Date: 2025-07-08

# Latent projection predictive feature selection

#### 2025-07-08

## Introduction

This vignette shows how to use the latent projection predictive feature selection from Catalina, Bürkner, and Vehtari (2021) in projpred. We recommend to read the main vignette first, as the latent-projection vignette presented here will skip some of the details explained in the main vignette.

### 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 `NA`s (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”).

## Example: Poisson distribution

In this example, we will illustrate that in case of a family supported by projpred’s traditional projection (here the `poisson()` family), the latent projection can improve runtime and results of the variable selection compared to projpred’s traditional projection, at least if the L1 search is used (see argument `method` of `varsel()` and `cv_varsel()`).

### 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 `NA`s 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.

## Example: Negative binomial distribution

In this example, we will illustrate the latent projection in case of the negative binomial family (more precisely, we will use the `rstanarm::neg_binomial_2()` family here) which is a family that is not supported by projpred’s traditional projection6.

### 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>$dis` consists of only `NA`s will not concern us here because we will only focus on response-scale post-processing. The message concerning `latent_ilink` can be safely ignored here (the internal default based on `family$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.

## References

Catalina, Alejandro, Paul Bürkner, and Aki Vehtari. 2021. “Latent Space Projection Predictive Inference.” arXiv. https://doi.org/10.48550/arXiv.2109.04702.
Cover, Thomas M., and Joy A. Thomas. 1991. Elements of Information Theory. New York, NY, USA: John Wiley & Sons, Ltd. https://doi.org/10.1002/0471200611.
Jørgensen, Bent. 1987. “Exponential Dispersion Models.” Journal of the Royal Statistical Society. Series B (Methodological) 49 (2): 127–62.
McCullagh, P., and J. A. Nelder. 1989. Generalized Linear Models. 2nd ed. London: Chapman & Hall.
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. 2017. “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.
Weber, Frank, Änne Glass, and Aki Vehtari. 2025. “Projection Predictive Variable Selection for Discrete Response Families with Finite Support.” Computational Statistics 40 (2): 701–21. https://doi.org/10.1007/s00180-024-01506-0.

1. Jørgensen (1987) himself only uses the term “exponential dispersion model”, but the discussion for that article mentions the term “ED [i.e., exponential dispersion] family”. Jørgensen (1987) also introduces the class of discrete exponential dispersion families (here abbreviated by “DED families”), see section “Example: Negative binomial distribution”.↩︎

2. The latent predictors are also known as the linear predictors, but “latent” is a more general term than “linear”.↩︎

3. If the `refmodel`-class object is not defined explicitly but implicitly by a call to a top-level function such as `project()`, `varsel()`, or `cv_varsel()`, then `latent = TRUE` and all other arguments related to the latent projection need to be set in each call to a top-level function.↩︎

4. More precisely, the latent projection replaces the KL divergence minimization problem in the original response space by a KL divergence minimization problem in the latent space and solves the latter.↩︎

5. Here, “exact” means apart from approximations and simplifications which are also undertaken for the traditional projection.↩︎

6. The negative binomial distribution belongs to the class of discrete exponential dispersion families (Jørgensen 1987) (here abbreviated by “DED families”). DED families are closely related to ED families (Jørgensen 1987), but strictly speaking, the class of DED families is not a subset of the class of ED families. GitHub issue #361 explains why the “traditional” projection onto a DED-family submodel is currently not implemented in projpred.↩︎

7. The suffix `_prec` in `refm_prec` stands for “precision” because here, we follow the Stan convention (see the Stan documentation for the `neg_binomial_2` distribution, the `brms::negbinomial()` documentation, and the brms vignette “Parameterization of Response Distributions in brms”) and prefer the term precision parameter for what is denoted by \(\phi\) there (confusingly, argument `size` in `?stats::NegBinomial`—which is the same as \(\phi\) from the Stan notation—is called the dispersion parameter there, although the variance is increased by its reciprocal).↩︎