Density-Ratio Backends and the proxymix Binding

The cross-package contract

kernR::estimate_density_ratio() is the entry point for every kernR estimator that needs to reweight observational samples to an interventional distribution — most notably bd_hsic_test() for the backdoor-HSIC causal test. The function offers four density-ratio backends behind a single signature:

Backend	Family	When to use
`logistic`	Noise-contrastive classifier	Default; robust on smooth, unimodal densities
`ranger`	Random-forest classifier	Flexible non-linear NCE; needs `ranger`
`xgboost`	Gradient-boosted classifier	Strong on tabular interactions; needs `xgboost`
`proxymix`	Gaussian-mixture density ratio	Multimodal/skewed densities; parametric alternative

The proxymix backend is the cross-package wedge between kernR (the distributional verdict layer of the UQ ag stack) and proxymix (the Gaussian-mixture proxy / KL-density-ratio bridge, Hoek & Elliott 2024). It fits one GMM to the joint sample cloud (x, z), a second GMM to the product-of-marginals cloud (x_perm, z), then evaluates the analytic ratio of the two mixture densities at each observation. No classifier calibration step — the ratio is closed-form in the fitted parameters.

Four backends, one problem

A toy confounded design: z is a 2-D Gaussian confounder; x is a linear-Gaussian function of z; y carries a real causal effect from x plus a confounded path through z.

suppressPackageStartupMessages(library(kernR))
set.seed(2026L)
n <- 200L
z <- matrix(rnorm(n * 2L), n, 2L)
x <- z[, 1L] + rnorm(n, sd = 0.5)
y <- 0.7 * x + z[, 2L] + rnorm(n, sd = 0.4)

Each backend produces a density_ratio_fit object exposing ESS and a weight vector. We tabulate them side-by-side.

dr_logistic <- estimate_density_ratio(x, z, method = "logistic", seed = 1L)
dr_ranger <- if (requireNamespace("ranger", quietly = TRUE)) {
  estimate_density_ratio(x, z, method = "ranger", seed = 1L)
} else NULL
dr_xgb <- if (requireNamespace("xgboost", quietly = TRUE)) {
  estimate_density_ratio(x, z, method = "xgboost", seed = 1L)
} else NULL

dr_proxymix <- estimate_density_ratio(
  x, z,
  method = "proxymix",
  proxymix_components = 2L,
  seed = 1L
)

Density-ratio diagnostics across backends.
backend	ess	min_weight	max_weight
logistic	199.4	0.8860	1.20
ranger	156.0	0.2380	3.21
xgboost	81.3	0.0918	7.08
proxymix	14.3	0.0171	47.00

bd-HSIC under each backend

res_logistic <- bd_hsic_test(
  x, y, z, density_ratio = "logistic",
  n_permutations = 199L, seed = 1L
)

res_proxymix <- bd_hsic_test(
  x, y, z, density_ratio = "proxymix",
  n_permutations = 199L, seed = 1L
)
#> Warning: bd_hsic_test(): ESS (3.4) is below 10% of n_test (100). The weighted
#> test statistic is dominated by a small number of high-weight observations; the
#> resulting p-value is not a reliable verdict. Increase n, switch density_ratio
#> backend, or tighten the design.

bd-HSIC test under each density-ratio backend.
backend	statistic	p_value	ess
logistic	0.0211	0.155	99.9
proxymix	0.0062	0.320	3.4

When to reach for proxymix

The classifier-based backends (logistic, ranger, xgboost) are the default for a reason: they tolerate misspecified densities, scale to high-dimensional z, and have well-understood calibration. Reach for method = "proxymix" when:

the joint density is plausibly multimodal (multi-regime climate, paddock × variety designs with distinct production zones, animal cohorts with separable subpopulations) — a 2-component GMM represents this cleanly where a classifier would smear across the modes;
you need a parametric density-ratio whose components you can inspect, hand off to a downstream Bayesian step (via proxymix::gmm_target_from_posterior()), or use as the seed of a KLD-EM refinement on a target you can evaluate but not sample from;
classifier calibration is unreliable on the cohort at hand (small n, sharp class imbalance after the joint-vs-marginal split, pathological feature scaling).

The proxymix package is GRDC-firewalled (MIT, no GRDC IP flows in) and ships its full Gaussian-mixture proxy API independently of kernR. kernR consumes it as a soft dependency via requireNamespace() — the binding is one-way and rebuildable from the local proxymix_*.tar.gz source.

References

Hoek, J. van der & Elliott, R. J. (2024). Mixtures of multivariate Gaussians. Stochastic Analysis and Applications. DOI: 10.1080/07362994.2024.2372605.
Hu, R., Sejdinovic, D. & Evans, R. J. (2024). A kernel test for causal association via noise contrastive backdoor adjustment. Journal of Machine Learning Research, 25(160), 1–56.

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