stats.inla | R Documentation |

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.

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

`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 |

A named numeric vector with the extracted statistics.

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))`

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

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. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1214/22-STS864")}.

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