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proxymix fits multivariate Gaussian-mixture proxies that
are Kullback–Leibler optimal to user-supplied target
densities on \(\mathbb{R}^p\). Three
regimes from van der Hoek and Elliott (2024) are unified under one
verb:
| Regime | When it applies | Method |
|---|---|---|
| (i) moment | \(N = 1\) component | Closed-form moment matching |
| (ii) sample | i.i.d. samples from the target are available | Classical EM |
| (iii) kld | target density f(x) can be evaluated but
not (cheaply) sampled |
KLD-EM with importance sampling |
Regime (iii) is the reason the package exists. The sample-based
mixture packages (mclust, mixtools,
flexmix) all assume i.i.d. draws from the target;
proxymix fits directly against an evaluable (possibly
unnormalised) log-density. The nearest CRAN neighbour is
AdMit, which adaptively fits a mixture of Student-t
distributions to an evaluable kernel as an importance/proposal density;
proxymix differs in fitting a Gaussian
mixture that is Kullback–Leibler optimal, precisely so that the fitted
object then supports the closed-form operator calculus (marginals,
conditionals, Bayes updates, products, convolutions, filtering) and
carries a fit-quality certificate through every operation.
Why not MCMC? If you can evaluate the unnormalised density you can always run a sampler and then fit a mixture to the draws. The regime-(iii) fit is the shortcut when what you want is the compact closed-form object: no chain tuning or convergence diagnostics, a deterministic pipeline given the seed, and a mixture whose marginals, conditionals, moments and samples are then available in closed form through the operator calculus. The trade-off is dimension: the importance sampling that drives regime (iii) loses effective sample size sharply beyond roughly \(p = 5\)–\(10\), and every fit reports its effective sample size so that limit is visible rather than silent. For high-dimensional posteriors, sample with your favourite MCMC and use regime (ii) on the draws.
From GitHub:
# install.packages("remotes")
remotes::install_github("max578/proxymix")or locally from the source tree:
R CMD build proxymix
R CMD INSTALL proxymix_*.tar.gzDocumentation site: https://max578.github.io/proxymix/.
library(proxymix)
## A target you can evaluate but not sample from -- a 2D "banana".
banana <- banana_target()
## Fit a 3-component Gaussian mixture proxy via KLD-EM with importance sampling.
fit <- fit_proxymix(banana, N = 3L, regime = "kld",
proposal = proposal_mvt(n_dim = 2L, df = 5),
is_size = 2000L, max_iter = 60L, seed = 1L)
print(fit)
#> <gmm_fit>: regime = "kld", K = 3, p = 2
#> target : banana
#> iterations : 37
#> converged : TRUE
#> [1] w = 0.6456, |mu| = 0.3299, tr(Sigma) = 1.3601
#> [2] w = 0.2473, |mu| = 1.2996, tr(Sigma) = 2.2312
#> [3] w = 0.1071, |mu| = 1.8882, tr(Sigma) = 4.9044
## Closed-form operations on the fitted mixture.
gmm_marginalise(fit, keep = 1L)
#> <marginalise(kld_em[N=3] on banana)>: K = 3 components in p = 1 dimensions
#> [1] w = 0.6456, |mu| = 0.1818, tr(Sigma) = 0.4620
#> [2] w = 0.2473, |mu| = 1.1518, tr(Sigma) = 0.4846
#> [3] w = 0.1071, |mu| = 1.5954, tr(Sigma) = 0.6321
gmm_conditionalise(fit, given = c(NA, 0.5))
#> <conditionalise(kld_em[N=3] on banana)>: K = 3 components in p = 1 dimensions
#> [1] w = 0.6730, |mu| = 0.2620, tr(Sigma) = 0.4524
#> [2] w = 0.2576, |mu| = 1.1133, tr(Sigma) = 0.2352
#> [3] w = 0.0694, |mu| = 1.4188, tr(Sigma) = 0.1199fit_proxymix(
target,
N = 1L,
regime = c("auto", "moment", "sample", "kld"),
...
)target is an S7 gmm_target produced by
gmm_target() (from a log-density),
gmm_target_from_samples() (from samples), or one of the
built-in factories (banana_target(),
donut_target(), mixture_target()).regime = "auto" picks the cheapest applicable regime
from the target’s contents.dgmm(x, fit) # density evaluation
rgmm(n, fit) # exact sampling
gmm_marginalise(fit, keep)
gmm_conditionalise(fit, given)
gmm_kld(fit_p, fit_q) # Monte Carlo + variational boundsfit <- from_objective(f, lower, upper, N = 10L) # a mixture map of the optima
gmm_modes(fit)$modes # the distinct optimafrom_objective() treats an objective f as
the Gibbs measure exp(-f / T) – a regime-(iii) target you
can evaluate but not sample – and returns a closed-form mixture over its
low regions, so a multimodal f is recovered as a whole
rather than one optimum at a time. gmm_modes() resolves the
fitted map into the recovered optima.
| You have | You want | Reach for |
|---|---|---|
| an evaluable (unnormalised) log-density | a compact closed-form proxy | gmm_target() then
fit_proxymix(regime = "kld") |
| i.i.d. samples | a mixture fit | gmm_target_from_samples() then
fit_proxymix() |
| an objective function | a map of its optima | from_objective(), gmm_modes() |
| a kernel density estimate | a small closed-form surrogate | from_kde() |
| a fitted mixture | marginals, conditionals, updates | gmm_marginalise(), gmm_conditionalise(),
gmm_observe(), gmm_affine() |
| data with holes | multiple imputation | gmm_impute() with mar() /
mnar() / censored() |
| a time series + state-space model | filtering / stability testing | gmm_filter(), gmm_eos_test() |
| a fitted mixture | information-theoretic diagnostics | gmm_entropy(), gmm_divergence(),
gmm_mutual_information() |
| a fitted mixture + treatment data | causal / decision quantities | gmm_intervene(), fit_uplift(),
proxy_cate() |
| an unnormalised posterior | the marginal likelihood | gmm_evidence() |
| a fitted proxy | error bars on any functional | gmm_fit_ensemble(),
proxy_functional_ci() |
| an evaluable-only target | the component count | select_N() |
| a fit collapsing in high dimension | an adaptive proposal | fit_kld_em(adapt = "pmc") |
The package ships with twelve vignettes:
quickstart – one-page tour.posterior_proxy – the flagship
workflow: a real unnormalised Bayesian posterior compressed to a proxy,
with the evidence, closed-form reads, and bootstrap error bars.three_regimes – a walk-through of
regimes (i)–(iii) on toy 2-D targets, including the agreement of (i) and
(iii) at \(N = 1\).density_shapes – the regime-(iii)
demonstration: banana, donut, three-mixture targets fit by
importance-sampled KLD-EM.operator_calculus – closed-form
pushforward, Bayesian update, aggregation and conditioning on a fitted
mixture.from_kde – compressing a kernel
density estimate into a Gaussian-mixture proxy.many_methods – one fitted mixture in
place of regression, clustering, PCA and ridge regression.entropy – closed-form entropy,
divergence and mutual-information diagnostics.calibration – mapping the optima of an
objective via its Gibbs measure.missing_data – multiple imputation by
conditioning the fitted mixture (missing at random).missing_data_mnar – imputation under
value-dependent missingness and censoring, with sensitivity
analysis.end_of_sample – testing whether the
last few observations of a series are consistent with a fitted
state-space model.A standalone, single-file HTML primer for proxymix is
hosted at
The page is targeted at Adelaide University Mathematics and
Statistics undergraduates (Guided depth, default) and postgraduate / PhD
readers (Technical depth, via the in-page dial). It runs in the browser
with no R install required, and works the same way offline by
double-click. Source at max578/proxymix-tutorial.
Hoek, J. van der and Elliott, R. J. (2024). Mixtures of multivariate Gaussians. Stochastic Analysis and Applications. doi:10.1080/07362994.2024.2372605.
Bug reports and suggestions are welcome through the GitHub issues.
citation("proxymix")MIT © Max Moldovan.
These binaries (installable software) and packages are in development.
They may not be fully stable and should be used with caution. We make no claims about them.
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