Simulation-Based Inference with Quantile Regression

TL;DR

This work presents Neural Quantile Estimation (NQE), a novel Simulation-Based Inference method based on conditional quantile regression, offering substantially faster evaluation than the traditional Highest Posterior Density Region (HPDR).

Abstract

We present Neural Quantile Estimation (NQE), a novel Simulation-Based Inference (SBI) method based on conditional quantile regression. NQE autoregressively learns individual one dimensional quantiles for each posterior dimension, conditioned on the data and previous posterior dimensions. Posterior samples are obtained by interpolating the predicted quantiles using monotonic cubic Hermite spline, with specific treatment for the tail behavior and multi-modal distributions. We introduce an alternative definition for the Bayesian credible region using the local Cumulative Density Function (CDF), offering substantially faster evaluation than the traditional Highest Posterior Density Region (HPDR). In case of limited simulation budget and/or known model misspecification, a post-processing calibration step can be integrated into NQE to ensure the unbiasedness of the posterior estimation with negligible additional computational cost. We demonstrate that NQE achieves state-of-the-art performance on a variety of benchmark problems.

Figures (16)

• Figure 1: (Top) Network architecture of our NQE method, which autoregressively learns 1-dim conditional quantiles for each posterior dimension. The estimated quantiles are then interpolated to reconstruct the full distribution. (Bottom) A post-processing calibration step can be employed to ensure the unbiasedness of NQE inference results.
• Figure 2: (1st row) Interpolation of Gaussian and Gaussian Mixture distributions. While the original PCHIP algorithm shows significant interpolation artifacts, our modified PCHIP-ET scheme decently reconstructs the distributions with only $\sim 15$ quantiles. (2nd row) Comparison of the 68.3% and 95.4% credible regions for a mixture of two asymmetric modes, evaluated with HPDR ($p$-coverage) and QMCR ($q$-coverage). (3rd row) Broadening of the interpolated posterior, with the broadening factors indicated in the legend. (4th row) The bijective mapping $f_{\rm qm}$ establishes a one-to-one correspondence between ${\bm \theta}$ and ${\bm \theta}'$ with the same 1-dim conditional CDF across all the $\theta^{(i)}$ dimensions. The $p-$coverage and $q-$coverage are based on the ranking of $p({\bm \theta})$ and $q_{\rm aux}({\bm \theta}')$, respectively.
• Figure 3: Probability density estimation for two toy examples from grathwohl2018ffjord. Despite the intricate multimodal structures, NQE is able to faithfully reconstruct both distributions.
• Figure 4: Comparison of C2ST as a function of simulation budget for the six benchmark problems, with lower C2ST values representing better performance of the algorithm. The error bars are estimated using the 25%, 50% and 75% quantiles of C2ST over ten realizations for each problem. (Uncalibrated) NQE achieves state-of-the-art performance across all the examples.
• Figure 5: (Top) NQE $q-$coverage for the benchmark problems. Like other SBI methods, with limited simulation budgets, NQE may predict biased posteriors. (Bottom) Calibrated NQE predicts unbiased posteriors for all the problems. Errorbars are small and thus not plotted. See \ref{['app:cover']} for a convergence test and \ref{['fig:p-cover']} for a similar plot with $p-$coverage.