Latent projection predictive feature selection

General idea

The response families used in GLMs (McCullagh and Nelder 1989, chap. 2) (and in GLMMs, GAMs, and GAMMs) 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 analytically². 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 predictors³ 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, the new 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, for the latent projection, we recommend to define the reference model object of class refmodel explicitly (as illustrated in the examples below) to avoid repetitions⁴.

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 calculation of the performance statistics in summary.vsel()) can be performed on the original response scale. For this, extend_family() has gained several new arguments accepting R functions responsible for the inverse-link transformation from latent scale to response scale (latent_ilink), for the calculation of log-likelihood values on response scale (latent_ll_oscale), and for drawing from the (posterior-projection) predictive distribution on response scale (latent_ppd_oscale). For some families, these 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 to these arguments. When creating a reference model object for a family of the latter category (i.e., lacking full response-scale support by default), projpred will throw messages stating whether (and which) features will be unavailable unless at least some of these 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 the new argument resp_oscale to FALSE, with the exception of predict.refmodel() and proj_linpred() where the existing arguments type and transform serve this purpose (see the documentation).

Apart from the arguments mentioned above, extend_family() has also gained a new 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 space⁵, the augmented-data projection (Weber, Glass, and Vehtari 2023) gives the exact⁶ 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 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))) )
# 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,
  ###
  QR = TRUE, refresh = 0
)

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)
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 = FALSE,
  ###
  ### For comparability with varsel() based on the traditional projection:
  seed = 95930
  ###
))

print(time_lat)

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

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

( gg_lat <- plot(vs_lat, stats = "mlpd", deltas = TRUE) )

Although the submodels’ MLPDs seem to be very close to the reference model’s MLPD from a submodel size of 6 on, a zoomed plot reveals that there is still some discrepancy at sizes 6 to 11 and that size 12 would be a better choice (further down below in the summary() output, we will also see that on absolute scale, the discrepancy at sizes 6 to 11 is not negligible):

gg_lat + ggplot2::coord_cartesian(ylim = c(-10, 0.05))

Thus, we decide for a submodel size of 12:

size_decided_lat <- 12

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

suggest_size(vs_lat, stat = "mlpd")

To obtain the results from the varsel() run in tabular form, we call summary.vsel():

smmry_lat <- summary(vs_lat, stats = "mlpd",
                     type = c("mean", "lower", "upper", "diff"))
print(smmry_lat, digits = 2)

On absolute scale (column mlpd), we see that submodel size leads to an MLPD of , i.e., a geometric mean predictive density (GMPD; due to the discrete response family, the "density" values are probabilities here, so we will report the GMPD as percentage) of which is ca. whereas size leads to a GMPD of ca. . This is a considerable improvement from size to size , so another justification for size . (Size would have resulted in about the same GMPD as size .)

In the predictor ranking up to the selected size of , we can see that apart from the noise term xn.6, projpred has correctly selected the truly relevant predictors first and only then the noise predictors. We can see this more clearly using the following code:

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

We will skip post-selection inference here (see the main vignette for a demonstration of post-selection inference), but note that proj_predict() has gained a new argument resp_oscale and that analogous response-scale 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:

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 = 14,
  ### In interactive use, we recommend not to deactivate the verbose mode:
  verbose = FALSE,
  ###
  ### For comparability with varsel() based on the latent projection:
  seed = 95930
  ###
))

print(time_trad)
( gg_trad <- plot(vs_trad, stats = "mlpd", deltas = TRUE) )
smmry_trad <- summary(vs_trad, stats = "mlpd",
                      type = c("mean", "lower", "upper", "diff"))
print(smmry_trad, digits = 2)

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 predictive performance plot is much more unstable and the predictor ranking contains several noise terms before truly relevant ones.

Conclusion

In conclusion, this example showed that the latent projection can be advantageous also for families supported by projpred’s traditional projection by improving the runtime and the stability of the variable selection, eventually also leading to better variable selection results.

An important point is that we have used the L1 search here. In case of the latent projection, a forward search would have given only slightly different results (in particular, a slightly smoother predictive performance plot). However, in case of the traditional projection, a forward search would have given markedly better results (in particular, the predictive performance plot would have been much smoother and all of the noise terms would have been selected after the truly relevant ones). Thus, the conclusions made for the L1 search here cannot be transmitted easily to the 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) doesn’t list the negative binomial distribution, this would first require a manual derivation of the pseudo-variance σ̃².

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,
  ###
  QR = TRUE, refresh = 0
)

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) which will be passed to extend_family()⁸:

refm_prec <- as.matrix(refm_fit_nebin)[, "reciprocal_dispersion", drop = FALSE]
latent_ll_oscale_nebin <- function(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, 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)
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 = FALSE
  ###
)

Again, the message telling that <refmodel>$dis consists of only NAs will not concern us here because we will only be focusing on response-scale post-processing. The message concerning latent_ilink can be safely ignored here (the internal default based on family$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 = "mlpd", deltas = TRUE) )

Again, a zoomed plot is more helpful:

gg_nebin + ggplot2::coord_cartesian(ylim = c(-2.5, 0.25))

Although the submodels’ MLPDs approach the reference model’s already from a submodel size of 9 on (taking into account the uncertainty bars), the curve levels off only from a submodel size of 11 on. For a more informed decision, the GMPD could be taken into account again, but for the sake of brevity, we will simply decide for a submodel size of 11 here:

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 = "mlpd")

Again, we call summary.vsel() to obtain the results from the varsel() run in tabular form:

smmry_nebin <- summary(vs_nebin, stats = "mlpd",
                       type = c("mean", "lower", "upper", "diff"))
print(smmry_nebin, digits = 2)

As we can see from the predictor ranking included in the plot and in the summary table, our selected 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) )

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. (In the future, this vignette will be extended to demonstrate how both, latent and augmented-data projection, can be applied to those non-ED families which are supported by both.)