Latent projection predictive feature selection

General idea

The response families used in GLMs (McCullagh and Nelder 1989, chap. 2), GLMMs, GAMs, and GAMMs (in particular, the gaussian(), the binomial(), and the poisson() family which are supported by projpred’s traditional projection) may be termed exponential dispersion (ED) families (Jørgensen 1987)¹. For a response family that is not an ED family, the Kullback-Leibler (KL) divergence minimization problem (see Piironen, Paasiniemi, and Vehtari 2020) is often not easy to solve analytically (exceptions are non-ED families that are discrete and have finite support; see the comment on the augmented-data projection in section “Implementation”). In order to bypass this issue, the latent projection (Catalina, Bürkner, and Vehtari 2021) solves the KL minimization problem in the predictive space of the latent predictors² instead of in the predictive space of the original response values.

To this end, the latent predictor is assumed to have a Gaussian distribution, since it (i) constitutes a combination of predictor data and regression parameters which is often linear (in the parameters, but—less—often also in the predictor data) or at least additive (across the predictor terms) and (ii) has the complete real line as support. Furthermore, the Gaussian distribution has the highest differential entropy among all distributions with two finite moments and with the real line as support (see, e.g., Cover and Thomas 1991). In some cases, e.g., for the probit link, the Gaussian distribution is even part of the original statistical model. In case of the logit link, the Gaussian distribution with a standard deviation of 1.6 approximates the logistic distribution (with a scale parameter of 1).

The assumption of a Gaussian distribution for the latent predictors makes things a lot easier because it allows us to make use of projpred’s traditional projection.

As illustrated by the Poisson example below, the latent projection can not only be used for families not supported by projpred’s traditional projection, but it can also be beneficial for families supported by it.

Implementation

To use the latent projection in projpred, argument latent of extend_family() needs to be set to TRUE. Since extend_family() is called by init_refmodel() which in turn is called by get_refmodel() (more precisely, by the get_refmodel() methods) which in turn is called at the beginning of the top-level functions project(), varsel(), and cv_varsel(), it is possible to pass latent = TRUE from such a top-level function down to extend_family() via the ellipsis (...). However, we recommend to define the reference model object of class refmodel explicitly (as illustrated in the examples below) to avoid repetitive and inefficient code³.

After performing the projection (either as a stand-alone feature via project() or embedded in a variable selection via varsel() or cv_varsel()), the post-processing (e.g., the estimation of the performance statistics in summary.vsel()) can be performed on the original response scale. For this, there are three arguments of extend_family() which accept R functions: latent_ilink (responsible for the inverse-link transformation from latent scale to response scale), latent_ll_oscale (responsible for the calculation of log-likelihood values on response scale), and latent_ppd_oscale (responsible for drawing from the (posterior-projection) predictive distribution on response scale). For some families, these three arguments have internal defaults implemented natively in projpred. These families are listed in the main vignette (section “Supported types of models”). For all other families, projpred either tries to infer a reasonable function internally (in case of latent_ilink) or uses a dummy function returning only NAs (in case of latent_ll_oscale and latent_ppd_oscale), unless the user supplies custom functions. When creating a reference model object for a family that lacks projpred’s native support for full response-scale post-processing, projpred will throw messages stating whether (and which) features will be unavailable unless at least some of these three arguments are provided by the user. Again, the ellipsis (...) can be used to pass these arguments from a top-level function such as cv_varsel() down to extend_family(). In the post-processing functions, response-scale analyses can usually be deactivated by setting argument resp_oscale to FALSE, with the exception of predict.refmodel() and proj_linpred() where arguments type and transform serve this purpose (see the documentation).

Apart from the arguments mentioned above, extend_family() also features the latent-projection argument latent_y_unqs whose purpose is described in the documentation.

While the latent projection is an approximate solution to the KL divergence minimization problem in the original response space⁴, the augmented-data projection (Weber, Glass, and Vehtari 2025) gives the exact⁵ solution for some non-ED families, namely those where the response distribution has finite support. However, the augmented-data projection comes with a higher runtime than the latent projection. The families currently supported by projpred’s augmented-data projection are also listed in the main vignette (again section “Supported types of models”).

Data

First, we generate a training and a test dataset with a Poisson-distributed response:

# Number of observations in the training dataset (= number of observations in
# the test dataset):
N <- 71
# Data-generating function:
sim_poiss <- function(nobs = 2 * N, ncon = 10, ncats = 4, nnoise = 39) {
  # Regression coefficients for continuous predictors:
  coefs_con <- rnorm(ncon)
  # Continuous predictors:
  dat_sim <- matrix(rnorm(nobs * ncon), ncol = ncon)
  # Start linear predictor:
  linpred <- 2.1 + dat_sim %*% coefs_con
  
  # Categorical predictor:
  dat_sim <- data.frame(
    x = dat_sim,
    xcat = gl(n = ncats, k = nobs %/% ncats, length = nobs,
              labels = paste0("cat", seq_len(ncats)))
  )
  # Regression coefficients for the categorical predictor:
  coefs_cat <- rnorm(ncats)
  # Continue linear predictor:
  linpred <- linpred + coefs_cat[dat_sim$xcat]
  
  # Noise predictors:
  dat_sim <- data.frame(
    dat_sim,
    xn = matrix(rnorm(nobs * nnoise), ncol = nnoise)
  )
  
  # Poisson response, using the log link (i.e., exp() as inverse link):
  dat_sim$y <- rpois(nobs, lambda = exp(linpred))
  # Shuffle order of observations:
  dat_sim <- dat_sim[sample.int(nobs), , drop = FALSE]
  # Drop the shuffled original row names:
  rownames(dat_sim) <- NULL
  return(dat_sim)
}
# Generate data:
set.seed(300417)
dat_poiss <- sim_poiss()
dat_poiss_train <- head(dat_poiss, N)
dat_poiss_test <- tail(dat_poiss, N)

Reference model

Next, we fit the reference model that we consider as the best model (in terms of predictive performance) that we can construct (here, we assume that we don’t know about the true data-generating process even though the dataset was simulated):

library(rstanarm)

# Number of regression coefficients:
( D <- sum(grepl("^x", names(dat_poiss_train))) )

[1] 50

# Prior guess for the number of relevant (i.e., non-zero) regression
# coefficients:
p0 <- 10
# Prior guess for the overall magnitude of the response values, see Table 1 of
# Piironen and Vehtari (2017, DOI: 10.1214/17-EJS1337SI):
mu_prior <- 100
# Hyperprior scale for tau, the global shrinkage parameter:
tau0 <- p0 / (D - p0) / sqrt(mu_prior) / sqrt(N)
# 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)
refm_fml <- as.formula(paste("y", "~", paste(
  grep("^x", names(dat_poiss_train), value = TRUE),
  collapse = " + "
)))
refm_fit_poiss <- stan_glm(
  formula = refm_fml,
  family = poisson(),
  data = dat_poiss_train,
  prior = hs(global_scale = tau0, slab_df = 100, slab_scale = 1),
  ### Only for the sake of speed (not recommended in general):
  chains = 2, iter = 1000,
  ###
  refresh = 0
)

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

Due to the technical reasons for which we reduced chains and iter in this vignette, we ignore the bulk-ESS warning here.

Variable selection using the latent projection

Within projpred, we define the reference model object explicitly and set latent = TRUE in the corresponding get_refmodel() call (see section “Implementation”) so that the latent projection is used in downstream functions. Since we have a hold-out test dataset available, we can use varsel() with argument d_test instead of cv_varsel(). Furthermore, we measure the runtime to be able to compare it to the traditional projection’s later:

library(projpred)

d_test_lat_poiss <- list(
  data = dat_poiss_test,
  offset = rep(0, nrow(dat_poiss_test)),
  weights = rep(1, nrow(dat_poiss_test)),
  ### Here, we are not interested in latent-scale post-processing, so we can set
  ### element `y` to a vector of `NA`s:
  y = rep(NA, nrow(dat_poiss_test)),
  ###
  y_oscale = dat_poiss_test$y
)
refm_poiss <- get_refmodel(refm_fit_poiss, latent = TRUE)

Since `<refmodel>$dis` will consist of only `NA`s, downstream analyses based on this reference model object won't be able to use log predictive density (LPD) values on latent scale. Furthermore, proj_predict() won't be able to draw from the latent Gaussian distribution.

time_lat <- system.time(vs_lat <- varsel(
  refm_poiss,
  d_test = d_test_lat_poiss,
  ### Only for demonstrating an issue with the traditional projection in the
  ### next step (not recommended in general):
  method = "L1",
  ###
  ### Only for the sake of speed (not recommended in general):
  nclusters_pred = 20,
  ###
  nterms_max = 14,
  ### In interactive use, we recommend not to deactivate the verbose mode:
  verbose = 0,
  ###
  ### For comparability with varsel() based on the traditional projection:
  seed = 95930
  ###
))

print(time_lat)

   user  system elapsed 
  1.912   0.147   2.059 

The message telling that <refmodel>$dis consists of only NAs will not concern us here because we will only focus on response-scale post-processing.

In order to decide for a submodel size, we first inspect the plot() results. In contrast to the main vignette where we used the mean log predictive density (MLPD) as predictive performance statistic for a gaussian() family reference model (and gaussian() submodels), we have a discrete family (poisson()) here, so it makes sense to exponentiate the MLPD to obtain the geometric mean predictive density (GMPD; in case of a discrete response, the predictive density values are actually predictive probabilities and hence the GMPD is bounded by 0 and 1). As in the main vignette, we plot with deltas = TRUE (in case of the GMPD, this means that the ratio of the submodel GMPD vs. the reference model GMPD is shown). Via global option projpred.plot_vsel_size_position, we set argument size_position of plot.vsel() to "secondary_x" to make the submodel sizes readable in all of the plots in this vignette.

options(projpred.plot_vsel_size_position = "secondary_x")
( gg_lat <- plot(vs_lat, stats = "gmpd", deltas = TRUE) )

Based on this plot, we decide for a submodel size of 11:

size_decided_lat <- 11

This is also the size that suggest_size() would suggest:

suggest_size(vs_lat, stat = "gmpd")

[1] 11

In the predictor ranking up to the selected size of 11, we can see that projpred has correctly selected the truly relevant predictors first and only then the noise predictors:

rk_lat <- ranking(vs_lat)
( predictors_final_lat <- head(rk_lat[["fulldata"]], size_decided_lat) )

 [1] "x.4"  "x.6"  "x.10" "x.3"  "x.1"  "x.2"  "x.8"  "x.7"  "xcat" "x.5" 
[11] "x.9" 

We will skip post-selection inference here (see the main vignette for a demonstration of post-selection inference), but note that proj_predict() has argument resp_oscale for controlling whether to draw from the posterior-projection predictive distributions on the original response scale (TRUE, the default) or on latent scale (FALSE) and that analogous functionality is available in proj_linpred() (argument transform) and predict.refmodel() (argument type).

Variable selection using the traditional projection

We will now look at what projpred’s traditional projection would have given. For this, we increase nterms_max because this will reveal an issue with this approach:

d_test_trad_poiss <- d_test_lat_poiss
d_test_trad_poiss$y <- d_test_trad_poiss$y_oscale
d_test_trad_poiss$y_oscale <- NULL
time_trad <- system.time(vs_trad <- varsel(
  refm_fit_poiss,
  d_test = d_test_trad_poiss,
  ### Only for demonstrating an issue with the traditional projection (not
  ### recommended in general):
  method = "L1",
  ###
  ### Only for the sake of speed (not recommended in general):
  nclusters_pred = 20,
  ###
  nterms_max = 30,
  ### In interactive use, we recommend not to deactivate the verbose mode:
  verbose = 0,
  ###
  ### For comparability with varsel() based on the latent projection:
  seed = 95930
  ###
))

print(time_trad)

   user  system elapsed 
  3.503   0.000   3.505 

( gg_trad <- plot(vs_trad, stats = "gmpd", deltas = TRUE) )

As these results show, the traditional projection takes longer than the latent projection, although the difference is rather small on absolute scale (which is due to the fact that the L1 search is already quite fast). More importantly however, the predictor ranking contains several noise terms before truly relevant ones, causing the predictive performance of the reference model not to be reached before submodel size 28.

Conclusion

This example showed that the latent projection can be advantageous also for families supported by projpred’s traditional projection by improving the runtime as well as the results of the variable selection.

An important point is that we have used L1 search here. In case of the latent projection, a forward search would have given only slightly different results. However, in case of the traditional projection, a forward search would have given markedly better results (in particular, all of the noise terms would have been selected after the truly relevant ones). Thus, the conclusions made here for L1 search cannot be transmitted easily to forward search.

Data

We will re-use the data generated above in the Poisson example.

Reference model

We now fit a reference model with the negative binomial distribution as response family. For the sake of simplicity, we won’t adjust tau0 to this new family, but in a real-world example, such an adjustment would be necessary. However, since Table 1 of Piironen and Vehtari (2017) does not list the negative binomial distribution, this would first require a manual derivation of the pseudo-variance \(\tilde{\sigma}^2\).

refm_fit_nebin <- stan_glm(
  formula = refm_fml,
  family = neg_binomial_2(),
  data = dat_poiss_train,
  prior = hs(global_scale = tau0, slab_df = 100, slab_scale = 1),
  ### Only for the sake of speed (not recommended in general):
  chains = 2, iter = 1000,
  ###
  refresh = 0
)

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

Again, we ignore the bulk-ESS warning due to the technical reasons for which we reduced chains and iter in this vignette.

Variable selection using the latent projection

To request the latent projection with latent = TRUE, we now need to specify more arguments (latent_ll_oscale and latent_ppd_oscale; the internal default for latent_ilink works correctly in this example) which will be passed to extend_family()⁷:

refm_prec <- as.matrix(refm_fit_nebin)[, "reciprocal_dispersion", drop = FALSE]
latent_ll_oscale_nebin <- function(ilpreds,
                                   dis = rep(NA, nrow(ilpreds)),
                                   y_oscale,
                                   wobs = rep(1, length(y_oscale)),
                                   cl_ref,
                                   wdraws_ref = rep(1, length(cl_ref))) {
  y_oscale_mat <- matrix(y_oscale, nrow = nrow(ilpreds), ncol = ncol(ilpreds),
                         byrow = TRUE)
  wobs_mat <- matrix(wobs, nrow = nrow(ilpreds), ncol = ncol(ilpreds),
                     byrow = TRUE)
  refm_prec_agg <- cl_agg(refm_prec, cl = cl_ref, wdraws = wdraws_ref)
  ll_unw <- dnbinom(y_oscale_mat, size = refm_prec_agg, mu = ilpreds, log = TRUE)
  return(wobs_mat * ll_unw)
}
latent_ppd_oscale_nebin <- function(ilpreds_resamp,
                                    dis_resamp = rep(NA, nrow(ilpreds_resamp)),
                                    wobs,
                                    cl_ref,
                                    wdraws_ref = rep(1, length(cl_ref)),
                                    idxs_prjdraws) {
  refm_prec_agg <- cl_agg(refm_prec, cl = cl_ref, wdraws = wdraws_ref)
  refm_prec_agg_resamp <- refm_prec_agg[idxs_prjdraws, , drop = FALSE]
  ppd <- rnbinom(prod(dim(ilpreds_resamp)), size = refm_prec_agg_resamp,
                 mu = ilpreds_resamp)
  ppd <- matrix(ppd, nrow = nrow(ilpreds_resamp), ncol = ncol(ilpreds_resamp))
  return(ppd)
}
refm_nebin <- get_refmodel(refm_fit_nebin, latent = TRUE,
                           latent_ll_oscale = latent_ll_oscale_nebin,
                           latent_ppd_oscale = latent_ppd_oscale_nebin)

Defining `latent_ilink` as a function which calls `family$linkinv`, but there is no guarantee that this will work for all families. If relying on `family$linkinv` is not appropriate or if this raises an error in downstream functions, supply a custom `latent_ilink` function (which is also allowed to return only `NA`s if response-scale post-processing is not needed).

Since `<refmodel>$dis` will consist of only `NA`s, downstream analyses based on this reference model object won't be able to use log predictive density (LPD) values on latent scale. Furthermore, proj_predict() won't be able to draw from the latent Gaussian distribution.

vs_nebin <- varsel(
  refm_nebin,
  d_test = d_test_lat_poiss,
  ### Only for the sake of speed (not recommended in general):
  method = "L1",
  nclusters_pred = 20,
  ###
  nterms_max = 14,
  ### In interactive use, we recommend not to deactivate the verbose mode:
  verbose = 0
  ###
)

Again, the message telling that <refmodel>$dis consists of only NAs will not concern us here because we will only focus on response-scale post-processing. The message concerning latent_ilink can be safely ignored here (the internal default based on family$linkinv works correctly in this case).

Again, we first inspect the plot() results to decide for a submodel size:

( gg_nebin <- plot(vs_nebin, stats = "gmpd", deltas = TRUE) )

For the decision of the final submodel size, we act as if we preferred accuracy over sparsity in their trade-off mentioned in the main vignette, so we decide for a submodel size of 11:

size_decided_nebin <- 11

This is not the size that suggest_size() would suggest, but as mentioned in the main vignette and in the documentation, suggest_size() provides only a quite heuristic decision (so we stick with our manual decision here):

suggest_size(vs_nebin, stat = "gmpd")

[1] 10

As we can see from the predictor ranking included in the plot, our selected 11 predictor terms lack one truly relevant predictor (x.9) and include one noise term (xn.29). More explicitly, our selected predictor terms are:

rk_nebin <- ranking(vs_nebin)
( predictors_final_nebin <- head(rk_nebin[["fulldata"]],
                                 size_decided_nebin) )

 [1] "x.4"   "x.6"   "x.10"  "x.3"   "x.1"   "x.2"   "x.8"   "xn.29" "x.7"  
[10] "xcat"  "x.5"  

Again, we will skip post-selection inference here (see the main vignette for a demonstration of post-selection inference).

Conclusion

This example demonstrated how the latent projection can be used for those families which are neither supported by projpred’s traditional nor by projpred’s augmented-data projection, which reflects the flexibility of the latent approach.