Modeling local predictive ability using power-transformed Gaussian processes

A Gaussian process is proposed as a model for the posterior distribution of the local predictive ability of a model or expert, conditional on a vector of covariates, from historical predictions in the form of log predictive scores. Assuming Gaussian expert predictions and a Gaussian data generating process, a linear transformation of the predictive score follows a noncentral chi-squared distribution with one degree of freedom. Motivated by this we develop a noncentral chi-squared Gaussian process regression to flexibly model local predictive ability, with the posterior distribution of the latent GP function and kernel hyperparameters sampled by Hamiltonian Monte Carlo. We show that a cube-root transformation of the log scores is approximately Gaussian with homoscedastic variance, making it possible to estimate the model much faster by marginalizing the latent GP function analytically. A multi-output Gaussian process regression is also introduced to model the dependence in predictive ability between experts, both for inference and prediction purposes. Linear pools based on learned local predictive ability are applied to predict daily bike usage in Washington DC.