[Paper Review] Calibrated and Sharp Uncertainties in Deep Learning via Density Estimation

Volodymyr Kuleshov, Shachi Deshpande|arXiv (Cornell University)|Dec 14, 2021

Gaussian Processes and Bayesian Inference4 citations

TL;DR

This paper proposes a simple, general-purpose recalibration method for deep learning models that achieves distribution calibration via low-dimensional density estimation, ensuring probabilistic predictions are both well-calibrated and sharp. The approach uses quantile function regression to remap predictive distributions, outperforming prior methods in calibration while maintaining high accuracy across classification and regression tasks.

ABSTRACT

Accurate probabilistic predictions can be characterized by two properties -- calibration and sharpness. However, standard maximum likelihood training yields models that are poorly calibrated and thus inaccurate -- a 90% confidence interval typically does not contain the true outcome 90% of the time. This paper argues that calibration is important in practice and is easy to maintain by performing low-dimensional density estimation. We introduce a simple training procedure based on recalibration that yields calibrated models without sacrificing overall performance; unlike previous approaches, ours ensures the most general property of distribution calibration and applies to any model, including neural networks. We formally prove the correctness of our procedure assuming that we can estimate densities in low dimensions and we establish uniform convergence bounds. Our results yield empirical performance improvements on linear and deep Bayesian models and suggest that calibration should be increasingly leveraged across machine learning. We release a library that implements our methods along with a blog post here: https://shachideshpande.github.io/blog-distribution-calibration/.

Motivation & Objective

• To address the widespread issue of miscalibrated uncertainty in deep learning models trained via maximum likelihood.
• To develop a general, simple, and theoretically grounded method for achieving distribution calibration across diverse models, including neural networks.
• To improve both calibration and sharpness of predictive uncertainty without sacrificing model performance.
• To formalize and empirically validate a framework that guarantees asymptotic distribution calibration with vanishing regret.
• To demonstrate that distribution calibration is more accessible and practical than previously thought, encouraging broader adoption in machine learning.

Proposed method

• The method performs recalibration using a learned mapping from predicted quantiles to observed calibration statistics via quantile function regression.
• It estimates the conditional density P(Y | F) where F is the forecast distribution, enabling distribution calibration by adjusting uncertainty per forecast feature.
• A neural network or simple regressor R maps predicted quantiles to recalibrated ones, minimizing a calibration error objective.
• The approach assumes low-dimensional density estimation is feasible and uses uniform convergence bounds to ensure theoretical correctness.
• It generalizes Platt scaling and isotonic regression by enabling per-forecast variance adjustment, not just global recalibration.
• The framework supports both classification and regression and can be implemented in a few lines of code, making it broadly accessible.

Experimental results

Research questions

• RQ1 Can distribution calibration be achieved efficiently and generally across different deep learning models without sacrificing predictive performance?
• RQ2 How does low-dimensional density estimation enable better calibration than existing quantile-based recalibration methods?
• RQ3 What theoretical guarantees can be provided for recalibration procedures that ensure asymptotic distribution calibration?
• RQ4 Can a simple, end-to-end trainable method outperform complex variational approximations in uncertainty calibration?
• RQ5 To what extent does distribution calibration improve uncertainty quality in Bayesian and deep learning models?

Key findings

• The proposed recalibration method achieves superior distribution calibration compared to Platt scaling, temperature scaling, and variational inference-based methods like those of Song et al. (2019).
• Empirical results show consistent improvements in calibration metrics across linear models, Bayesian neural networks, and convolutional neural networks on classification tasks.
• The method maintains high predictive accuracy while significantly improving uncertainty quality, as measured by proper scoring rules and calibration curves.
• The approach enables tighter, more reliable credible intervals—e.g., the 90% interval correctly covers the true outcome ~90% of the time, with reduced width compared to baseline methods.
• Theoretical analysis establishes uniform convergence bounds and proves that the method asymptotically achieves distribution calibration with vanishing regret.
• The method is computationally efficient and easily implementable, requiring only a few lines of code to integrate into existing models.