Efficient Interpolation of Computationally Expensive Posterior Densities With Variable Parameter Costs

Area of Science:

• Computational Statistics
• Bayesian Inference
• Numerical Analysis

Background:

• Markov chain Monte Carlo (MCMC) is standard for Bayesian posterior density integration.
• Evaluating posterior densities can be computationally intractable, hindering MCMC applications.
• Identifying subvectors responsible for high computational cost is crucial.

Purpose of the Study:

• To develop novel interpolation methods (DOSKA and INDA) for computationally expensive Bayesian inference.
• To mitigate the curse of dimensionality in MCMC by exploiting problem structure.
• To reduce the number of expensive function evaluations required for accurate posterior approximations.

Main Methods:

• Proposed DOSKA and INDA interpolation approaches.
• Derived a Gaussian processes interpolant reducing effective dimensionality.
• Applied methods to a spatio-temporal linear model for air pollution data.

Main Results:

• Gaussian processes interpolant provably improves over existing methods.
• Reduced effective interpolation dimension from dim(η) to dim(β).
• Demonstrated significant reduction in necessary expensive evaluations for high-dimensional problems with low-dimensional sensitive components.

Conclusions:

• DOSKA and INDA offer efficient approximations for intractable posterior densities.
• The proposed Gaussian processes interpolant effectively reduces computational burden.
• These methods enhance the feasibility of Bayesian inference in complex, high-dimensional models.