This vignette illustrates the main functionalities of the
**projpred** package, which implements the projection
predictive variable selection for various regression models (see section
“Supported types of models” below for more
details on supported model types). What is special about the projection
predictive variable selection is that it not only performs a variable
selection, but also allows for (approximately) valid post-selection
inference.

The projection predictive variable selection is based on the ideas of
Goutis and Robert (1998) and Dupuis and Robert (2003). The methods
implemented in **projpred** are described in detail in
Piironen, Paasiniemi, and Vehtari (2020), Catalina, Bürkner, and Vehtari (2022), Weber, Glass, and Vehtari (2023), and Catalina, Bürkner, and Vehtari (2021). A comparison to many
other methods may also be found in Piironen and
Vehtari (2017a). For
details on how to cite **projpred**, see the projpred
citation info on CRAN^{1}.

For this vignette, we use **projpred**’s
`df_gaussian`

data. It contains 100 observations of 20
continuous predictor variables `X1`

, …, `X20`

(originally stored in a sub-matrix; we turn them into separate columns
below) and one continuous response variable `y`

.

First, we have to construct a reference model for the projection predictive variable selection. This model is considered as the best (“reference”) solution to the prediction task. The aim of the projection predictive variable selection is to find a subset of a set of candidate predictors which is as small as possible but achieves a predictive performance as close as possible to that of the reference model.

Usually (and this is also the case in this vignette), the reference
model will be an **rstanarm** or **brms**
fit. To our knowledge, **rstanarm** and
**brms** are currently the only packages for which a
`get_refmodel()`

method (which establishes the compatibility
with **projpred**) exists. Creating a reference model
object via one of these methods `get_refmodel.stanreg()`

or
`brms::get_refmodel.brmsfit()`

(either implicitly by a call
to a top-level function such as `project()`

,
`varsel()`

, and `cv_varsel()`

, as done below, or
explicitly by a call to `get_refmodel()`

) leads to a
“typical” reference model object. In that case, all candidate models are
actual *sub*models of the reference model. In general, however,
this assumption is not necessary for a projection predictive variable
selection (see, e.g., Piironen, Paasiniemi, and Vehtari
2020). This is why “custom” (i.e., non-“typical”) reference
model objects allow to avoid this assumption (although the candidate
models of a “custom” reference model object will still be actual
*sub*models of the full `formula`

used by the search
procedure—which does not have to be the same as the reference model’s
`formula`

, if the reference model possesses a
`formula`

at all). Such “custom” reference model objects can
be constructed via `init_refmodel()`

(or
`get_refmodel.default()`

), as shown in section “Examples” of
the `?init_refmodel`

help^{2}.

Here, we use the **rstanarm** package to fit the
reference model. If you want to use the **brms** package,
simply replace the **rstanarm** fit (of class
`stanreg`

) in all the code below by your
**brms** fit (of class `brmsfit`

).

For our **rstanarm** reference model, we use the
Gaussian distribution as the `family`

for our response. With
respect to the predictors, we only include the linear main effects of
all 20 predictor variables. Compared to the more complex types of
reference models supported by **projpred** (see section “Supported types of models” below), this is a quite
simple reference model which is sufficient, however, to demonstrate the
interplay of **projpred**’s functions.

We use **rstanarm**’s default priors in our reference
model, except for the regression coefficients for which we use a
regularized horseshoe prior (Piironen and Vehtari
2017c) with the hyperprior for its global shrinkage parameter
following Piironen and Vehtari (2017b) and Piironen and Vehtari (2017c). In R code, these
are the preparation steps for the regularized horseshoe prior:

`[1] 20`

```
# Prior guess for the number of relevant (i.e., non-zero) regression
# coefficients:
p0 <- 5
# Number of observations:
N <- nrow(dat_gauss)
# Hyperprior scale for tau, the global shrinkage parameter (note that for the
# Gaussian family, 'rstanarm' will automatically scale this by the residual
# standard deviation):
tau0 <- p0 / (D - p0) * 1 / sqrt(N)
```

We now fit the reference model to the data. To make this vignette
build faster, we use only 2 MCMC chains and 1000 iterations per chain
(with half of them being discarded as warmup draws). In practice, 4
chains and 2000 iterations per chain are reasonable defaults.
Furthermore, we make use of **rstan**’s
parallelization, which means to run each chain on a separate CPU core^{3}. If you
run the following code yourself, you can either rely on an automatic
mechanism to detect the number of CPU cores (like the
`parallel::detectCores()`

function shown below) or adapt
`ncores`

manually to your system.

```
# Set this manually if desired:
ncores <- parallel::detectCores(logical = FALSE)
### Only for technical reasons in this vignette (you can omit this when running
### the code yourself):
ncores <- min(ncores, 2L)
###
options(mc.cores = ncores)
set.seed(5078022)
refm_fit <- stan_glm(
y ~ X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X10 + X11 + X12 + X13 + X14 +
X15 + X16 + X17 + X18 + X19 + X20,
family = gaussian(),
data = dat_gauss,
prior = hs(global_scale = tau0),
### Only for the sake of speed (not recommended in general):
chains = 2, iter = 1000,
###
QR = TRUE, refresh = 0
)
```

```
Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
Running the chains for more iterations may help. See
https://mc-stan.org/misc/warnings.html#tail-ess
```

Usually, we would now have to check the convergence diagnostics (see,
e.g., `?posterior::diagnostics`

and
`?posterior::default_convergence_measures`

; the tail-ESS
warning already indicates a problem). However, due to the technical
reasons for which we reduced `chains`

and `iter`

,
we skip this step here (and hence ignore the tail-ESS warning).

Now, **projpred** comes into play.

From the reference model fit (called `refm_fit`

here), we
create a reference model object (i.e., an object of class
`refmodel`

) since this avoids redundant calculations in the
remainder of this vignette^{4}:

In **projpred**, the projection predictive variable
selection relies on a so-called *search* part and a so-called
*evaluation* part. The search part determines the predictor
ranking (also known as solution path), i.e., the best submodel for each
submodel size (the size is given by the number of predictor terms). The
evaluation part determines the predictive performance of the
increasingly complex submodels along the predictor ranking.

There are two functions for running the combination of search and
evaluation: `varsel()`

and `cv_varsel()`

. In
contrast to `varsel()`

, `cv_varsel()`

performs a
cross-validation (CV). With `cv_method = "LOO"`

(the
default), `cv_varsel()`

runs a Pareto-smoothed importance
sampling leave-one-out CV (PSIS-LOO CV, see Vehtari, Gelman, and Gabry 2017;
Vehtari et al. 2022). With
`cv_method = "kfold"`

, `cv_varsel()`

runs a \(K\)-fold CV. The extent of the CV depends
on `cv_varsel()`

’s argument `validate_search`

: If
`validate_search = TRUE`

(the default), the search part is
run with the training data of each CV fold separately and the evaluation
part is run with the corresponding test data of each CV fold. If
`validate_search = FALSE`

, the search is excluded from the CV
so that only a single full-data search is run. Because of its most
thorough protection against overfitting^{5},
`cv_varsel()`

with `validate_search = TRUE`

is
recommended over `varsel()`

and `cv_varsel()`

with
`validate_search = FALSE`

. Nonetheless, a preliminary and
comparatively fast run of `varsel()`

or
`cv_varsel()`

with `validate_search = FALSE`

can
give a rough idea of the performance of the submodels and can be used
for finding a suitable value for argument `nterms_max`

in
subsequent runs (argument `nterms_max`

imposes a limit on the
submodel size up to which the search is continued and is thus able to
reduce the runtime considerably).

`cv_varsel()`

runTo illustrate a preliminary `cv_varsel()`

run with
`validate_search = FALSE`

, we set `nterms_max`

to
the number of predictor terms in the full model, i.e.,
`nterms_max = 20`

. To speed up the building of the vignette
(this is not recommended in general), we choose the `"L1"`

search `method`

and set `refit_prj`

to `FALSE`

.

```
# Preliminary cv_varsel() run:
cvvs_fast <- cv_varsel(
refm_obj,
validate_search = FALSE,
### Only for the sake of speed (not recommended in general):
method = "L1",
refit_prj = FALSE,
###
nterms_max = 20,
### In interactive use, we recommend not to deactivate the verbose mode:
verbose = FALSE
###
)
```

`Warning: Some Pareto k diagnostic values are slightly high. See help('pareto-k-diagnostic') for details.`

```
Warning in warn_pareto(n07 = sum(pareto_k > 0.7), n05 = sum(0.7 >= pareto_k & :
In the calculation of the reference model's PSIS-LOO CV weights, 11 (out of
100) Pareto k-values are in the interval (0.5, 0.7]. Moment matching (see the
loo package), mixture importance sampling (see the loo package), and
`reloo`-ing (see the brms package) are not supported by projpred. If these
techniques (run outside of projpred, i.e., for the reference model only; note
that `reloo`-ing may be computationally costly) result in a markedly different
reference model ELPD estimate than ordinary PSIS-LOO CV does, we recommend to
use K-fold CV within projpred.
```

In this case, we ignore the Pareto \(\hat{k}\) warnings due to the reduced
values for `chains`

and `iter`

in the reference
model fit.

To find a suitable value for `nterms_max`

in subsequent
`cv_varsel()`

runs, we take a look at a plot of at least one
predictive performance statistic in dependence of the submodel size.
Here, we choose the mean log predictive density (MLPD; see the
documentation for argument `stats`

of
`summary.vsel()`

for details) as the only performance
statistic. Since we will be using the following plot only to determine
`nterms_max`

for subsequent `cv_varsel()`

runs, we
can omit the predictor ranking from the plot by setting
`ranking_nterms_max`

to `NA`

:

This plot suggests that the submodel MLPD levels off for the first time
from submodel size 9 to 10 and that it increases only slightly
afterwards. However, we used L1 search with
`refit_prj = FALSE`

, which means that the projections
employed for the predictive performance evaluation are L1-penalized,
which is usually undesired (Piironen, Paasiniemi, and Vehtari
2020, sec. 4). Thus, to investigate the impact of
`refit_prj = FALSE`

, we re-run `cv_varsel()`

, but
this time with the default of `refit_prj = TRUE`

and re-using
the search results (as well as the CV-related arguments such as
`validate_search`

; see section “Usage” in
`?cv_varsel.vsel`

) from the former `cv_varsel()`

call (this is done by applying `cv_varsel()`

to
`cvvs_fast`

instead of `refm_obj`

so that the
`cv_varsel()`

generic dispatches to the
`cv_varsel.vsel()`

method that was introduced in
**projpred** 2.8.0^{6}). To save time, we also set
`nclusters_pred`

to a comparatively low value of
`20`

:

```
# Preliminary cv_varsel() run with `refit_prj = TRUE`:
cvvs_fast_refit <- cv_varsel(
cvvs_fast,
### Only for the sake of speed (not recommended in general):
nclusters_pred = 20,
###
### In interactive use, we recommend not to deactivate the verbose mode:
verbose = FALSE
###
)
```

`Warning: Some Pareto k diagnostic values are slightly high. See help('pareto-k-diagnostic') for details.`

```
Warning in warn_pareto(n07 = sum(pareto_k > 0.7), n05 = sum(0.7 >= pareto_k & :
In the calculation of the reference model's PSIS-LOO CV weights, 11 (out of
100) Pareto k-values are in the interval (0.5, 0.7]. Moment matching (see the
loo package), mixture importance sampling (see the loo package), and
`reloo`-ing (see the brms package) are not supported by projpred. If these
techniques (run outside of projpred, i.e., for the reference model only; note
that `reloo`-ing may be computationally costly) result in a markedly different
reference model ELPD estimate than ordinary PSIS-LOO CV does, we recommend to
use K-fold CV within projpred.
```

```
Warning in loo_varsel(refmodel = refmodel, method = method, nterms_max =
nterms_max, : The projected draws used for the performance evaluation have
different (i.e., nonconstant) weights, so using standard importance sampling
(SIS) instead of Pareto-smoothed importance sampling (PSIS). In general, PSIS
is recommended over SIS.
```

As explained above, we ignore the Pareto \(\hat{k}\) warnings
here. We
also ignore the warning that SIS is used instead of PSIS (this is due to
`nclusters_pred = 20`

which we used only to speed up the
building of the vignette).

With the `refit_prj = TRUE`

results, the predictive
performance plot from above now looks as follows:

This refined plot shows that the submodel MLPD does not change much
after submodel size 8, so in our final `cv_varsel()`

run, we
set `nterms_max`

to a value slightly higher than 8 (here: 9)
to ensure that we see the MLPD leveling off. The search results from the
initial `cvvs_fast`

object could now be re-used again (via
`cv_varsel.vsel()`

) for investigating the sensitivity of the
results to changes in `nclusters_pred`

(or
`ndraws_pred`

). Here, we skip this for the sake of brevity
and instead head over to the final `cv_varsel()`

run.

`cv_varsel()`

runFor this final `cv_varsel()`

run (with
`validate_search = TRUE`

, as recommended), we use a \(K\)-fold CV with a small number of folds
(`K = 2`

) to make this vignette build faster. In practice, we
recommend using either the default of `cv_method = "LOO"`

or
a larger value for `K`

if this is possible in terms of
computation time.
Here, we also perform the \(K\)
reference model refits outside of `cv_varsel()`

. Although not
strictly necessary here, this is helpful in practice because often,
`cv_varsel()`

needs to be re-run multiple times in order to
try out different argument settings. We also illustrate how
**projpred**’s CV (i.e., the CV comprising search and
performance evaluation, after refitting the reference model \(K\) times) can be parallelized, even though
this is of little use here (we have only `K = 2`

folds and
the fold-wise searches and performance evaluations are quite fast, so
the parallelization overhead eats up any runtime improvements). Note
that for this final `cv_varsel()`

run, we cannot make use of
`cv_varsel.vsel()`

(by applying `cv_varsel()`

to
`cvvs_fast`

or `cvvs_fast_refit`

instead of
`refm_obj`

) because of the change in `nterms_max`

(here: 9, for `cvvs_fast`

and `cvvs_fast_refit`

:
20).

```
# Refit the reference model K times:
cv_fits <- run_cvfun(
refm_obj,
### Only for the sake of speed (not recommended in general):
K = 2
###
)
# For running projpred's CV in parallel (see cv_varsel()'s argument `parallel`):
doParallel::registerDoParallel(ncores)
# Final cv_varsel() run:
cvvs <- cv_varsel(
refm_obj,
cv_method = "kfold",
cvfits = cv_fits,
### Only for the sake of speed (not recommended in general):
method = "L1",
nclusters_pred = 20,
###
nterms_max = 9,
parallel = TRUE,
### In interactive use, we recommend not to deactivate the verbose mode:
verbose = FALSE
###
)
# Tear down the CV parallelization setup:
doParallel::stopImplicitCluster()
foreach::registerDoSEQ()
```

We can now select a final submodel size by looking at a predictive
performance plot similar to the one created for the preliminary
`cv_varsel()`

run above. By default, the performance
statistics are plotted on their original scale, but with
`deltas = TRUE`

, they are plotted as differences^{7} from a
baseline model (which is the reference model by default, at least in the
most common cases). Since the differences^{8} and the (frequentist)
uncertainty in their estimation are usually of more interest than the
original-scale performance statistics (at least with regard to the
decision for a final submodel size), we directly plot with
`deltas = TRUE`

here:

Based on that final predictive performance plot, we decide for a
submodel size. Usually, the aim is to find the smallest submodel size
where the predictive performance of the submodels levels off and is
close enough to the reference model’s predictive performance (the dashed
red horizontal line). Sometimes (as here), the plot may be ambiguous
because after reaching the reference model’s performance, the submodels’
performance may keep increasing (and hence become even better than the
reference model’s performance^{9}). In that case, one has to find a suitable
trade-off between predictive performance (accuracy) and model size
(sparsity) in the context of subject-matter knowledge. Here, we assume
that the focus of our predictive model is sparsity (not accuracy).
Hence, based on the plot, we decide for a submodel size of 6 because
this is the smallest size where the submodel MLPD is close enough to the
reference model MLPD:

If the focus of our predictive model had been accuracy (not sparsity), size 7 (or even 8) would have been a natural choice because at size 7 (at the latest at size 8), the submodel MLPD can be considered to “level off”. Further down, the predictor ranking and the (CV) ranking proportions that are shown in the plot (below the submodel sizes on the x-axis) are explained in detail—and also how they could have been incorporated into our decision for a submodel size.

The `suggest_size()`

function offered by
**projpred** may help in the decision for a submodel size,
but this is a rather heuristic method and needs to be interpreted with
caution (see `?suggest_size`

):

`[1] 6`

With this heuristic, we would get the same final submodel size
(`6`

) as by our manual (sparsity-based) decision.

`summary()`

method for `vsel`

objectsA tabular representation of the plot created by
`plot.vsel()`

can be achieved via
`summary.vsel()`

. For the output of
`summary.vsel()`

, there is a sophisticated
`print()`

method (`print.vselsummary()`

) which is
also called by the shortcut method `print.vsel()`

^{10}.
Specifically, to create the summary table matching the predictive
performance plot above as closely as possible (and to also adjust the
minimum number of printed significant digits), we may call
`summary.vsel()`

and `print.vselsummary()`

as
follows:

```
smmry <- summary(cvvs, stats = "mlpd", type = c("mean", "lower", "upper"),
deltas = TRUE)
print(smmry, digits = 1)
```

```
Family: gaussian
Link function: identity
Formula: y ~ X1 + X2 + X3 + X4 + X5 + X6 + X7 + X8 + X9 + X10 + X11 +
X12 + X13 + X14 + X15 + X16 + X17 + X18 + X19 + X20
Observations: 100
Projection method: traditional
CV method: K-fold CV with K = 2 and search included (i.e., fold-wise searches)
Search method: L1
Maximum submodel size for the search: 9
Number of projected draws in the search: 1 (from clustered projection)
Number of projected draws in the performance evaluation: 20 (from clustered projection)
Argument `refit_prj`: TRUE
Submodel performance evaluation summary with `deltas = TRUE` and `cumulate = FALSE`:
size ranking_fulldata cv_proportions_diag mlpd mlpd.lower mlpd.upper
0 (Intercept) NA -0.90 -0.980 -0.82
1 X1 1.0 -0.70 -0.787 -0.62
2 X14 0.5 -0.59 -0.669 -0.50
3 X5 0.5 -0.43 -0.521 -0.35
4 X20 0.5 -0.34 -0.418 -0.26
5 X6 0.0 -0.11 -0.175 -0.05
6 X3 0.5 0.05 0.003 0.11
7 X8 1.0 0.13 0.082 0.18
8 X11 0.5 0.14 0.094 0.18
9 X10 0.0 0.12 0.086 0.16
Reference model performance evaluation summary with `deltas = TRUE`:
mlpd mlpd.lower mlpd.upper
0 0 0
```

As indicated by its column name, the predictor ranking from column
`ranking_fulldata`

of the `summary.vsel()`

output
is based on the full-data search. In case of `cv_varsel()`

with `validate_search = TRUE`

, there is not only the
full-data search, but also fold-wise searches, implying that there are
also fold-wise predictor rankings. All of these predictor rankings (the
full-data one and—if available—the fold-wise ones) can be retrieved via
`ranking()`

:

In addition to inspecting the full-data predictor ranking, it usually
makes sense to investigate the ranking proportions derived from the
fold-wise predictor rankings (only available in case of
`cv_varsel()`

with `validate_search = TRUE`

, which
we have here) in order to get a sense for the variability in the ranking
of the predictors. For a given predictor \(x\) and a given submodel size \(j\), the ranking proportion is the
proportion of CV folds which have predictor \(x\) at position \(j\) of their predictor ranking. To compute
these ranking proportions, we use `cv_proportions()`

:

```
predictor
size X1 X14 X5 X20 X6 X3 X8 X11 X10
1 1 0.0 0.0 0.0 0.0 0.0 0 0.0 0
2 0 0.5 0.0 0.0 0.0 0.5 0 0.0 0
3 0 0.5 0.5 0.0 0.0 0.0 0 0.0 0
4 0 0.0 0.0 0.5 0.5 0.0 0 0.0 0
5 0 0.0 0.5 0.5 0.0 0.0 0 0.0 0
6 0 0.0 0.0 0.0 0.5 0.5 0 0.0 0
7 0 0.0 0.0 0.0 0.0 0.0 1 0.0 0
8 0 0.0 0.0 0.0 0.0 0.0 0 0.5 0
9 0 0.0 0.0 0.0 0.0 0.0 0 0.5 0
attr(,"class")
[1] "cv_proportions"
```

Here, the ranking proportions are of little use as we have used
`K = 2`

(in the final `cv_varsel()`

call above)
for the sake of speed. Nevertheless, we can see that the two CV folds
agree on the most relevant predictor term (`X1`

). Since the
column names of the matrix returned by `cv_proportions()`

follow the full-data predictor ranking, we can infer that
`X1`

is also the most relevant predictor term in the
full-data predictor ranking. To see this more explicitly, we can access
element `fulldata`

of the `ranking()`

output:

`[1] "X1" "X14" "X5" "X20" "X6" "X3" "X8" "X11" "X10"`

This is the same as column `ranking_fulldata`

in the
`summary.vsel()`

output above (apart from the intercept). (As
also stated by the message thrown by `print.vselsummary()`

,
column `cv_proportions_diag`

of the
`summary.vsel()`

output contains the main diagonal of the
matrix that we stored manually as `pr_rk`

.)

Note that we have cut off the search at `nterms_max = 9`

(which is smaller than the number of predictor terms in the full model,
20 here), so the ranking proportions in the `pr_rk`

matrix do
not need to sum to 100 % (neither column-wise nor row-wise).

The *transposed* matrix of ranking proportions can be
visualized via `plot.cv_proportions()`

:

Apart from visualizing the variability in the ranking of the predictors
(here, this is of little use because of `K = 2`

), this plot
will be helpful later.

To retrieve the predictor terms of the final submodel (except for the intercept which is always included in the submodels), we combine the chosen submodel size of 6 with the full-data predictor ranking:

`[1] "X1" "X14" "X5" "X20" "X6" "X3" `

At this place, it is again helpful to take the ranking proportions into account, but now in a cumulated fashion:

This plot shows that the two fold-wise searches (as well as the
full-data search, whose predictor ranking determines the order of the
predictors on the y-axis) agree on the *set* of the 6 most
relevant predictors: When looking at `<=6`

on the x-axis,
all tiles above and including the 6^{th} main diagonal element
are at 100 %. Similarly, the two CV folds agree on the *set* of
the 7 most relevant predictors (and also on the set of the most relevant
predictor, which is a singleton).

Although not demonstrated here, the cumulated ranking proportions
could also have guided the decision for a submodel size (if we had not
been willing to follow a strict rule based on accuracy or sparsity):
From their plot, we can see that size 9 might have been an unfortunate
choice because `X10`

(which—by cutting off the full-data
predictor ranking at size 9—would then have been selected as the
9^{th} predictor in the final submodel) is not included among
the first 9 terms by any CV fold. However, since `K = 2`

is
too small for reliable statements regarding the variability of the
predictor ranking, we did not take the cumulated ranking proportions
into account when we made our decision for a submodel size above.

In a real-world application, we might also be able to incorporate the full-data predictor ranking into our decision for a submodel size (usually, this requires to also take into account the variability of the predictor ranking, as reflected by the—possibly cumulated—ranking proportions). For example, the predictors might be associated with different measurement costs, so that we might want to select a costly predictor only if the submodel size at which it would be selected (according to the full-data predictor ranking, but taking into account that there might be variability in the ranking of the predictors) comes with a considerable increase in predictive performance.

The `project()`

function returns an object of class
`projection`

which forms the basis for convenient
post-selection inference. By the following `project()`

call,
we project the reference model onto the final submodel once again^{11}:

```
prj <- project(
refm_obj,
predictor_terms = predictors_final,
### In interactive use, we recommend not to deactivate the verbose mode:
verbose = FALSE
###
)
```

Next, we create a matrix containing the projected posterior draws
stored in the depths of `project()`

’s output:

This matrix is all we need for post-selection inference. It can be
used like any matrix of draws from MCMC procedures, except that it
doesn’t reflect a typical posterior distribution, but rather a projected
posterior distribution, i.e., the distribution arising from the
deterministic projection of the reference model’s posterior distribution
onto the parameter space of the final submodel^{12}. Beware that in case
of clustered projection (i.e., a non-`NULL`

argument
`nclusters`

in the `project()`

call), the
projected draws have different (i.e., nonconstant) weights, which needs
to be taken into account when performing post-selection (or, more
generally, post-projection) inference, see
`as_draws_matrix.projection()`

(`proj_linpred()`

and `proj_predict()`

offer similar functionality via
arguments `return_draws_matrix`

and
`nresample_clusters`

, respectively^{13}).

The **posterior**
package provides a general way to deal with posterior distributions, so
it can also be applied to our projected posterior. For example, to
calculate summary statistics for the marginals of the projected
posterior:

```
prj_drws <- as_draws_matrix(prj_mat)
prj_smmry <- summarize_draws(
prj_drws,
"median", "mad", function(x) quantile(x, probs = c(0.025, 0.975))
)
# Coerce to a `data.frame` because pkgdown versions > 1.6.1 don't print the
# tibble correctly:
prj_smmry <- as.data.frame(prj_smmry)
print(prj_smmry, digits = 1)
```

```
variable median mad 2.5% 97.5%
1 (Intercept) 0.04 0.10 -0.2 0.2
2 X1 1.38 0.09 1.2 1.6
3 X14 -1.12 0.09 -1.3 -0.9
4 X5 -0.92 0.11 -1.2 -0.7
5 X20 -1.13 0.12 -1.3 -0.9
6 X6 0.54 0.09 0.3 0.7
7 X3 0.79 0.10 0.6 1.0
8 sigma 1.13 0.09 1.0 1.3
```

A visualization of the projected posterior can be achieved with the
**bayesplot**
package, for example using its `mcmc_intervals()`

function:

```
bayesplot_theme_set(ggplot2::theme_bw())
mcmc_intervals(prj_mat) +
ggplot2::coord_cartesian(xlim = c(-1.5, 1.6))
```

Note that we only visualize the *1-dimensional* marginals of
the projected posterior here. To gain a more complete picture, we would
have to visualize at least some *2-dimensional* marginals of the
projected posterior (i.e., marginals for pairs of parameters).

For comparison, consider the marginal posteriors of the corresponding parameters in the reference model:

```
refm_mat <- as.matrix(refm_fit)
mcmc_intervals(refm_mat, pars = colnames(prj_mat)) +
ggplot2::coord_cartesian(xlim = c(-1.5, 1.6))
```

Here, the reference model’s marginal posteriors differ only slightly from the marginals of the projected posterior. This does not necessarily have to be the case.

Predictions from the final submodel can be made by
`proj_linpred()`

and `proj_predict()`

.

We start with `proj_linpred()`

. For example, suppose we
have the following new observations:

```
( dat_gauss_new <- setNames(
as.data.frame(replicate(length(predictors_final), c(-1, 0, 1))),
predictors_final
) )
```

```
X1 X14 X5 X20 X6 X3
1 -1 -1 -1 -1 -1 -1
2 0 0 0 0 0 0
3 1 1 1 1 1 1
```

Then `proj_linpred()`

can calculate the linear
predictors^{14} for all new observations from
`dat_gauss_new`

. Depending on argument
`integrated`

, these linear predictors can be averaged across
the projected draws (within each new observation). For instance, the
following computes the expected values of the new observations’
predictive distributions (beware that the following code refers to the
Gaussian family with the identity link function; for other
families—which usually come in combination with a different link
function—one would typically have to use `transform = TRUE`

in order to achieve such expected values):

```
prj_linpred <- proj_linpred(prj, newdata = dat_gauss_new, integrated = TRUE)
cbind(dat_gauss_new, linpred = as.vector(prj_linpred$pred))
```

```
X1 X14 X5 X20 X6 X3 linpred
1 -1 -1 -1 -1 -1 -1 0.48904558
2 0 0 0 0 0 0 0.03647046
3 1 1 1 1 1 1 -0.41610466
```

If `dat_gauss_new`

also contained response values (i.e.,
`y`

values in this example), then `proj_linpred()`

would also evaluate the log predictive density at these (conditional on
each of the projected parameter draws if `integrated = FALSE`

and integrated over the projected parameter draws—before taking the
logarithm—if `integrated = TRUE`

).

With `proj_predict()`

, we can obtain draws from predictive
distributions based on the final submodel. In contrast to
`proj_linpred(<...>, integrated = FALSE)`

, this
encompasses not only the uncertainty arising from parameter estimation,
but also the uncertainty arising from the observation (or “sampling”)
model for the response^{15}. This is useful for what is usually
termed a posterior predictive check (PPC), but would have to be termed
something like a posterior-projection predictive check (PPPC) here:

```
prj_predict <- proj_predict(prj)
# Using the 'bayesplot' package:
ppc_dens_overlay(y = dat_gauss$y, yrep = prj_predict)
```

This PPPC shows that our final projection is able to generate predictions similar to the observed response values, which indicates that this model is reasonable, at least in this regard.

In principle, the projection predictive variable selection requires only little information about the form of the reference model. Although many aspects of the reference model coincide with those from the submodels if a “typical” reference model object is used, this does not need to be the case if a “custom” reference model object is used (see section “Reference model” above for the definition of “typical” and “custom” reference model objects). This explains why in general, the following remarks refer to the submodels and not to the reference model.

In the following and throughout **projpred**’s
documentation, the term “traditional projection” is used whenever the
projection type is neither “augmented-data” nor “latent” (see below for
a description of these).

Apart from the `gaussian()`

response family used in this
vignette, **projpred**’s traditional projection also
supports the `binomial()`

^{16} and the `poisson()`

family.

The families supported by **projpred**’s augmented-data
projection (Weber, Glass, and Vehtari
2023) are `binomial()`

^{17} ^{18},
`brms::cumulative()`

, `rstanarm::stan_polr()`

fits, and `brms::categorical()`

^{19} ^{20}. See
`?extend_family`

(which is called by
`init_refmodel()`

) for an explanation how to apply the
augmented-data projection to “custom” reference model objects. For
“typical” reference model objects (i.e., those created by
`get_refmodel.stanreg()`

or
`brms::get_refmodel.brmsfit()`

), the augmented-data
projection is applied automatically if the family is supported by the
augmented-data projection and neither `binomial()`

nor
`brms::bernoulli()`

. For applying the augmented-data
projection to the `binomial()`

(or
`brms::bernoulli()`

) family, see `?extend_family`

as well as `?augdat_link_binom`

and
`?augdat_ilink_binom`

. Finally, we note that there are some
restrictions with respect to the augmented-data projection;
**projpred** will throw an informative error if a requested
feature is currently not supported for the augmented-data
projection.

The latent projection (Catalina, Bürkner, and Vehtari
2021) is a quite general principle for extending
**projpred**’s traditional projection to more response
families. The latent projection is applied when setting argument
`latent`

of `extend_family()`

(which is called by
`init_refmodel()`

) to `TRUE`

. The families for
which full latent-projection functionality (in particular,
`resp_oscale = TRUE`

, i.e., post-processing on the original
response scale) is currently available are `binomial()`

^{21} ^{22},
`poisson()`

, `brms::cumulative()`

, and
`rstanarm::stan_polr()`

fits^{23}. For all other
families, you can try to use the latent projection (by setting
`latent = TRUE`

) and **projpred** should tell
you if any features are not available and how to make them available.
More details concerning the latent projection are given in the
corresponding latent-projection
vignette. Note that there are some restrictions with respect to the
latent projection; **projpred** will throw an informative
error if a requested feature is currently not supported for the latent
projection.

On the side of the predictors, **projpred** not only
supports linear main effects as shown in this vignette, but also
interactions, multilevel^{24}, and—as an experimental
feature—additive^{25} terms.

Transferring this vignette to such more complex problems is
straightforward (also because this vignette employs a “typical”
reference model object): Basically, only the code for fitting the
reference model via **rstanarm** or **brms**
needs to be adapted. The **projpred** code stays almost the
same. Only note that in case of multilevel or additive reference models,
some **projpred** functions then have slightly different
options for a few arguments. See the documentation for details.

For example, to apply **projpred** to the
`VerbAgg`

dataset from the **lme4**
package, a corresponding multilevel reference model for the binary
response `r2`

could be created by the following code:

```
data("VerbAgg", package = "lme4")
refm_fit <- stan_glmer(
r2 ~ btype + situ + mode + (btype + situ + mode | id),
family = binomial(),
data = VerbAgg,
QR = TRUE, refresh = 0
)
```

As an example for an additive (non-multilevel) reference model,
consider the `lasrosas.corn`

dataset from the **agridat**
package. A corresponding reference model for the continuous response
`yield`

could be created by the following code (note that
`pp_check(refm_fit)`

gives a bad PPC in this case, so there’s
still room for improvement):

```
data("lasrosas.corn", package = "agridat")
# Convert `year` to a `factor` (this could also be solved by using
# `factor(year)` in the formula, but we avoid that here to put more emphasis on
# the demonstration of the smooth term):
lasrosas.corn$year <- as.factor(lasrosas.corn$year)
refm_fit <- stan_gamm4(
yield ~ year + topo + t2(nitro, bv),
family = gaussian(),
data = lasrosas.corn,
QR = TRUE, refresh = 0
)
```

As an example for an additive multilevel reference model, consider
the `gumpertz.pepper`

dataset from the
**agridat** package. A corresponding reference model for
the binary response `disease`

could be created by the
following code:

```
data("gumpertz.pepper", package = "agridat")
refm_fit <- stan_gamm4(
disease ~ field + leaf + s(water),
random = ~ (1 | row) + (1 | quadrat),
family = binomial(),
data = gumpertz.pepper,
QR = TRUE, refresh = 0
)
```

In case of multilevel models (currently only non-additive ones),
**projpred** has two global options that may be relevant
for users: `projpred.mlvl_pred_new`

and
`projpred.mlvl_proj_ref_new`

. These are explained in detail
in the general package documentation (available online
or by typing `?`projpred-package``

).

Sometimes, the predictor ranking makes sense, but for an increasing submodel size, the predictive performance of the submodels does not approach the reference model’s predictive performance so that the submodels exhibit a predictive performance that stays worse than the reference model’s. There are different reasons that can explain this behavior (the following list might not be exhaustive, though):

- The reference model’s posterior may be so wide that the default
`ndraws_pred`

could be too small. Usually, this comes in combination with a difference in predictive performance which is comparatively small. Increasing`ndraws_pred`

should help, but it also increases the computational cost. Refitting the reference model and thereby ensuring a narrower posterior (usually by employing a stronger sparsifying prior) should have a similar effect. - For non-Gaussian models, the discrepancy may be due to the fact that the penalized iteratively reweighted least squares (PIRLS) algorithm might have convergence issues (Catalina, Bürkner, and Vehtari 2021). In this case, the latent-space approach by Catalina, Bürkner, and Vehtari (2021) might help, see also the latent-projection vignette linked above.

If you are using `varsel()`

, then the lack of CV in
`varsel()`

may lead to overconfident and overfitted results.
In this case, try running `cv_varsel()`

instead of
`varsel()`

(which you should in any case for your final
results).

Similarly, `cv_varsel()`

with
`validate_search = FALSE`

is more prone to overfitting than
`cv_varsel()`

with `validate_search = TRUE`

.

For multilevel binomial models, the traditional projection may not
work properly and give suboptimal results, see #353 on GitHub
(the underlying issue is described in **lme4** issue #682). With
suboptimality of the results, we mean that in such cases, the relevance
of the group-level terms can be underestimated. According to the
simulation-based case study from #353, the
latent projection should be considered as a currently available
remedy.

For multilevel Poisson models, the traditional projection may take very long, see #353. According to the simulation-based case study from #353, the latent projection should be considered as a currently available remedy.

Finally, as illustrated in the Poisson example of the latent-projection vignette, the latent projection can be beneficial for non-multilevel models with a (non-Gaussian) family that is also supported by the traditional projection, at least in case of the Poisson family and L1 search.

For multilevel models, the augmented-data projection seems to suffer
from the same issue as the traditional projection for the binomial
family (see above), i.e., it may not work properly and give suboptimal
results, see #353 (the
underlying issue is probably similar to the one described in
**lme4** issue #682). With
suboptimality of the results, we mean that in such cases, the relevance
of the group-level terms can be underestimated. According to the
simulation-based case study from #353, the
latent projection should be considered as a remedy in such cases.

There are many ways to speed up **projpred**, but in
general, such speed-ups lead to results that are less accurate and hence
should only be considered as *preliminary* results. Some speed-up
possibilities are:

Using

`cv_varsel()`

with`validate_search = FALSE`

instead of`validate_search = TRUE`

. In case of`cv_method = "LOO"`

^{26},`cv_varsel()`

with`validate_search = FALSE`

has comparable runtime to`varsel()`

, but accounts for some overfitting, namely that induced by`varsel()`

’s in-sample predictions during the predictive performance evaluation. However, as explained in section “Variable selection” (see also section “Overfitting”),`cv_varsel()`

with`validate_search = FALSE`

is more prone to overfitting than`cv_varsel()`

with`validate_search = TRUE`

.Using

`cv_varsel()`

with \(K\)-fold CV instead of PSIS-LOO CV. Whether this provides a speed improvement mainly depends on the number of observations and the complexity of the reference model. Note that PSIS-LOO CV is often more accurate than \(K\)-fold CV if argument`K`

is (much) smaller than the number of observations.Using a “custom” reference model object with a dimension reduction technique for the predictor data (e.g., by computing principal components from the original predictors, using these principal components as predictors when fitting the reference model, and then performing the variable selection in terms of the

*original*predictor terms). Examples are given in Piironen, Paasiniemi, and Vehtari (2020) and Pavone et al. (2022). This approach makes sense if there is a large number of predictor variables, in which case this aims at improving the runtime required for fitting the reference model and hence improving the runtime of \(K\)-fold CV.Using

`varsel()`

with its argument`d_test`

for evaluating predictive performance on a hold-out dataset instead of doing this with`cv_varsel()`

’s CV approach. Typically, the hold-out approach requires a large amount of data.Reducing

`nterms_max`

in`varsel()`

or`cv_varsel()`

. The resulting predictive performance plot(s) should be inspected to check that the search is not terminated too early (i.e., before the submodel performance levels off), which would indicate that`nterms_max`

has been reduced too much.Reducing argument

`nclusters`

(of`varsel()`

or`cv_varsel()`

) below`20`

and/or setting`nclusters_pred`

to some non-`NULL`

(and smaller than`400`

, the default for`ndraws_pred`

) value. If setting`nclusters_pred`

as low as`nclusters`

(and using forward search),`refit_prj`

can instead be set to`FALSE`

, see below.Using L1 search (see argument

`method`

of`varsel()`

or`cv_varsel()`

) instead of forward search. Note that L1 search implies`nclusters = 1`

and is not always supported. In general, forward search is more accurate than L1 search and hence more desirable (see section “Details” in`?varsel`

or`?cv_varsel`

for a more detailed comparison of the two). The issue demonstrated in the Poisson example from the latent-projection vignette is related to this.Setting argument

`refit_prj`

(of`varsel()`

or`cv_varsel()`

) to`FALSE`

, which basically means to set`ndraws_pred = ndraws`

and`nclusters_pred = nclusters`

, but in a more efficient (i.e., faster) way. In case of L1 search, this means that the L1-penalized projections of the regression coefficients are used for the predictive performance evaluation, which is usually undesired (Piironen, Paasiniemi, and Vehtari 2020, sec. 4). In case of forward search, this issue does not exist.Parallelizing costly parts of the CV implied by

`cv_varsel()`

(this was demonstrated in the example above; see argument`parallel`

of`cv_varsel()`

). When using`project()`

, parallelizing the projection might also help (see the general package documentation available online or by typing`?`projpred-package``

).Using

`varsel.vsel()`

or`cv_varsel.vsel()`

to re-use previous search results for new performance evaluation(s). This is helpful if the performance evaluation part is run multiple times based on the same search results (e.g., when only arguments`ndraws_pred`

or`nclusters_pred`

of`varsel()`

or`cv_varsel()`

are changed). In the example above, this was illustrated when`cv_varsel()`

was applied to`cvvs_fast`

instead of`refm_obj`

to yield`cvvs_fast_refit`

. In that example, search and performance evaluation were effectively run*separately*by the`cv_varsel()`

calls yielding`cvvs_fast`

and`cvvs_fast_refit`

, respectively. This is because`cv_varsel()`

with`refit_prj = FALSE`

(used for`cvvs_fast`

) has almost only the computational cost of the search (the performance evaluation with`refit_prj = FALSE`

has almost no computational cost) and`cv_varsel.vsel()`

(used for`cvvs_fast_refit`

) has almost only the computational cost of the performance evaluation (the search in`cv_varsel.vsel()`

has no computational cost at all).Using

`run_cvfun()`

in case of repeated \(K\)-fold CV with the same \(K\) reference model refits. The output of`run_cvfun()`

is typically used as input for argument`cvfits`

of`cv_varsel.refmodel()`

(so in order to have a speed improvement, the output of`run_cvfun()`

needs to be assigned to an object which is then re-used in multiple`cv_varsel()`

calls).

Catalina, Alejandro, Paul-Christian Bürkner, and Aki Vehtari. 2022.
“Projection Predictive Inference for Generalized Linear and
Additive Multilevel Models.” In *Proceedings of
The 25th International Conference on
Artificial Intelligence and Statistics*,
edited by Gustau Camps-Valls, Francisco J. R. Ruiz, and Isabel Valera,
151:4446–61. Proceedings of Machine Learning Research.
PMLR. https://proceedings.mlr.press/v151/catalina22a.html.

Catalina, Alejandro, Paul Bürkner, and Aki Vehtari. 2021. “Latent
Space Projection Predictive Inference.” arXiv. https://doi.org/10.48550/arXiv.2109.04702.

Dupuis, Jérome A., and Christian P. Robert. 2003. “Variable
Selection in Qualitative Models via an Entropic Explanatory
Power.” *Journal of Statistical Planning and Inference*
111 (1–2): 77–94. https://doi.org/10.1016/S0378-3758(02)00286-0.

Goutis, Constantinos, and Christian P. Robert. 1998. “Model Choice
in Generalised Linear Models: A Bayesian Approach via
Kullback-Leibler Projections.” *Biometrika*
85 (1): 29–37.

Pavone, Federico, Juho Piironen, Paul-Christian Bürkner, and Aki
Vehtari. 2022. “Using Reference Models in Variable
Selection.” *Computational Statistics*. https://doi.org/10.1007/s00180-022-01231-6.

Piironen, Juho, Markus Paasiniemi, and Aki Vehtari. 2020.
“Projective Inference in High-Dimensional Problems:
Prediction and Feature Selection.” *Electronic
Journal of Statistics* 14 (1): 2155–97. https://doi.org/10.1214/20-EJS1711.

Piironen, Juho, and Aki Vehtari. 2017a. “Comparison of
Bayesian Predictive Methods for Model Selection.”
*Statistics and Computing* 27 (3): 711–35. https://doi.org/10.1007/s11222-016-9649-y.

———. 2017b. “On the Hyperprior Choice for the Global Shrinkage
Parameter in the Horseshoe Prior.” In *Proceedings of the 20th
International Conference on Artificial
Intelligence and Statistics*, edited by Aarti
Singh and Jerry Zhu, 54:905–13. Proceedings of Machine Learning
Research. PMLR. https://proceedings.mlr.press/v54/piironen17a.html.

———. 2017c. “Sparsity Information and Regularization in the
Horseshoe and Other Shrinkage Priors.” *Electronic Journal of
Statistics* 11 (2): 5018–51. https://doi.org/10.1214/17-EJS1337SI.

Vehtari, Aki, Andrew Gelman, and Jonah Gabry. 2017. “Practical
Bayesian Model Evaluation Using Leave-One-Out
Cross-Validation and WAIC.” *Statistics and
Computing* 27 (5): 1413–32. https://doi.org/10.1007/s11222-016-9696-4.

Vehtari, Aki, Daniel Simpson, Andrew Gelman, Yuling Yao, and Jonah
Gabry. 2022. “Pareto Smoothed Importance Sampling.”
arXiv. https://doi.org/10.48550/arXiv.1507.02646.

Weber, Frank, Änne Glass, and Aki Vehtari. 2023. “Projection
Predictive Variable Selection for Discrete Response Families with Finite
Support.” arXiv. https://doi.org/10.48550/arXiv.2301.01660.

The citation information can be accessed offline by typing

`print(citation("projpred"), bibtex = TRUE)`

within R.↩︎We will cover custom reference models more deeply in a future vignette.↩︎

More generally, the number of chains is split up as evenly as possible among the number of CPU cores.↩︎

Here, the

`get_refmodel()`

call is quite fast, but in other situations, it may take a while, so in general, it is better to create the`refmodel`

object once explicitly and then to re-use it.↩︎Currently, neither

`varsel()`

nor`cv_varsel()`

(not even`cv_varsel()`

with`validate_search = TRUE`

) guard against overfitting in the selection of the submodel*size*. This is why we added “approximately” to “valid post-selection inference” in section “Introduction”. Typically, however, the overfitting induced by the size selection should be comparatively small (Piironen and Vehtari 2017a).↩︎Analogous functionality has been implemented for

`varsel()`

, namely in`varsel.vsel()`

.↩︎For the geometric mean predictive density (GMPD, see argument

`stats`

of`summary.vsel()`

),`deltas = TRUE`

means to calculate the GMPD*ratio*(not difference) vs. the baseline model.↩︎For the GMPD, this is again a ratio.↩︎

In general, only

`cv_varsel()`

with`validate_search = TRUE`

(which we have here) allows to judge whether the submodels perform better than the reference model or not. Such a judgment is not possible with`varsel()`

or`cv_varsel()`

with`validate_search = FALSE`

(in general).↩︎`print.vsel()`

is the method that is called when simply printing an object resulting from`varsel()`

or`cv_varsel()`

.↩︎During the search, the reference model is projected onto all candidate models (this is where arguments

`ndraws`

and`nclusters`

of`varsel()`

and`cv_varsel()`

come into play; note that in case of an L1 search, the projection is L1-penalized). For the evaluation of the submodels along the predictor ranking returned by the search, the reference model is projected onto these submodels again (this is where arguments`ndraws_pred`

and`nclusters_pred`

of`varsel()`

and`cv_varsel()`

come into play; note that this only holds if argument`refit_prj`

of`varsel()`

and`cv_varsel()`

is set to`TRUE`

, as by default). Within`project()`

,`refit_prj = FALSE`

allows to re-use the submodel fits (that is, the projections) from the full-data search of a`vsel`

object, but usually, the search relies on a rather coarse clustering or thinning of the reference model’s posterior draws (by default,`varsel()`

and`cv_varsel()`

use`nclusters = 20`

—or`nclusters = 1`

in case of L1 search), which would then imply the same coarseness for a`project()`

call where`refit_prj`

is set to`FALSE`

. In general, we want the final projection (that post-selection inference is based on) to be as accurate as possible, so here we call`project()`

with the defaults of`refit_prj = TRUE`

and`ndraws = 400`

. For more accurate results, we could increase argument`ndraws`

of`project()`

(up to the number of posterior draws in the reference model). However, this would increase the runtime, which we don’t want in this vignette.↩︎In general, this implies that projected regression coefficients do not reflect isolated effects of the predictors. For example, especially in case of highly correlated predictors, it is possible that projected regression coefficients “absorb” effects from predictors that have been excluded in the projection.↩︎

`proj_predict()`

also has an argument`return_draws_matrix`

, but it simply converts the return value type. In`proj_predict()`

, different weights of the projected draws are taken into account via argument`nresample_clusters`

.↩︎`proj_linpred()`

can also transform the linear predictor to response scale, but here, this is the same as the linear predictor scale (because of the identity link function).↩︎In case of the Gaussian family we are using here, the uncertainty arising from the observation model is the uncertainty due to the residual standard deviation.↩︎

Via

`brms::get_refmodel.brmsfit()`

, the`brms::bernoulli()`

family is supported as well.↩︎Currently, the augmented-data support for the

`binomial()`

family does not include binomial distributions with more than one trial. In such a case, a workaround is to de-aggregate the Bernoulli trials which belong to the same (aggregated) observation, i.e., to use a “long” dataset.↩︎Like the traditional projection, the augmented-data projection also supports the

`brms::bernoulli()`

family via`brms::get_refmodel.brmsfit()`

.↩︎For the augmented-data projection based on a “typical”

**brms**reference model object,**brms**version 2.17.0 or later is needed.↩︎More

**brms**families might be supported in the future.↩︎Currently, the latent-projection support for the

`binomial()`

family does not include binomial distributions with more than one trial. In such a case, a workaround is to de-aggregate the Bernoulli trials which belong to the same (aggregated) observation, i.e., to use a “long” dataset.↩︎Like the traditional projection, the latent projection also supports the

`brms::bernoulli()`

family via`brms::get_refmodel.brmsfit()`

.↩︎For the latent projection based on a “typical”

**brms**reference model object,**brms**version 2.19.0 or later is needed.↩︎Multilevel models are also known as

*hierarchical*models or models with*partially pooled*,*group-level*, or—in frequentist terms—*random*effects.↩︎Additive terms are also known as

*smooth*terms.↩︎In case of

`cv_method = "kfold"`

, the runtime advantage of`validate_search = FALSE`

compared to`validate_search = TRUE`

is not as large as in case of`cv_method = "LOO"`

, but even for`cv_method = "kfold"`

, such a runtime advantage still exists.↩︎