LaplaceMetropolis_gaussian: Laplace-Metropolis estimator of log marginal likelihood

Description

As pointed out by Raftery (1996), the Laplace-Metropolis estimator performs well in calculating log marginal likelihood among other methods considered.

Usage

LaplaceMetropolis_gaussian(theta, data = NULL, data_y, prior_p, prior_st,  method = c("likelihood","L1center","median"))

Arguments

theta

MCMC output

data

Regressors

data_y

Response

prior_p

Hyperparameter of the inverse-gamma prior

prior_st

Hyperparameter of the inverse-gamma prior

method

Computing method. L1center and median are computationally fast

Value

Log marginal likelihood

Details

The idea of the Laplace-Metropolis estimator is to avoid the limitations of the Laplace method by using posterior simulation to estimate the quantities it needs. The Laplace method for integrals is based on a Taylor series expansion of the real-valued function $f(u)$ of the $d$-dimensional vector $u$, and yields the approximation $P(D)\approx (2*pi)^(d/2)|A|^(1/2)P(D|\theta)P(\theta)$, where $\theta$ is the posterior mode of $h(\theta)=log(P(D|\theta)P(\theta))$, $A$ is minus the inverse Hessian of $h(\theta)$ evaluated at $theta$, and $d$ is the dimension of $\theta$.

The simplest way to estimate $\theta$ from posterior simulation output, and probably the most accurate, is to compute $h(\theta^(t))$ for each $t=1,\dots,T$ and take the value for which it is largest.

References

I. Ntzoufras (2009) Bayesian Modeling Using WinBUGS. John Wiley and Sons, Inc. New Jersey.

A. E. Raftery (1996) Hypothesis testing and model selection, in Markov Chain Monte Carlo In Practice by W. R. Gilks, S. Richardson and D. J. Spiegelhalter, Chapman and Hall, London.