This function computes the expected log pointwise predictive density (elppd)
for an object of class lmBayes. Given a test (held-out) set,
\(\{y_{\text{test},i},\,
\mathbf{x}_{\text{test},i}\}_{i=1}^{n_{\text{test}}}\), the elppd is defined
as
$$\text{elppd} = \frac{1}{n_{\text{test}}}\sum_{i=1}^{n_{\text{test}}}
\log\left(\frac{1}{S}\sum_{s=1}^{S}p\left({y}_{\text{test}, i} |
\mathbf{x}_{\text{test}, i}^{_{'}}\boldsymbol{\beta}_{(s)},
\sigma^{2}_{(s)}\right)\right),$$
where \(p()\) is the likelihood of the observed data and
\(\boldsymbol{\beta}_{(s)}\) and \(\sigma^{2}_{(s)}\) are the
\(s\)-th draws of the coefficient vector and the sampling variance in the
MCMC algorithm, respectively.