Description Usage Arguments Details Value Author(s) References See Also Examples

Efficient approximate leave-one-out cross validation (LOO), deviance information criterion (DIC) and widely applicable information criterion (WAIC) for Bayesian models, calculated on the observed data.

1 | ```
pic(x, criterion = "dic", module = "total")
``` |

`x` |
A |

`criterion` |
type of information criteria to be produced. Available choices are |

`module` |
The modules with respect to which the information criteria should be computed. Available choices are |

The Deviance Information Criterion (DIC), Leave-One-Out Information Criterion (LOOIC) and the Widely Applicable Information Criterion (WAIC) are methods for estimating
out-of-sample predictive accuracy from a Bayesian model using the log-likelihood evaluated at the posterior simulations of the parameters.
DIC is computationally simple to calculate but it is known to have some problems, arising in part from it not being fully Bayesian in that it is based on a point esitmate.
LOOIC can be computationally expensive but can be easily approximated using importance weights that are smoothed by fitting a generalised Pareto distribution to the upper tail
of the distribution of the importance weights. For more details about the methods used to compute LOOIC see the PSIS-LOO section in `loo-package`

.
WAIC is fully Bayesian and closely approximates Bayesian cross-validation. Unlike DIC, WAIC is invariant to parameterisation and also works for singular models.
In finite cases, WAIC and LOO give similar esitmates, but for influential observations WAIC underestimates the effect of leaving out one observation.

A named list containing different predictive information criteria results and quantities according to the value of `criterion`

. In all cases, the measures are
computed on the observed data for the specific modules of the model selected in `module`

.

- d_bar
Posterior mean deviance (only if

`criterion`

is`'dic'`

).- pD
Effective number of parameters calculated with the formula used by

`JAGS`

(only if`criterion`

is`'dic'`

)

.

- dic
Deviance Information Criterion calculated with the formula used by

`JAGS`

(only if`criterion`

is`'dic'`

)

.

- d_hat
Deviance evaluated at the posterior mean of the parameters and calculated with the formula used by

`JAGS`

(only if`criterion`

is`'dic'`

)- elpd, elpd_se
Expected log pointwise predictive density and standard error calculated on the observed data for the model nodes indicated in

`module`

(only if`criterion`

is`'waic'`

or`'loo'`

).- p, p_se
Effective number of parameters and standard error calculated on the observed data for the model nodes indicated in

`module`

(only if`criterion`

is`'waic'`

or`'loo'`

).- looic, looic_se
The leave-one-out information criterion and standard error calculated on the observed data for the model nodes indicated in

`module`

(only if`criterion`

is`'loo'`

).- waic, waic_se
The widely applicable information criterion and standard error calculated on the observed data for the model nodes indicated in

`module`

(only if`criterion`

is`'waic'`

).- pointwise
A matrix containing the pointwise contributions of each of the above measures calculated on the observed data for the model nodes indicated in

`module`

(only if`criterion`

is`'waic'`

or`'loo'`

).- pareto_k
A vector containing the estimates of the shape parameter

*k*for the generalised Pareto fit to the importance ratios for each leave-one-out distribution calculated on the observed data for the model nodes indicated in`module`

(only if`criterion`

is`'loo'`

). See`loo`

for details about interpreting*k*.

Andrea Gabrio

Plummer, M. *JAGS: A program for analysis of Bayesian graphical models using Gibbs sampling.* (2003).

Vehtari, A. Gelman, A. Gabry, J. (2016a) Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC.
*Statistics and Computing*. Advance online publication.

Vehtari, A. Gelman, A. Gabry, J. (2016b) Pareto smoothed importance sampling. *ArXiv* preprint.

Gelman, A. Hwang, J. Vehtari, A. (2014) Understanding predictive information criteria for Bayesian models.
*Statistics and Computing* 24, 997-1016.

Watanable, S. (2010). Asymptotic equivalence of Bayes cross validation and widely application information
criterion in singular learning theory. *Journal of Machine Learning Research* 11, 3571-3594.

1 2 3 4 | ```
#For examples see the function selection, pattern or hurdle
#
#
``` |

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