stats.inla {INLAspacetime}

To retrieve goodness of fit statistics for a specific model class.

Description

Extracts dic, waic and log-cpo from an output returned by the inla function from the INLA package or by the bru function from the inlabru package, and computes log-po, mse, mae, crps and scrps for a given input. A summary is applied considering the user imputed function, which by default is the mean.

Usage

stats.inla(m, i = NULL, y, fsummarize = mean)

Arguments

m

an inla output object.

i

an index to subset the estimated values.

y

observed to compare against.

fsummarize

the summary function, the default is base::mean().

Value

A named numeric vector with the extracted statistics.

Details

It assumes Gaussian posterior predictive distributions! Considering the defaults, for n observations, y_i, i = 1, 2, ..., n, we have

. dic

\sum_i d_i/n

where d_i is the dic computed for observation i.

. waic

\sum_i w_i/n

where w_i is the waic computed for observation i.

. lcpo

-\sum_i \log(p_i)/n

where p_i is the cpo computed for observation i.

For the log-po, crps, and scrps scores it assumes a Gaussian predictive distribution for each observation y_i which the following definitions: z_i = (y_i-\mu_i)/\sigma_i, \mu_i is the posterior mean for the linear predictor, \sigma_i = \sqrt{v_i + 1/\tau_y}, \tau_y is the observation posterior mean, v_i is the posterior variance of the linear predictor for y_i.

Then we consider \phi() the density of a standard Gaussian variable and \psi() the corresponding Cumulative Probability Distribution.

. lpo

-\sum_i \log(\phi(z_i))/n

. crps

\sum_i r_i/n

where

r_i=\sigma_i/\sqrt{\pi} - 2\sigma_i\phi(z_i) + (y_i-\mu_i)(1-2\psi(z_i))

. scrps

\sum_i s_i/n

where

s_i=-\log(2\sigma_i/\sqrt{\pi})/2 -\sqrt{\pi}(\phi(z_i)-\sigma_iz_i/2+z_i\psi(z_i))

Warning

All the scores are negatively oriented which means that smaller scores are better.

References

Held, L. and Schrödle, B. and Rue, H. (2009). Posterior and Cross-validatory Predictive Checks: A Comparison of MCMC and INLA. Statistical Modelling and Regression Structures pp 91–110. https://link.springer.com/chapter/10.1007/978-3-7908-2413-1_6.

Bolin, D. and Wallin, J. (2022) Local scale invariance and robustness of proper scoring rules. Statistical Science. doi:10.1214/22-STS864.