Sequential Sparse Gaussian Process Quantile Regression Developed.

Quantile regression is a powerful statistical tool for estimating conditional quantiles, offering a more complete picture than just mean estimation. However, in a Bayesian context, Gaussian process quantile regression (GPQR) faces significant computational hurdles due to the non-conjugacy of its likelihood and the cost of posterior inference. This research proposes a novel sequential sparse Gaussian process framework to overcome these issues. The approach represents the quantile function using a reduced set of inducing variables, and employs a Laplace approximation for efficient posterior inference. A key innovation is the decomposition of predictive uncertainty into prior and posterior components, which drives two adaptive mechanisms: intelligently adding inducing inputs and strategically acquiring new data. This sequential algorithm efficiently allocates computational effort to the dominant source of uncertainty, dynamically controlling model complexity. Numerical experiments confirm the accuracy and effectiveness of this strategy.