01: A Framework for Loss Scale Determination in General Bayesian Updating

Music City Center 

General Bayesian updating (GBU) is a framework for updating prior beliefs about the parameter of interest to a posterior distribution via a loss function without imposing the distribution assumption on data. In recent years, the asymptotic distribution of the loss-likelihood bootstrap (LLB) sample has been a standard for determining the loss scale parameter which controls the relative weight of the loss function to the prior in GBU. However, the existing method fails to consider the prior distribution since it relies on the asymptotic equivalence between GBU and LLB. To address this limitation, we propose a new finite-sample-based approach to loss scale determination using the Bayesian generalized method of moments (BGMM) as a reference. We develop an efficient algorithm that determines the loss scale parameter by minimizing the Kullback-Leibler divergence between the exact posteriors of GBU and BGMM. We prove the convexity of our objective function to ensure a unique solution. Asymptotic properties of the proposed method are established to demonstrate its generalizability. We demonstrate the performance of our proposed method through a simulation study and a real data application.

General Bayesian updating

Loss-likelihood bootstrap

Generalized method of moments

Kullback-Leibler divergence

Monte Carlo Newton-Raphson method 

Korean International Statistical Society