Accurately assessing the convergence of Monte Carlo Markov Chains is crucial for ensuring their validity in Bayesian inference problems. However, convergence assessment in high dimensions is known to be difficult, due to most convergence diagnostics becoming insensitive beyond a few dozen parameters. Conventional convergence statistics analyze marginal distributions which in high dimensions neglects increasing levels of information from the joint and conditional probability distributions. Thus we propose NFDiag, which taps into the information of the full multivariate joint posterior with a normalizing flow (NF). Our convergence diagnostic thereby also analyzes information from conditional distributions. In other words, by working in the full multivariate space, it is sensitive with respect to the geometry of the chain in high dimensions. Our diagnostic utilizes that NFs assume independent and identically distributed (iid) training samples, which is a testable assumption that will be violated by a non-converged or strongly correlated chain. Our diagnostic trains NFs on separate parts of a chain, and then assesses whether these correspond to the same multivariate target distribution. We test our diagnostic in up to 250 dimensions, and demonstrate it is universal and unbiased, and detects problematic chain quality also when Gelman-Rubin R^ and the multivarite effective sample size do not.
Figures & tables
Figure 1: Sample density histograms H1 and H2 for two Gamma distributions f(x,a) for a=[2,4] . We outline the overlap region of the two histograms with dark purple. As the volume of the overlap region increases, dB decreases.
Hyperparameter
Value
Hyperparameter
Value
Flow layers
3
Batch size
5k
NN layers
3
Spline knots
20
Neurons/layer
512
Dropout
0.45
Learning rate
10−3
Weight decay
0.005
Table 1: Hyperparameters shared between all NF models and training regimens for the experiments in Sec. 5 .
Figure 2: Evaluating NFD for iid samples in increasing dimensions, and for several distributions. The error bars are standard deviations obtained from 5 repetitions. The multivariate Student’s t and matrix-variate Wishart distributions are indicated with triangles and circles respectively. With increasing L , the NFD approaches 1, indicating that it is unbiased. The Gaussian Mixture (GM) chains indicated with crosses are all of L=100000 . Since any probability distribution can be approximated through a GM, we conclude that the NFD is universally applicable to any target distribution as we see here that it performs reliably even for these multimodal distributions.
Figure 3: NFD of chains from different samplers, vs. mean R^ and mESS. The points indicate the mESS with the true correlation matrix, while the right-pointing error bars are the MCSE estimator with a batch size of L1/3=47 . The dashed lines are provided to navigate the plot by connecting points of the same dimension. The vertical blue region corresponds to chains of allegedly identical convergence quality as assessed by the mESS and R^ . Going downwards within these blue regions, one sees that these statistics become insensitive in higher dimensions, whereas NFD continues to resolve difference in convergence quality. As such, we see that mESS and R^ incorrectly indicated convergence high dimensions.
Figure 4: Marginal plot of, in order, iid samples, MALA chains with ϵ=[0.7,0.2] and MH chain with l=2.38 , together with their mESS, R^ and NFD . The points are colored from orange to black in the sampling order of the chain. All chains are considered converged by mESS and R^ , but visually inspecting the marginals reveals convergence problems in the latter two, which would be exacerbated in the full multivariate geometry. NFD is very sensitive to this, and accurately flags the chains as not converged.
Figure 5: Histograms of the training loss Lt and validation loss Lv at Epoch 4 for the NF model trained on ( Top ) 100000Csignal,ℓ samples and ( Bottom ) for 100000 thinned samples of Csignal,ℓ from a longer chain of length 1 M from our HMC sampler.
Appendix figures & tables3 assets
Supplementary material from the paper’s appendix.
Appendix
Figure 6: Effective sample size and R^ for the 100000 sample chain.
Figure 7: Hanson’s diagnostic and FMI for the 100000 sample chain.
Figure 8: Histograms of the training loss Lt and validation loss Lv at Epoch 6 for the NF model trained on 1 M Csignal,ℓ samples from our HMC sampler.
This work shows that diffusion models learned with standard denoising loss can provide effective global MCMC proposals for complex high-dimensional target densities. The method is motivated by the observation that sequentially applying a forward and reverse diffusion process defines a Markov chain with a target stationary distribution for an ideal denoiser trained on samples of the target distribution. This observation can be made exact for any denoiser by applying a Metropolis-Hastings step whose acceptance ratio includes the density of the forward and reverse paths of a discrete time SDE approximation. We therefore propose to train denoising diffusion models on locally convergent MALA samples to learn global MCMC proposals. We call the composition of the global denoiser-based path sampler and a local MALA sampler Denoising Diffusion Monte Carlo (DDMC). Experiments show that DDMC can provide global proposals with high acceptance across a variety of complex target densities. Our results offer preliminary evidence that the established scaling behavior of standard diffusion training transfers directly to exact sampling from high-dimensional unnormalized densities.
Transport MCMC trains a normalizing flow to precondition Metropolis--Hastings proposals, achieving high empirical efficiency on challenging posteriors; yet no prior work produces a numerically non-vacuous, rigorous spectral-gap bound for such samplers. We establish the first such bounds. For independence MH on the banana family we certify (γ^\ast = 0.828) at (D = 2) (covering in the original space) and (γ^\ast \ge 7.6\times 10^{-4}) at (D = 5) (covering in an analytically unwarped Gaussian space with a grid-certified gradient bound under the stated numerical Lipschitz certification), both rigorous at 95% confidence. The framework rests on three pillars: (i) spectral normalization with reduced scale clips constrains the flow Lipschitz constant from (10^{47}) to (10^4); (ii) a coverage-based empirical oscillation bound replaces the vacuous analytical bound with a data-dependent certificate; and (iii) oscillation-regularised training cuts the empirical oscillation by 60--90% at no cost to density fit, extending practical certificates through (D = 20) ((γ^\ast \ge 1.7\times 10^{-4})). Tests on four further targets (Gaussian mixture, shear-building, Neal's funnel, Bayesian logistic regression) identify three precise barriers: boundary curvature, target stiffness, and tail-coverage mismatch. An affine-vs-spline comparison shows that simpler architectures yield tighter certificates at identical NLL, inverting the usual expressiveness hierarchy.
Jun Hu
China Sanya Science and Education Innovation Park, Wuhan University of Technology, Sanya 572025, China · School of Civil Engineering and Architecture, Wuhan University of Technology, Wuhan 430070, China
In Bayesian mixture models and other exchangeable-component models, the posterior is invariant under permutation of component labels, creating m! equivalent modes-the label-switching problem. Standard MCMC methods either mix poorly across these modes or rely on post-hoc relabelling that cannot guarantee the sampler has converged. We propose Folded Transport MCMC (FolT-MCMC), which eliminates label switching before sampling by restricting the Markov chain to a fundamental domain-a sorted or reflected subspace containing exactly one representative from each symmetric mode. The proposal is a learned normalising flow whose density is symmetrised over the group orbits, ensuring correct targeting on the reduced space. We show that this construction preserves a computable convergence diagnostic based on the oscillation of the log-density ratio, and that the diagnostic becomes sharper on the fundamental domain whenever the original-space flow under-covers one or more symmetric modes. Experiments on Gaussian mixtures (d=2-20), label-switching targets (up to 24 equivalent modes), a standard Bayesian three-component mixture posterior, and real accelerometer data from a supertall building show improvement ratios of 2x to 145x, with the folded diagnostic stable across dimensions while the unfolded diagnostic collapses.