From 34d70fd35f518abcde60aa90a93f84cd470aa2ab Mon Sep 17 00:00:00 2001
From: Aki Vehtari <Aki.Vehtari@aalto.fi>
Date: Thu, 15 Oct 2026 11:04:48 +0300
Subject: [PATCH 1/2] Use a relative tolerance in is_constant

---
 R/helpers.R                     |  7 ++++++-
 tests/testthat/test_0_helpers.R | 12 ++++++++++++
 tests/testthat/test_E_loo.R     | 12 ++++++++++++
 3 files changed, 30 insertions(+), 1 deletion(-)

diff --git a/R/helpers.R b/R/helpers.R
index 37d407a8..962e93bc 100644
--- a/R/helpers.R
+++ b/R/helpers.R
@@ -277,6 +277,11 @@ release_questions <- function() {
 }
 # nocov end
 
+# `x` is constant if `max(x) - min(x)` is at most `tol` times the largest
+# absolute value in `x`. Default is `.Machine$double.eps`.
 is_constant <- function(x, tol = .Machine$double.eps) {
-  abs(max(x) - min(x)) < tol
+  r <- range(x)
+  spread <- r[2] - r[1]
+  # the equality check covers all-zero and all-infinite x
+  r[1] == r[2] || (is.finite(spread) && spread <= tol * max(abs(r)))
 }
diff --git a/tests/testthat/test_0_helpers.R b/tests/testthat/test_0_helpers.R
index a0add421..01a808d6 100644
--- a/tests/testthat/test_0_helpers.R
+++ b/tests/testthat/test_0_helpers.R
@@ -91,3 +91,15 @@ test_that("loo_cores works", {
   expect_warning(expect_equal(loo_cores(10), 2), "deprecated")
   options(loo.cores = NULL)
 })
+
+test_that("is_constant uses a tolerance relative to the magnitude of x", {
+  expect_true(is_constant(rep(0, 5)))
+  expect_true(is_constant(rep(1e-300, 5)))
+  expect_true(is_constant(rep(-Inf, 5)))
+  expect_true(is_constant(c(1, 1 + .Machine$double.eps)))
+  expect_false(is_constant(c(1e-20, 2e-20)))
+  expect_false(is_constant(c(1e-300, 2e-300)))
+  expect_false(is_constant(c(-Inf, 2)))
+  expect_false(is_constant(c(1, 2)))
+  expect_true(is.na(is_constant(c(1, NA))))
+})
diff --git a/tests/testthat/test_E_loo.R b/tests/testthat/test_E_loo.R
index 913ca9aa..cc7f75c1 100644
--- a/tests/testthat/test_E_loo.R
+++ b/tests/testthat/test_E_loo.R
@@ -245,3 +245,15 @@ test_that("weighted variance works", {
   x <- 1e8 + c(-1, 0, 1)
   expect_equal(.wvar(x, rep(1 / 3, 3)), 1)
 })
+
+test_that("E_loo pareto_k does not depend on the scale of x", {
+  # posterior::pareto_khat() of h * r also needs the relative tolerance
+  # with an older posterior it returns NA here
+  skip_if(posterior:::is_constant(c(1e-20, 2e-20)),
+          "posterior::is_constant() has an absolute tolerance")
+  k <- E_loo(x, psis_mat, type = "mean", log_ratios = log_rats)$pareto_k
+  for (s in c(1e-20, 1e-200)) {
+    k_s <- E_loo(s * x, psis_mat, type = "mean", log_ratios = log_rats)$pareto_k
+    expect_equal(k_s, k, tolerance = 1e-10)
+  }
+})

From 65dc87ed642c22ffa10c17a72c8622fe2a4926ac Mon Sep 17 00:00:00 2001
From: Florence Bockting <florence.bockting@aalto.fi>
Date: Fri, 16 Oct 2026 11:41:02 +0300
Subject: [PATCH 2/2] chore: add NEWS.md

---
 NEWS.md | 2 ++
 1 file changed, 2 insertions(+)

diff --git a/NEWS.md b/NEWS.md
index 877c1afe..544c0891 100644
--- a/NEWS.md
+++ b/NEWS.md
@@ -9,6 +9,8 @@ comparison-model minus reference-model by @florence-bockting in #391
 * Fix `crps()` and `scrps()` ignoring the `permutations` argument for numeric
 input by @thanasibakis in #398
 * Speed up the Bayesian bootstrap in `pseudobma_weights()` by @VisruthSK in #368
+* Use a tolerance relative to the magnitude of `x` in `is_constant()` by
+@avehtari in #407
 
 # loo 2.10.1