Proper Bayesian estimating equation based on Hilbert space method

This paper uses Hilbert space method to introduce and investigate the validity of Bayesian estimating equation. A validity for Hilbert-based Bayesian estimating function is established via the Hilbert-based unbiasedness and information unbiasedness. As an application, the newly proposed method is adopted to construct an estimating equation for nonlinear regression model. Furthermore, the new notion is employed to lay a theoretical foundation for the penalty-based methods such as penalized likelihood and penalized least squares.