Conditionally Calibrated Predictive Distributions by Probability-Probability Map: Application to Galaxy Redshift Estimation and Probabilistic Forecasting

• URL: http://arxiv.org/abs/2205.14568v4
• Date: Mon, 17 Jul 2023 16:58:54 GMT
• Title: Conditionally Calibrated Predictive Distributions by Probability-Probability Map: Application to Galaxy Redshift Estimation and Probabilistic Forecasting
• Authors: Biprateep Dey and David Zhao and Jeffrey A. Newman and Brett H. Andrews and Rafael Izbicki and Ann B. Lee
• Abstract summary: Uncertainty is crucial for assessing the predictive ability of AI algorithms. We propose textttCal-PIT, a method that addresses both PD diagnostics and recalibration. We benchmark our corrected prediction bands against oracle bands and state-of-the-art predictive inference algorithms.
• Score: 4.186140302617659
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Uncertainty quantification is crucial for assessing the predictive ability of AI algorithms. Much research has been devoted to describing the predictive distribution (PD) $F(y|\mathbf{x})$ of a target variable $y \in \mathbb{R}$ given complex input features $\mathbf{x} \in \mathcal{X}$. However, off-the-shelf PDs (from, e.g., normalizing flows and Bayesian neural networks) often lack conditional calibration with the probability of occurrence of an event given input $\mathbf{x}$ being significantly different from the predicted probability. Current calibration methods do not fully assess and enforce conditionally calibrated PDs. Here we propose \texttt{Cal-PIT}, a method that addresses both PD diagnostics and recalibration by learning a single probability-probability map from calibration data. The key idea is to regress probability integral transform scores against $\mathbf{x}$. The estimated regression provides interpretable diagnostics of conditional coverage across the feature space. The same regression function morphs the misspecified PD to a re-calibrated PD for all $\mathbf{x}$. We benchmark our corrected prediction bands (a by-product of corrected PDs) against oracle bands and state-of-the-art predictive inference algorithms for synthetic data. We also provide results for two applications: (i) probabilistic nowcasting given sequences of satellite images, and (ii) conditional density estimation of galaxy distances given imaging data (so-called photometric redshift estimation). Our code is available as a Python package https://github.com/lee-group-cmu/Cal-PIT .

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