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

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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