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Neural Conditional Probability for Uncertainty Quantification
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Proposes Neural Conditional Probability (NCP), which learns a truncated SVD of the conditional expectation operator via a loss on separable neural features, enabling conditional probabilities, quantiles and confidence regions from a single unconditional training.
Problem
Estimating conditional distributions and derived statistics (conditional quantiles, moments, confidence regions) from samples without retraining for each conditioning value.
Approach
Learn the conditional expectation operator's truncated SVD with two MLP embeddings and singular values, using a loss derived from the deflated operator plus an orthonormality penalty; then compute any conditional statistic analytically from the trained model, with whitening post-processing.
Key results
- NCP-W has the best KS distance on 4 of 6 benchmark datasets (e.g., 0.010 on LinearGaussian, 0.047 on LGGMD), tied or second on others.
- On Laplace confidence regions, CCP achieves lower conditional mean mse (3.6e-3) than NCP (8.3e-3), but CCP intervals are unreliable for most conditionings.
- Coverage on Cauchy test set: 88% for NCP, 99% for NF, 79% for CCP.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use NCP-W instead of NF | Conditional density estimation · LinearGaussian | KS distance: 0.008 -> 0.010 | Hurt |
| Use NCP-W instead of NF | Conditional density estimation · EconDensity | KS distance: 0.006 -> 0.005 | Helped |
| Use NCP-W instead of NNKCDE | Conditional density estimation · ArmaJump | KS distance: 0.030 -> 0.010 | Helped |
| Use NCP-W instead of CKDE | Conditional density estimation · SkewNormal | KS distance: 0.023 -> 0.008 | Helped |
| Use NCP-W instead of FC | Conditional density estimation · GaussianMixture | KS distance: 0.016 -> 0.015 | Helped |
| Use NCP-W instead of FC | Conditional density estimation · LGGMD | KS distance: 0.047 -> 0.047 | No clear effect |
Surprising findings
- On the Cauchy confidence-region experiment, NF and CCP collapse, while NCP remains reasonable.
- NCP needs a relatively large sample size (n≳10^4) before it outperforms other methods.
Limitations
- Requires relatively large sample sizes (n≳10^4) to outperform baselines.
- Confidence regions can be overly conservative/overestimated for some conditionings (e.g., near x=0 in Laplace experiment).
- Under-coverage for larger x in Cauchy experiment.
Experimental setup
Datasets: LinearGaussian EconDensity ArmaJump SkewNormal GaussianMixture LGGMD synthetic Laplace and Cauchy confidence region data synthetic high-dimensional data protein folding molecular dynamics trajectories
Benchmarks: conditional density benchmark (KS distance over six synthetic models) Laplace/Cauchy confidence regions high-dimensional synthetic experiment molecular dynamics transition probabilities
Metrics: Kolmogorov-Smirnov distance mse of conditional mean coverage
Baselines: DDPM NF KMN MDN LSCDE CKDE NNKCDE RFCDE FlexCode LCDE CCP RFSCP
Builds on: DeepCCA Eckart-Young-Mirsky theorem conformal prediction
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