Local Derivative-Free Approximation of Computationally Expensive Posterior Densities

Area of Science:

• Computational Statistics
• Bayesian Inference

Background:

• Markov chain Monte Carlo (MCMC) methods are computationally intensive for expensive posterior density evaluations.
• Existing global approximation methods can waste computational resources on low-probability regions.

Purpose of the Study:

• To develop a derivative-free algorithm (GRIMA) for accurate approximation of posterior densities.
• To improve computational efficiency in Bayesian inference by focusing on high-probability regions.

Main Methods:

• GRIMA employs a local interpolation approach over the high-probability density (HPD) region.
• It iteratively refines a surrogate posterior using sequential knot selection and MCMC sampling.
• The algorithm does not require derivatives of the posterior density.

Main Results:

• GRIMA accurately approximates general unnormalized posterior densities.
• Simulation experiments demonstrate effectiveness across various density characteristics (dependence, skewness, multimodality).
• The method was successfully applied to calibrate a nonlinear regression model in a watershed case study.

Conclusions:

• GRIMA offers a computationally efficient and accurate alternative for Bayesian inference with expensive posterior evaluations.
• The derivative-free nature and focus on HPD regions enhance its applicability.
• This algorithm can facilitate the use of MCMC in complex, real-world modeling scenarios.