Last updated: 2025-04-22

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Introduction

In this tutorial, we examine several diagnostic tools that can be used to assess the convergence of BOSS.

For now, let’s assume the following true (log) posterior:

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"))

lower = 0; upper = 10

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")

Version Author Date
b5d1ce2 Ziang Zhang 2025-04-22
6bb1cbf Ziang Zhang 2025-04-22 
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
plot(exp(true_log_post(x)) ~ x, type = "l", cex.lab = 1.5, cex.axis = 1.5, 
     xlab = "x", ylab = "Posterior density", lwd = 2, col = "blue")

Version Author Date
b5d1ce2 Ziang Zhang 2025-04-22

KL Divergence

Let \(f_t\) and \(f_{t-j}\) be the corresponding surrogate density at time \(t\) and \(t-j\), respectively. We can compute the KL divergence between \(f_t\) and \(f_{t-j}\) as follows: \[Q_t = KL(f_t,f_{t-j}) = \int \log \frac{f_t(x)}{f_{t-j}(x)}f_{t}(x)dx.\]

For one-dimensional problems, this can be done efficiently through numerical integration. For higher-dimensional problems, sampling-based methods can be used to approximate the KL divergence.

result_ad <- BOSS(
  func = eval_once, initial_design = 5,
  update_step = 5, max_iter = 30,
  opt.lengthscale.grid = 100, opt.grid = 1000,
  delta = 0.01, noise_var = noise_var,
  lower = lower, upper = upper,
  verbose = 0,
  KL_iter_check = 1, KL_check_warmup = 5, KL_eps = 0, criterion = "KL"
)

plot(result_ad$KL_result$KL ~ result_ad$KL_result$i,
     xlab = "Iteration", ylab = "KL statistic",
     main = "KL statistic over iterations", 
     log = "y",
     pch = 19, col = "blue")

Version Author Date
6bb1cbf Ziang Zhang 2025-04-22

Based on the KL divergence, it seems like the algorithm has converged around 30 iterations.

KS Statistics

The Kolmogorov-Smirnov (KS) statistic measures the maximum difference between the cumulative distribution functions (CDFs) \(F_t\) and \(F_{t-j}\) of the surrogate densities \(f_t\) and \(f_{t-j}\), respectively. Specifically, for one dimensional problems, the KS statistic is defined as: \[Q_t = \max_x |F_t(x) - F_{t-j}(x)|.\]

result_ad <- BOSS(
  func = eval_once, initial_design = 5,
  update_step = 5, max_iter = 30,
  opt.lengthscale.grid = 100, opt.grid = 1000,
  delta = 0.01, noise_var = noise_var,
  lower = lower, upper = upper,
  verbose = 0,
  KS_iter_check = 1, KS_check_warmup = 5, KS_eps = 0, criterion = "KS"
)

plot(result_ad$KS_result$KS ~ result_ad$KS_result$i,
     xlab = "Iteration", ylab = "KS statistic",
     main = "KS statistic over iterations", 
     log = "y",
     pch = 19, col = "blue")

Version Author Date
6bb1cbf Ziang Zhang 2025-04-22

Based on the KS statistic, the conclusion is similar to that of the KL divergence. The KS statistics is very close to 0 after 30 iterations, indicating that the algorithm has likely converged.