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Simulation-Based Inference with Quantile Regression
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Proposes Neural Quantile Estimation (NQE), an autoregressive conditional-quantile-regression SBI method with PCHIP-ET interpolation, a quantile-based credible region, and a broadening calibration step.
Problem
Simulation-based inference needs accurate, unbiased posterior estimates from limited simulation budgets, and existing NPE/NLE/NRE methods can be biased and expensive to calibrate via coverage tests.
Approach
Learn 1-D conditional quantiles for each posterior dimension autoregressively with a weighted L1 (pinball) loss, interpolate CDFs with monotonic cubic Hermite splines with Gaussian tails, define a quantile-mapping credible region (QMCR) that needs only O(N_o) network calls, and calibrate by a global broadening factor.
Key results
- Uncalibrated NQE achieves state-of-the-art C2ST across all six SBI benchmark problems versus NPE, NLE, NRE baselines.
- Broadening calibration makes NQE q-coverage unbiased for all benchmarks using only about 10^3 validation simulations.
- QMCR coverage requires O(N_o) network calls versus O(N_o N_r) for HPDR-based p-coverage, making calibration nearly free.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use NQE with q-coverage rather than traditional p-coverage for broadening calibration | NQE · Broadening calibration | network calls: \mathcal{O}( N_i N_o N_r ) -> \mathcal{O}( N_o ) | Helped |
| Use NQE q-coverage instead of p-coverage with MCMC-based SBI methods | NQE calibration versus NLE, NPE, and NRE calibration · Broadening calibration | network calls: \mathcal{O}( N_i N_o N_r N_m ) -> \mathcal{O}( N_o ) | Helped |
| Apply global broadening calibration | NQE · Six SBI benchmark problems | empirical coverage | Helped |
| Apply global broadening calibration | NQE · Six SBI benchmark problems | C2ST | Mixed |
| Use quantile shifting instead of global broadening | NQE with known cosmology model misspecification · Cosmology | empirical coverage | Helped |
| Use quantile shifting instead of global broadening | NQE with known cosmology model misspecification · Cosmology | posterior width | Helped |
| Increase the number of quantile bins (ablation) | NQE · Two Moons (TM), large simulation budgets | C2ST | Helped |
| Change the ordering of posterior dimensions (ablation) | NQE · SBI benchmark problems | marginal C2ST | No clear effect |
| Use PCHIP-ET instead of original PCHIP interpolation | NQE · Unimodal and multimodal distribution reconstruction | interpolation artifacts | Helped |
Surprising findings
- No clear dependence of marginal C2ST on parameter dimension ordering, contrary to the expected error accumulation in autoregressive models.
- Calibration via global broadening typically leaves C2ST similar or slightly worse, since conservative posteriors are penalized similarly to biased ones.
Limitations
- Calibration only guarantees unbiasedness absent unknown model misspecification; partially known misspecification only.
- Global broadening can be over-conservative; quantile shifting scheme is not fully examined and its simulation needs depend on dimension.
- PCHIP-ET lacks continuous PDF derivatives.
Experimental setup
Datasets: Two Moons SLCP with Distractors Bernoulli GLM Raw Gaussian Mixture SIR Lotka-Volterra cosmology PM density fields (pmwd simulations)
Benchmarks: sbibm six benchmark problems (TM, SLCPD, BGLMR, GM, SIR, LV) cosmology model misspecification test
Metrics: C2ST p-coverage q-coverage
Baselines: NPE NLE NRE SNPE BNRE TMNRE
Builds on: PCHIP neural posterior estimation flow matching posterior estimation sbibm
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