Locally Adaptive Bayes Nonparametric Regression via Nested Gaussian Processes.

Area of Science:

• Statistics
• Machine Learning
• Computational Biology

Background:

• Bayesian nonparametric regression is crucial for flexible modeling.
• Existing methods may lack adaptivity to local variations in function smoothness.
• Gaussian processes are powerful priors but can be computationally intensive.

Purpose of the Study:

• To propose a novel nested Gaussian process (nGP) prior for Bayesian nonparametric regression.
• To develop a method that adaptively captures locally varying smoothness.
• To provide a theoretically grounded and computationally efficient Bayesian approach.

Main Methods:

• The nested Gaussian process (nGP) is specified via stochastic differential equations (SDEs).
• It imposes a Gaussian process prior on the function's mth-order derivative with a locally adaptive mean function.
• Posterior inference is performed using efficient Markov chain Monte Carlo (MCMC).

Main Results:

• The nGP prior's support is characterized within reproducing kernel Hilbert spaces.
• The posterior mean is equivalent to minimizing a nested penalized sum-of-squares.
• Simulation studies show superior performance compared to alternatives, with good scalability.

Conclusions:

• The nested Gaussian process (nGP) offers a powerful, locally adaptive prior for Bayesian nonparametric regression.
• The method is theoretically sound and computationally efficient for large-scale applications.
• The approach is validated through simulations and a real-world proteomics data analysis.