This may be used to predict either new, unobserved instances of
\(\textbf{y}\) (called \(\textbf{y}^{new}\)) or
replicates of \(\textbf{y}\) (called
\(\textbf{y}^{rep}\)), and then perform posterior
predictive checks. Either \(\textbf{y}^{new}\) or
\(\textbf{y}^{rep}\) is predicted given an object of
class `iterquad`

, the model specification, and data. This
function requires that posterior samples were produced with
`IterativeQuadrature`

.

```
# S3 method for iterquad
predict(object, Model, Data, CPUs=1, Type="PSOCK", …)
```

object

An object of class `iterquad`

is required.

Model

The model specification function is required.

Data

A data set in a list is required. The dependent
variable is required to be named either `y`

or `Y`

.

CPUs

This argument accepts an integer that specifies the number
of central processing units (CPUs) of the multicore computer or
computer cluster. This argument defaults to `CPUs=1`

, in which
parallel processing does not occur.

Type

This argument specifies the type of parallel processing to
perform, accepting either `Type="PSOCK"`

or
`Type="MPI"`

.

…

Additional arguments are unused.

This function returns an object of class `iterquad.ppc`

(where
``ppc'' stands for posterior predictive checks). The returned object
is a list with the following components:

This stores \(\textbf{y}\), the dependent variable.

This is a \(N \times S\) matrix, where \(N\) is the number of records of \(\textbf{y}\) and \(S\) is the number of posterior samples.

This is a vector of length \(S\), where \(S\) is the number of
independent posterior samples. Samples are obtained with the
sampling importance resampling algorithm, `SIR`

.

This is a \(N \times S\) matrix, where \(N\) is the
number of monitored variables and \(S\) is the number of independent
posterior samples. Samples are obtained with the sampling importance
resampling algorithm, `SIR`

.

Since iterative quadrature characterizes marginal posterior
distributions with means and variances, and posterior predictive
checks involve samples, the `predict.iterquad`

function requires
the use of independent samples of the marginal posterior
distributions, provided by `IterativeQuadrature`

when
`sir=TRUE`

.

The samples of the marginal posterior distributions of the target
distributions (the parameters) are passed along with the data to the
`Model`

specification and used to draw samples from the deviance
and monitored variables. At the same time, the fourth component in the
returned list, which is labeled `yhat`

, is a vector of
expectations of \(\textbf{y}\), given the samples, model
specification, and data. To predict \(\textbf{y}^{rep}\),
simply supply the data set used to estimate the model. To predict
\(\textbf{y}^{new}\), supply a new data set instead (though
for some model specifications, this cannot be done, and
\(\textbf{y}_{new}\) must be specified in the `Model`

function). If the new data set does not have \(\textbf{y}\), then
create `y`

in the list and set it equal to something sensible,
such as `mean(y)`

from the original data set.

The variable `y`

must be a vector. If instead it is matrix
`Y`

, then it will be converted to vector `y`

. The vectorized
length of `y`

or `Y`

must be equal to the vectorized length
of `yhat`

, the fourth component of the returned list of the
`Model`

function.

Parallel processing may be performed when the user specifies
`CPUs`

to be greater than one, implying that the specified number
of CPUs exists and is available. Parallelization may be performed on a
multicore computer or a computer cluster. Either a Simple Network of
Workstations (SNOW) or Message Passing Interface is used (MPI). With
small data sets and few samples, parallel processing may be slower,
due to computer network communication. With larger data sets and more
samples, the user should experience a faster run-time.

For more information on posterior predictive checks, see https://web.archive.org/web/20150215050702/http://www.bayesian-inference.com/posteriorpredictivechecks.

`IterativeQuadrature`

and
`SIR`

.