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CoLT: The conditional localization test for assessing the accuracy of neural posterior estimates
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Proposes CoLT, a conditional localization test that trains a localization function and embedding to find regions where a neural posterior deviates from the true posterior, using a single true draw per condition and a ball-probability rank statistic.
Problem
Validating whether a neural posterior estimate q(theta|x) accurately approximates the true posterior p(theta|x) across all conditioning inputs, when only one true draw per x is available.
Approach
Learn a localization function theta_l(x) and a metric embedding phi that maximally distort the ball probability rank statistic away from uniformity; the rank statistic is uniform iff p=q, giving a necessary-and-sufficient test and an IPM (ACLD) distance.
Key results
- On the tree toy example at alpha=1.5, CoLT achieves power 0.877 while C2ST gets 0.000; at alpha=4.0 CoLT reaches 1.000 vs C2ST 0.065.
- On the curved manifold benchmarks, CoLT Full outperforms C2ST, SBC and TARP, particularly under tail adjustments and anisotropic distortions.
- In the blind prior setting, TARP and SBC fail to detect the discrepancy while CoLT ID and CoLT Full detect it across all dimensions; C2ST degrades in higher dimensions.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use CoLT Full instead of C2ST | Tree toy example, α = 1.5 · Tree toy example, α = 1.5 | power: 0.000 -> 0.877 | Helped |
| Use CoLT Full instead of C2ST | Tree toy example, α = 4.0 · Tree toy example, α = 4.0 | power: 0.065 -> 1.000 | Helped |
| Jointly optimize a learned embedding instead of fixing the embedding to identity (ablation) | CoLT · Curved-manifold benchmarks, especially tail adjustments and anisotropic distortions | power | Helped |
| Fix the embedding to identity instead of learning it (ablation) | CoLT · Gaussian benchmarks, including covariance scaling and additional modes | power | No clear effect |
| Reduce the number of reverse diffusion steps from 20 | Diffusion posterior approximation · Diffusion benchmark | test statistic | Hurt |
Surprising findings
- Fixed Euclidean balls (CoLT ID) suffice for flat Gaussian posteriors, but a learned embedding is needed for curved manifold posteriors.
- C2ST power collapses in higher dimensions even on the blind prior case where it is effective in low dimensions.
Limitations
- Power depends on learned localization and embedding components; underparameterized or poorly trained networks may miss discrepancies.
- Rank statistic quality degrades with limited samples.
- Interpreting the IPM scalar in high dimensions is challenging because the function class is implicit.
Experimental setup
Datasets: NPTBench benchmark tasks (perturbed Gaussian and curved manifold posteriors) tree-shaped GMM toy posterior from EDM2 code diffusion benchmark theta=(x cos x, x sin x)
Benchmarks: NPTBench perturbed Gaussian tasks (mean shift, covariance scaling, anisotropic, t-tail, additional modes, mode collapse, NPTBench curved manifold tasks tree toy example
Metrics: statistical power vs perturbation alpha type I error at alpha=0
Baselines: C2ST SBC TARP
Builds on: classifier two-sample test simulation-based calibration TARP EDM
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