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Fast Wasserstein rates for estimating probability distributions of probabilistic graphical models

NeurIPS 2026 · Probabilistic and scientific modeling › Variational and generative sampling · approach: Exact inference for probabilistic programs · 0 citations (OpenAlex)

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Shows that estimating probabilistic graphical model distributions in Wasserstein distance achieves rates governed by local node-plus-parent dimensions, with a sharp n^{-2/(2+d_loc)} + n^{-1/d_max} rate under bidirectional TV-Lipschitz kernel conditions.

Problem

Estimating a high-dimensional distribution in Wasserstein distance suffers the curse of dimensionality; the question is whether structural knowledge of a DAG factorization can overcome it, with the set of graphical-model distributions alone (qualitative conditions) being statistically unhelpful.

Approach

Quantify compatibility of a measure with a DAG via Lipschitz continuity of transition kernels (Wasserstein-Lipschitz for forward kernels, or bidirectional TV-Lipschitz); build discrete estimators by local cell-wise averaging of samples (empirical-conditional smoothing, and a bidirectional variant that projects to cell midpoints), giving rates depending on local dimension d_loc rather than ambient

Key results

• Under Wasserstein-Lipschitz kernels, the estimator achieves rate n^{-1/d_loc} (up to log), compared with n^{-1/d} for the empirical measure; e.g., d=14, d_loc=4 gives n^{-1/4} instead of n^{-1/14}.
• Under bidirectional TV-Lipschitz kernels, the bidirectional estimator achieves n^{-2/(2+d_loc)} + n^{-1/d_max} and a matching lower bound (up to log) is proved for this rate, e.g., n^{-1/3} for the d_k=2, 1->2 example.

Surprising findings

• Qualitative graphical-model structure alone (without quantified kernel smoothness) gives no statistical benefit: P_G is dense and rate stays n^{-1/d} for many graphs.
• Adding edges to remove colliders (to satisfy the no-collider assumption) increases d_loc only modestly on average in random DAG experiments.

Limitations

• Upper bound under Wasserstein-Lipschitz kernels for the 1->2 graph is not shown to be sharp; the rate n^{-1/d} may be unavoidable there, only supported by numerics.
• Requires the no-collider assumption (parent sets fully connected), which may force adding edges and worsening rates.
• Graph-adaptive estimator is computationally less tractable.

Experimental setup

Metrics: Wasserstein distance convergence rate local dimension d_loc

Baselines: empirical measure

Builds on: empirical measure convergence rates in Wasserstein distance Lipschitz density estimation lower bounds Chow-Liu trees

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