_iter_check = 5, modal_check_warmup = 10, modal_k.nn = 5, modal_eps = 0, criterion = "modal"
)
saveRDS(result_ad, file = paste0(data_path, "/result_ad_easy.rds"))

BO_result_list <- list()
BO_result_original_list <- list()
result_ad <- readRDS(file = paste0(data_path, "/result_ad_easy.rds"))
for (i in 1:length(eval_num)) {
  eval_number <- eval_num[i]

  result_ad_selected <- list(x = result_ad$result$x[1:eval_number, ],
                             x_original = result_ad$result$x_original[1:eval_number, ],
                             y = result_ad$result$y[1:eval_number])
  
  data_to_smooth <- result_ad_selected
  BO_result_original_list[[i]] <- data_to_smooth

  ff <- list()
  ff$fn <- function(y) as.numeric(surrogate(pnorm(y), data_to_smooth = data_to_smooth) + dnorm(y, log = TRUE))
  fn_vals <- sapply(y, ff$fn)

  lognormal_const <- integrate_aghq(f = ff$fn)
  post_y <- data.frame(y = y, pos = exp(fn_vals - lognormal_const))
  post_x <- data.frame(x = pnorm(post_y$y) * (upper - lower) + lower, post = (post_y$pos / dnorm(post_y$y))/(upper - lower) )
  
  BO_result_list[[i]] <- post_x
}
saveRDS(BO_result_list, file = paste0(data_path, "/BO_result_list_easy.rds"))
saveRDS(BO_result_original_list, file = paste0(data_path, "/BO_result_original_list_easy.rds"))

Some illustrations

Here are some illustrations of the approximated posterior from BOSS with different numbers of BO iterations \(B\) (red: approximated posterior, black: true posterior).

BO_result_list <- readRDS(file = paste0(data_path, "/BO_result_list_easy.rds"))
BO_result_original_list <- readRDS(file = paste0(data_path, "/BO_result_original_list_easy.rds"))

B = 5

to_plot_data <- BO_result_list[[1]]
to_plot_data$logpos <- log(to_plot_data$pos)
to_plot_data_obs <- BO_result_original_list[[1]]
y_min <- -15; y_max <- 0
mar.default <- c(5,4,4,2)
par(mar = mar.default + c(0, 1, 0, 0), mfrow = c(2, 1))
plot(to_plot_data$logpos ~ to_plot_data$x, 
       type = "l", lty = "dashed", col = "red", lwd = 2,
       ylab = "Log Post", xlab = expression(alpha),
       ylim = c(y_min, y_max), cex.lab = 1.5, cex.axis = 1.5)
lines((true_log_post(x)) ~ x, lwd = 1)
y_offset <- 0.03 * (y_max - y_min) # adjust offset as needed
for(x_val in to_plot_data_obs$x_original) {
    segments(x_val, y_min, x_val, y_min - y_offset, col = "red")
}

plot(exp(to_plot_data$logpos) ~ to_plot_data$x, 
       type = "l", lty = "dashed", col = "red", lwd = 2,
       ylab = "Post", xlab = expression(alpha), ylim = c(0,2), cex.lab = 1.5, cex.axis = 1.5)
lines(exp(true_log_post(x)) ~ x, lwd = 1)
y_min <- -0.03
y_offset <- 0.01 * (2) # adjust offset as needed
for(x_val in to_plot_data_obs$x_original) {
    segments(x_val, y_min, x_val, y_min - y_offset, col = "red")
}

Past versions of unnamed-chunk-7-1.png

Version	Author	Date
d35a475	Ziang Zhang	2025-04-21

B = 10

to_plot_data <- BO_result_list[[2]]
to_plot_data$logpos <- log(to_plot_data$pos)
to_plot_data_obs <- BO_result_original_list[[2]]
y_min <- -15; y_max <- 0

mar.default <- c(5,4,4,2)
par(mar = mar.default + c(0, 1, 0, 0), mfrow = c(2, 1))
plot(to_plot_data$logpos ~ to_plot_data$x, 
       type = "l", lty = "dashed", col = "red", lwd = 2,
       ylab = "Log Post", xlab = expression(alpha),
       ylim = c(y_min, y_max), cex.lab = 1.5, cex.axis = 1.5)
lines((true_log_post(x)) ~ x, lwd = 1)
y_offset <- 0.03 * (y_max - y_min) # adjust offset as needed
for(x_val in to_plot_data_obs$x_original) {
    segments(x_val, y_min, x_val, y_min - y_offset, col = "red")
}

plot(exp(to_plot_data$logpos) ~ to_plot_data$x, 
       type = "l", lty = "dashed", col = "red", lwd = 2,
       ylab = "Post", xlab = expression(alpha), ylim = c(0,2), cex.lab = 1.5, cex.axis = 1.5)
lines(exp(true_log_post(x)) ~ x, lwd = 1)
y_min <- -0.03
y_offset <- 0.01 * (2) # adjust offset as needed
for(x_val in to_plot_data_obs$x_original) {
    segments(x_val, y_min, x_val, y_min - y_offset, col = "red")
}

Past versions of unnamed-chunk-8-1.png

Version	Author	Date
d35a475	Ziang Zhang	2025-04-21

B = 30

to_plot_data <- BO_result_list[[6]]
to_plot_data$logpos <- log(to_plot_data$pos)
to_plot_data_obs <- BO_result_original_list[[6]]
y_min <- -15; y_max <- 0

mar.default <- c(5,4,4,2)
par(mar = mar.default + c(0, 1, 0, 0), mfrow = c(2, 1))
plot(to_plot_data$logpos ~ to_plot_data$x, 
       type = "l", lty = "dashed", col = "red", lwd = 2,
       ylab = "Log Post", xlab = expression(alpha),
       ylim = c(y_min, y_max), cex.lab = 1.5, cex.axis = 1.5)
lines((true_log_post(x)) ~ x, lwd = 1)
y_offset <- 0.03 * (y_max - y_min) # adjust offset as needed
for(x_val in to_plot_data_obs$x_original) {
    segments(x_val, y_min, x_val, y_min - y_offset, col = "red")
}

plot(exp(to_plot_data$logpos) ~ to_plot_data$x, 
       type = "l", lty = "dashed", col = "red", lwd = 2,
       ylab = "Post", xlab = expression(alpha), ylim = c(0,2), cex.lab = 1.5, cex.axis = 1.5)
lines(exp(true_log_post(x)) ~ x, lwd = 1)
y_min <- -0.03
y_offset <- 0.01 * (2) # adjust offset as needed
for(x_val in to_plot_data_obs$x_original) {
    segments(x_val, y_min, x_val, y_min - y_offset, col = "red")
}

Past versions of unnamed-chunk-9-1.png

Version	Author	Date
d35a475	Ziang Zhang	2025-04-21

B = 100

to_plot_data <- BO_result_list[[20]]
to_plot_data$logpos <- log(to_plot_data$pos)
to_plot_data_obs <- BO_result_original_list[[20]]
y_min <- -15; y_max <- 0

mar.default <- c(5,4,4,2)
par(mar = mar.default + c(0, 1, 0, 0), mfrow = c(2, 1))
plot(to_plot_data$logpos ~ to_plot_data$x, 
       type = "l", lty = "dashed", col = "red", lwd = 2,
       ylab = "Log Post", xlab = expression(alpha),
       ylim = c(y_min, y_max), cex.lab = 1.5, cex.axis = 1.5)
lines((true_log_post(x)) ~ x, lwd = 1)
y_offset <- 0.03 * (y_max - y_min) # adjust offset as needed
for(x_val in to_plot_data_obs$x_original) {
    segments(x_val, y_min, x_val, y_min - y_offset, col = "red")
}

plot(exp(to_plot_data$logpos) ~ to_plot_data$x, 
       type = "l", lty = "dashed", col = "red", lwd = 2,
       ylab = "Post", xlab = expression(alpha), ylim = c(0,2), cex.lab = 1.5, cex.axis = 1.5)
lines(exp(true_log_post(x)) ~ x, lwd = 1)
y_min <- -0.03
y_offset <- 0.01 * (2) # adjust offset as needed
for(x_val in to_plot_data_obs$x_original) {
    segments(x_val, y_min, x_val, y_min - y_offset, col = "red")
}

Past versions of unnamed-chunk-10-1.png

Version	Author	Date
d35a475	Ziang Zhang	2025-04-21

KL and KS:

Let’s visualize the KL and KS divergence between the true posterior and the approximated posterior from BOSS.

KL_vec <- c()
for (i in 1:length(eval_num)) {
  KL_vec[i] <- Compute_KL(x = x, logpx = true_log_post(x), logqx = log(BO_result_list[[i]]$pos))
}
mar.default <- c(5,4,4,2)
par(mar = mar.default + c(0, 1, 0, 0))
plot((KL_vec) ~ eval_num, type = "o", ylab = "KL", xlab = "eval number: B", cex.lab = 1.5, cex.axis = 1.5)

KS_vec <- c()
for (i in 1:length(eval_num)) {
  KS_vec[i] <- Compute_KS(x = x, px = exp(true_log_post(x)), qx = BO_result_list[[i]]$pos)
}
mar.default <- c(5,4,4,2)
par(mar = mar.default + c(0, 1, 0, 0))
plot((KS_vec) ~ eval_num, type = "o", ylab = "KS", xlab = "eval number: B", ylim = c(0,1), cex.lab = 1.5, cex.axis = 1.5)

Based on the KL and KS divergence, we can the approximated posterior from BOSS is indistinguishable from the true posterior after 10 iterations in the simple example.

Medium Example:

Define the log posterior function:

log_prior <- function(x){
  1
}
log_likelihood <- function(x){
  log(x+1)*sin(x*2) - x*cos(x*2)
}
eval_once <- function(x){
  log_prior(x) + log_likelihood(x)
}
eval_once_mapped <- function(y){
  eval_once(pnorm(y) * (upper - lower) + lower) + dnorm(y, log = T) + log(upper - lower)
}
x <- seq(0.01,9.99, by = 0.01)
y <- qnorm((x - lower)/(upper - lower))
true_log_norm_constant <- integrate_aghq(f = function(y) eval_once_mapped(y), k = 100)
true_log_post_mapped <- function(y) {eval_once_mapped(y) - true_log_norm_constant}
plot((true_log_post_mapped(y)) ~ y, type = "l", cex.lab = 1.5, cex.axis = 1.5, 
     xlab = "y", ylab = "log density", lwd = 2, col = "blue")

Past versions of unnamed-chunk-14-1.png

Version	Author	Date
d35a475	Ziang Zhang	2025-04-21

true_log_post <- function(x) {true_log_post_mapped(qnorm((x - lower)/(upper - lower))) - dnorm(qnorm((x - lower)/(upper - lower)), log = T) - log(upper - lower)}
integrate(function(x) exp(true_log_post(x)), lower = 0, upper = 10)

1.001231 with absolute error < 2.3e-05

Let’s run BOSS with the above log posterior function:

objective_func <- eval_once

result_ad <- BOSS(
  func = objective_func, initial_design = 5,
  update_step = 5, max_iter = (max(eval_num) - 5),
  opt.lengthscale.grid = 100, opt.grid = 1000,
  delta = 0.01, noise_var = noise_var,
  lower = lower, upper = upper,
  modal_iter_check = 5, modal_check_warmup = 10, modal_k.nn = 5, modal_eps = 0, criterion = "modal"
)
saveRDS(result_ad, file = paste0(data_path, "/result_ad_med.rds"))

BO_result_list <- list()
BO_result_original_list <- list()
result_ad <- readRDS(file = paste0(data_path, "/result_ad_med.rds"))
for (i in 1:length(eval_num)) {
  eval_number <- eval_num[i]

  result_ad_selected <- list(x = result_ad$result$x[1:eval_number, ],
                             x_original = result_ad$result$x_original[1:eval_number, ],
                             y = result_ad$result$y[1:eval_number])
  
  data_to_smooth <- result_ad_selected
  BO_result_original_list[[i]] <- data_to_smooth

  ff <- list()
  ff$fn <- function(y) as.numeric(surrogate(pnorm(y), data_to_smooth = data_to_smooth) + dnorm(y, log = TRUE))
  fn_vals <- sapply(y, ff$fn)

  lognormal_const <- integrate_aghq(f = ff$fn)
  post_y <- data.frame(y = y, pos = exp(fn_vals - lognormal_const))
  post_x <- data.frame(x = pnorm(post_y$y) * (upper - lower) + lower, post = (post_y$pos / dnorm(post_y$y))/(upper - lower) )
  
  BO_result_list[[i]] <- post_x
}
saveRDS(BO_result_list, file = paste0(data_path, "/BO_result_list_med.rds"))
saveRDS(BO_result_original_list, file = paste0(data_path, "/BO_result_original_list_med.rds"))

Some illustrations

Here are some illustrations of the approximated posterior from BOSS with different numbers of BO iterations \(B\) (red: approximated posterior, black: true posterior).

BO_result_list <- readRDS(file = paste0(data_path, "/BO_result_list_med.rds"))
BO_result_original_list <- readRDS(file = paste0(data_path, "/BO_result_original_list_med.rds"))

B = 5

to_plot_data <- BO_result_list[[1]]
to_plot_data$logpos <- log(to_plot_data$pos)
to_plot_data_obs <- BO_result_original_list[[1]]
y_min <- -20; y_max <- 5
mar.default <- c(5,4,4,2)
par(mar = mar.default + c(0, 1, 0, 0), mfrow = c(2, 1))
plot(to_plot_data$logpos ~ to_plot_data$x, 
       type = "l", lty = "dashed", col = "red", lwd = 2,
       ylab = "Log Post", xlab = expression(alpha),
       ylim = c(y_min, y_max), cex.lab = 1.5, cex.axis = 1.5)
lines((true_log_post(x)) ~ x, lwd = 1)
y_offset <- 0.03 * (y_max - y_min) # adjust offset as needed
for(x_val in to_plot_data_obs$x_original) {
    segments(x_val, y_min, x_val, y_min - y_offset, col = "red")
}

plot(exp(to_plot_data$logpos) ~ to_plot_data$x, 
       type = "l", lty = "dashed", col = "red", lwd = 2,
       ylab = "Post", xlab = expression(alpha), ylim = c(0, 2.5), cex.lab = 1.5, cex.axis = 1.5)
lines(exp(true_log_post(x)) ~ x, lwd = 1)
y_min <- -0.03
y_offset <- 0.01 * (2) # adjust offset as needed
for(x_val in to_plot_data_obs$x_original) {
    segments(x_val, y_min, x_val, y_min - y_offset, col = "red")
}

Past versions of unnamed-chunk-19-1.png

Version	Author	Date
ff55aef	Ziang Zhang	2025-04-21
d35a475	Ziang Zhang	2025-04-21

B = 10

to_plot_data <- BO_result_list[[2]]
to_plot_data$logpos <- log(to_plot_data$pos)
to_plot_data_obs <- BO_result_original_list[[2]]
y_min <- -20; y_max <- 5

mar.default <- c(5,4,4,2)
par(mar = mar.default + c(0, 1, 0, 0), mfrow = c(2, 1))
plot(to_plot_data$logpos ~ to_plot_data$x, 
       type = "l", lty = "dashed", col = "red", lwd = 2,
       ylab = "Log Post", xlab = expression(alpha),
       ylim = c(y_min, y_max), cex.lab = 1.5, cex.axis = 1.5)
lines((true_log_post(x)) ~ x, lwd = 1)
y_offset <- 0.03 * (y_max - y_min) # adjust offset as needed
for(x_val in to_plot_data_obs$x_original) {
    segments(x_val, y_min, x_val, y_min - y_offset, col = "red")
}

plot(exp(to_plot_data$logpos) ~ to_plot_data$x, 
       type = "l", lty = "dashed", col = "red", lwd = 2,
       ylab = "Post", xlab = expression(alpha), ylim = c(0, 2.5), cex.lab = 1.5, cex.axis = 1.5)
lines(exp(true_log_post(x)) ~ x, lwd = 1)
y_min <- -0.03
y_offset <- 0.01 * (2) # adjust offset as needed
for(x_val in to_plot_data_obs$x_original) {
    segments(x_val, y_min, x_val, y_min - y_offset, col = "red")
}

Past versions of unnamed-chunk-20-1.png

Version	Author	Date
ff55aef	Ziang Zhang	2025-04-21
d35a475	Ziang Zhang	2025-04-21

B = 30

to_plot_data <- BO_result_list[[6]]
to_plot_data$logpos <- log(to_plot_data$pos)
to_plot_data_obs <- BO_result_original_list[[6]]
y_min <- -20; y_max <- 5

mar.default <- c(5,4,4,2)
par(mar = mar.default + c(0, 1, 0, 0), mfrow = c(2, 1))
plot(to_plot_data$logpos ~ to_plot_data$x, 
       type = "l", lty = "dashed", col = "red", lwd = 2,
       ylab = "Log Post", xlab = expression(alpha),
       ylim = c(y_min, y_max), cex.lab = 1.5, cex.axis = 1.5)
lines((true_log_post(x)) ~ x, lwd = 1)
y_offset <- 0.03 * (y_max - y_min) # adjust offset as needed
for(x_val in to_plot_data_obs$x_original) {
    segments(x_val, y_min, x_val, y_min - y_offset, col = "red")
}

plot(exp(to_plot_data$logpos) ~ to_plot_data$x, 
       type = "l", lty = "dashed", col = "red", lwd = 2,
       ylab = "Post", xlab = expression(alpha), ylim = c(0, 2.5), cex.lab = 1.5, cex.axis = 1.5)
lines(exp(true_log_post(x)) ~ x, lwd = 1)
y_min <- -0.03
y_offset <- 0.01 * (2) # adjust offset as needed
for(x_val in to_plot_data_obs$x_original) {
    segments(x_val, y_min, x_val, y_min - y_offset, col = "red")
}

Past versions of unnamed-chunk-21-1.png

Version	Author	Date
ff55aef	Ziang Zhang	2025-04-21
d35a475	Ziang Zhang	2025-04-21

B = 100

to_plot_data <- BO_result_list[[20]]
to_plot_data$logpos <- log(to_plot_data$pos)
to_plot_data_obs <- BO_result_original_list[[20]]
y_min <- -20; y_max <- 5

mar.default <- c(5,4,4,2)
par(mar = mar.default + c(0, 1, 0, 0), mfrow = c(2, 1))
plot(to_plot_data$logpos ~ to_plot_data$x, 
       type = "l", lty = "dashed", col = "red", lwd = 2,
       ylab = "Log Post", xlab = expression(alpha),
       ylim = c(y_min, y_max), cex.lab = 1.5, cex.axis = 1.5)
lines((true_log_post(x)) ~ x, lwd = 1)
y_offset <- 0.03 * (y_max - y_min) # adjust offset as needed
for(x_val in to_plot_data_obs$x_original) {
    segments(x_val, y_min, x_val, y_min - y_offset, col = "red")
}

plot(exp(to_plot_data$logpos) ~ to_plot_data$x, 
       type = "l", lty = "dashed", col = "red", lwd = 2,
       ylab = "Post", xlab = expression(alpha), ylim = c(0, 2.5), cex.lab = 1.5, cex.axis = 1.5)
lines(exp(true_log_post(x)) ~ x, lwd = 1)
y_min <- -0.03
y_offset <- 0.01 * (2) # adjust offset as needed
for(x_val in to_plot_data_obs$x_original) {
    segments(x_val, y_min, x_val, y_min - y_offset, col = "red")
}

Past versions of unnamed-chunk-22-1.png

Version	Author	Date
ff55aef	Ziang Zhang	2025-04-21
d35a475	Ziang Zhang	2025-04-21

KL and KS:

Let’s visualize the KL and KS divergence between the true posterior and the approximated posterior from BOSS.

KL_vec <- c()
for (i in 1:length(eval_num)) {
  KL_vec[i] <- Compute_KL(x = x, logpx = true_log_post(x), logqx = log(BO_result_list[[i]]$pos))
}
mar.default <- c(5,4,4,2)
par(mar = mar.default + c(0, 1, 0, 0))
plot((KL_vec) ~ eval_num, type = "o", ylab = "KL", xlab = "eval number: B", cex.lab = 1.5, cex.axis = 1.5)

Past versions of unnamed-chunk-24-1.png

Version	Author	Date
d35a475	Ziang Zhang	2025-04-21

KS_vec <- c()
for (i in 1:length(eval_num)) {
  KS_vec[i] <- Compute_KS(x = x, px = exp(true_log_post(x)), qx = BO_result_list[[i]]$pos)
}
mar.default <- c(5,4,4,2)
par(mar = mar.default + c(0, 1, 0, 0))
plot((KS_vec) ~ eval_num, type = "o", ylab = "KS", xlab = "eval number: B", ylim = c(0,1), cex.lab = 1.5, cex.axis = 1.5)

Past versions of unnamed-chunk-24-2.png

Version	Author	Date
d35a475	Ziang Zhang	2025-04-21

Based on the KL and KS divergence, we can the approximated posterior from BOSS is indistinguishable from the true posterior after 20 iterations of BO in the medium example.

Hard Example:

Define the log posterior function:

log_prior <- function(x){
  1
}
log_likelihood <- function(x){
  log(x + 1) * (sin(x * 4) + cos(x * 2))
}
eval_once <- function(x){
  log_prior(x) + log_likelihood(x)
}
eval_once_mapped <- function(y){
  eval_once(pnorm(y) * (upper - lower) + lower) + dnorm(y, log = T) + log(upper - lower)
}
x <- seq(0.01,9.99, by = 0.01)
y <- qnorm((x - lower)/(upper - lower))
true_log_norm_constant <- log(integrate(f = function(y) exp(eval_once_mapped(y)), lower = -Inf, upper = Inf)$value)
true_log_post_mapped <- function(y) {eval_once_mapped(y) - true_log_norm_constant}
plot((true_log_post_mapped(y)) ~ y, type = "l", cex.lab = 1.5, cex.axis = 1.5, 
     xlab = "y", ylab = "log density", lwd = 2, col = "blue")

Past versions of unnamed-chunk-26-1.png

Version	Author	Date
d35a475	Ziang Zhang	2025-04-21

true_log_post <- function(x) {true_log_post_mapped(qnorm((x - lower)/(upper - lower))) - dnorm(qnorm((x - lower)/(upper - lower)), log = T) - log(upper - lower)}
integrate(function(x) exp(true_log_post(x)), lower = 0, upper = 10)

1 with absolute error < 9.1e-05

Let’s run BOSS with the above log posterior function:

objective_func <- eval_once

result_ad <- BOSS(
  func = objective_func, initial_design = 5,
  update_step = 5, max_iter = (max(eval_num) - 5),
  opt.lengthscale.grid = 100, opt.grid = 1000,
  delta = 0.01, noise_var = noise_var,
  lower = lower, upper = upper,
  AGHQ_iter_check = Inf, AGHQ_eps = 0
)
saveRDS(result_ad, file = paste0(data_path, "/result_ad_hard.rds"))

BO_result_list <- list()
BO_result_original_list <- list()
result_ad <- readRDS(file = paste0(data_path, "/result_ad_hard.rds"))
for (i in 1:length(eval_num)) {
  eval_number <- eval_num[i]

  result_ad_selected <- list(x = result_ad$result$x[1:eval_number, ],
                             x_original = result_ad$result$x_original[1:eval_number, ],
                             y = result_ad$result$y[1:eval_number])
  
  data_to_smooth <- result_ad_selected
  BO_result_original_list[[i]] <- data_to_smooth

  ff <- list()
  ff$fn <- function(y) as.numeric(surrogate(pnorm(y), data_to_smooth = data_to_smooth) + dnorm(y, log = TRUE))
  fn_vals <- sapply(y, ff$fn)

  lognormal_const <- log(integrate(f = function(y) exp(ff$fn(y)), lower = -Inf, upper = Inf)$value)
  post_y <- data.frame(y = y, pos = exp(fn_vals - lognormal_const))
  post_x <- data.frame(x = pnorm(post_y$y) * (upper - lower) + lower, post = (post_y$pos / dnorm(post_y$y))/(upper - lower) )
  
  BO_result_list[[i]] <- post_x
}
saveRDS(BO_result_list, file = paste0(data_path, "/BO_result_list_hard.rds"))
saveRDS(BO_result_original_list, file = paste0(data_path, "/BO_result_original_list_hard.rds"))

Some illustrations

Here are some illustrations of the approximated posterior from BOSS with different numbers of BO iterations \(B\) (red: approximated posterior, black: true posterior).

BO_result_list <- readRDS(file = paste0(data_path, "/BO_result_list_hard.rds"))
BO_result_original_list <- readRDS(file = paste0(data_path, "/BO_result_original_list_hard.rds"))

B = 5

to_plot_data <- BO_result_list[[1]]
to_plot_data$logpos <- log(to_plot_data$pos)
to_plot_data_obs <- BO_result_original_list[[1]]
y_min <- -20; y_max <- 5
mar.default <- c(5,4,4,2)
par(mar = mar.default + c(0, 1, 0, 0), mfrow = c(2, 1))
plot(to_plot_data$logpos ~ to_plot_data$x, 
       type = "l", lty = "dashed", col = "red", lwd = 2,
       ylab = "Log Post", xlab = expression(alpha),
       ylim = c(y_min, y_max), cex.lab = 1.5, cex.axis = 1.5)
lines((true_log_post(x)) ~ x, lwd = 1)
y_offset <- 0.03 * (y_max - y_min) # adjust offset as needed
for(x_val in to_plot_data_obs$x_original) {
    segments(x_val, y_min, x_val, y_min - y_offset, col = "red")
}

plot(exp(to_plot_data$logpos) ~ to_plot_data$x, 
       type = "l", lty = "dashed", col = "red", lwd = 2,
       ylab = "Post", xlab = expression(alpha), ylim = c(0,2), cex.lab = 1.5, cex.axis = 1.5)
lines(exp(true_log_post(x)) ~ x, lwd = 1)
y_min <- -0.03
y_offset <- 0.01 * (2) # adjust offset as needed
for(x_val in to_plot_data_obs$x_original) {
    segments(x_val, y_min, x_val, y_min - y_offset, col = "red")
}

Past versions of unnamed-chunk-31-1.png

Version	Author	Date
ff55aef	Ziang Zhang	2025-04-21
d35a475	Ziang Zhang	2025-04-21

B = 10

to_plot_data <- BO_result_list[[2]]
to_plot_data$logpos <- log(to_plot_data$pos)
to_plot_data_obs <- BO_result_original_list[[2]]
y_min <- -20; y_max <- 5
mar.default <- c(5,4,4,2)
par(mar = mar.default + c(0, 1, 0, 0), mfrow = c(2, 1))
plot(to_plot_data$logpos ~ to_plot_data$x, 
       type = "l", lty = "dashed", col = "red", lwd = 2,
       ylab = "Log Post", xlab = expression(alpha),
       ylim = c(y_min, y_max), cex.lab = 1.5, cex.axis = 1.5)
lines((true_log_post(x)) ~ x, lwd = 1)
y_offset <- 0.03 * (y_max - y_min) # adjust offset as needed
for(x_val in to_plot_data_obs$x_original) {
    segments(x_val, y_min, x_val, y_min - y_offset, col = "red")
}

plot(exp(to_plot_data$logpos) ~ to_plot_data$x, 
       type = "l", lty = "dashed", col = "red", lwd = 2,
       ylab = "Post", xlab = expression(alpha), ylim = c(0,2), cex.lab = 1.5, cex.axis = 1.5)
lines(exp(true_log_post(x)) ~ x, lwd = 1)
y_min <- -0.03
y_offset <- 0.01 * (2) # adjust offset as needed
for(x_val in to_plot_data_obs$x_original) {
    segments(x_val, y_min, x_val, y_min - y_offset, col = "red")
}

Past versions of unnamed-chunk-32-1.png

Version	Author	Date
ff55aef	Ziang Zhang	2025-04-21
d35a475	Ziang Zhang	2025-04-21

B = 30

to_plot_data <- BO_result_list[[6]]
to_plot_data$logpos <- log(to_plot_data$pos)
to_plot_data_obs <- BO_result_original_list[[6]]
y_min <- -20; y_max <- 5
mar.default <- c(5,4,4,2)
par(mar = mar.default + c(0, 1, 0, 0), mfrow = c(2, 1))
plot(to_plot_data$logpos ~ to_plot_data$x, 
       type = "l", lty = "dashed", col = "red", lwd = 2,
       ylab = "Log Post", xlab = expression(alpha),
       ylim = c(y_min, y_max), cex.lab = 1.5, cex.axis = 1.5)
lines((true_log_post(x)) ~ x, lwd = 1)
y_offset <- 0.03 * (y_max - y_min) # adjust offset as needed
for(x_val in to_plot_data_obs$x_original) {
    segments(x_val, y_min, x_val, y_min - y_offset, col = "red")
}

plot(exp(to_plot_data$logpos) ~ to_plot_data$x, 
       type = "l", lty = "dashed", col = "red", lwd = 2,
       ylab = "Post", xlab = expression(alpha), ylim = c(0,2), cex.lab = 1.5, cex.axis = 1.5)
lines(exp(true_log_post(x)) ~ x, lwd = 1)
y_min <- -0.03
y_offset <- 0.01 * (2) # adjust offset as needed
for(x_val in to_plot_data_obs$x_original) {
    segments(x_val, y_min, x_val, y_min - y_offset, col = "red")
}

Past versions of unnamed-chunk-33-1.png

Version	Author	Date
ff55aef	Ziang Zhang	2025-04-21
d35a475	Ziang Zhang	2025-04-21

B = 100

to_plot_data <- BO_result_list[[20]]
to_plot_data$logpos <- log(to_plot_data$pos)
to_plot_data_obs <- BO_result_original_list[[20]]
y_min <- -20; y_max <- 5
mar.default <- c(5,4,4,2)
par(mar = mar.default + c(0, 1, 0, 0), mfrow = c(2, 1))
plot(to_plot_data$logpos ~ to_plot_data$x, 
       type = "l", lty = "dashed", col = "red", lwd = 2,
       ylab = "Log Post", xlab = expression(alpha),
       ylim = c(y_min, y_max), cex.lab = 1.5, cex.axis = 1.5)
lines((true_log_post(x)) ~ x, lwd = 1)
y_offset <- 0.03 * (y_max - y_min) # adjust offset as needed
for(x_val in to_plot_data_obs$x_original) {
    segments(x_val, y_min, x_val, y_min - y_offset, col = "red")
}
y_min <- -20; y_max <- 5
plot(exp(to_plot_data$logpos) ~ to_plot_data$x, 
       type = "l", lty = "dashed", col = "red", lwd = 2,
       ylab = "Post", xlab = expression(alpha), ylim = c(0,2), cex.lab = 1.5, cex.axis = 1.5)
lines(exp(true_log_post(x)) ~ x, lwd = 1)
y_min <- -0.03
y_offset <- 0.01 * (2) # adjust offset as needed
for(x_val in to_plot_data_obs$x_original) {
    segments(x_val, y_min, x_val, y_min - y_offset, col = "red")
}

Past versions of unnamed-chunk-34-1.png

Version	Author	Date
ff55aef	Ziang Zhang	2025-04-21
d35a475	Ziang Zhang	2025-04-21

Compute KL and KS:

Let’s visualize the KL and KS divergence between the true posterior and the approximated posterior from BOSS.

KL_vec <- c()
for (i in 1:length(eval_num)) {
  KL_vec[i] <- Compute_KL(x = x, logpx = true_log_post(x), logqx = log(BO_result_list[[i]]$pos))
}

mar.default <- c(5,4,4,2)
par(mar = mar.default + c(0, 1, 0, 0))
plot((KL_vec) ~ eval_num, type = "o", ylab = "KL", xlab = "eval number: B", cex.lab = 1.5, cex.axis = 1.5)

Past versions of unnamed-chunk-36-1.png

Version	Author	Date
ff55aef	Ziang Zhang	2025-04-21
d35a475	Ziang Zhang	2025-04-21

KS_vec <- c()
for (i in 1:length(eval_num)) {
  KS_vec[i] <- Compute_KS(x = x, px = exp(true_log_post(x)), qx = BO_result_list[[i]]$pos)
}
mar.default <- c(5,4,4,2)
par(mar = mar.default + c(0, 1, 0, 0))
plot((KS_vec) ~ eval_num, type = "o", ylab = "KS", xlab = "eval number: B", ylim = c(0,1), cex.lab = 1.5, cex.axis = 1.5)

Past versions of unnamed-chunk-36-2.png

Version	Author	Date
ff55aef	Ziang Zhang	2025-04-21
d35a475	Ziang Zhang	2025-04-21

Based on the KL and KS divergence, we can the approximated posterior from BOSS is indistinguishable from the true posterior after around 20 iterations in this hard example.

Session information

sessionInfo()

R version 4.3.1 (2023-06-16)
Platform: aarch64-apple-darwin20 (64-bit)
Running under: macOS Monterey 12.7.4

Matrix products: default
BLAS:   /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/lib/libRblas.0.dylib 
LAPACK: /Library/Frameworks/R.framework/Versions/4.3-arm64/Resources/lib/libRlapack.dylib;  LAPACK version 3.11.0

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/Chicago
tzcode source: internal

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

other attached packages:
[1] aghq_0.4.1      ggplot2_3.5.1   npreg_1.1.0     workflowr_1.7.1

loaded via a namespace (and not attached):
 [1] sass_0.4.9          utf8_1.2.4          generics_0.1.3     
 [4] lattice_0.22-6      stringi_1.8.4       digest_0.6.37      
 [7] magrittr_2.0.3      evaluate_1.0.1      grid_4.3.1         
[10] fastmap_1.2.0       rprojroot_2.0.4     jsonlite_1.8.9     
[13] Matrix_1.6-4        processx_3.8.4      whisker_0.4.1      
[16] ps_1.8.0            promises_1.3.0      httr_1.4.7         
[19] fansi_1.0.6         scales_1.3.0        numDeriv_2016.8-1.1
[22] jquerylib_0.1.4     cli_3.6.3           rlang_1.1.4        
[25] munsell_0.5.1       withr_3.0.2         cachem_1.1.0       
[28] yaml_2.3.10         tools_4.3.1         dplyr_1.1.4        
[31] colorspace_2.1-1    httpuv_1.6.15       vctrs_0.6.5        
[34] R6_2.5.1            lifecycle_1.0.4     git2r_0.33.0       
[37] stringr_1.5.1       fs_1.6.4            pkgconfig_2.0.3    
[40] callr_3.7.6         pillar_1.9.0        bslib_0.8.0        
[43] later_1.3.2         gtable_0.3.6        data.table_1.16.2  
[46] glue_1.8.0          Rcpp_1.0.13-1       statmod_1.5.0      
[49] mvQuad_1.0-8        highr_0.11          xfun_0.48          
[52] tibble_3.2.1        tidyselect_1.2.1    rstudioapi_0.16.0  
[55] knitr_1.48          htmltools_0.5.8.1   rmarkdown_2.28     
[58] compiler_4.3.1      getPass_0.2-4