Multi-Parameter GAMs: Full Distributional Modeling

Updated 4 July 2026

Multi-parameter GAMs are regression models that extend conventional GAMs by modeling multiple parameters (e.g., location, scale, shape) of a response distribution.
They generalize mean regression to full distributional modeling, allowing covariates to influence variance, skewness, tail behavior, and dependence.
These models employ advanced penalization, smoothing, and estimation techniques—including Bayesian and empirical Bayes approaches—for robust inference.

Searching arXiv for the specified papers to ground the article in the cited literature. Multi-parameter Generalised Additive Models are regression models in which several parameters of a response distribution depend on covariates through additive predictors, rather than restricting modeling to a single mean parameter. In the literature summarized here, they appear under closely related labels including generalized additive models for location, scale and shape, distributional regression, multiple generalized additive models, and structured additive distributional regression. Their defining feature is parameter-wise regression: location, scale, shape, tail, dependence, or other distributional parameters can each have their own additive predictor, their own smooth terms, and their own regularization. This architecture generalizes ordinary GAMs from mean regression to full conditional distribution modeling and, in more recent work, to bivariate and truly multivariate dependence modeling, as well as to embedded covariate transformations learned jointly with the model (Marra et al., 2016).

1. Definition and scope

In the formulation emphasized for “multiple generalized additive models,” the response distribution depends on a vector of $D$ parameters,

$Y_i \sim F_i(y_i;\theta_i), \qquad \theta_i=(\theta_i^{(1)},\ldots,\theta_i^{(D)})\in\mathbb{R}^D,$

and each parameter may have its own additive predictor (El-Bachir et al., 2018). This is the central distinction from an ordinary mean-only GAM: explanatory variables can affect not only the mean but also variance, scale, skewness, tail behavior, and shape parameters (El-Bachir et al., 2018).

A general multi-parameter formulation given in the recent covariate-transformation framework writes

$g_j(\theta_{ji}) = \eta_{ji} = \left(\mathbf Z^0_{ji}\right)\boldsymbol\gamma_{j0} + \sum_{k=1}^{K_j} f_{jk}\!\left(\mathbf x_i^{S_{jk}}\right) + \sum_{u=1}^{U_j} s_{ju}\!\left(\tilde s_{ju}\!\left(\mathbf x_i^{\tilde S_{ju}}\right)\right), \qquad j=1,\dots,m,$

so that each of the $m$ distributional parameters has its own additive predictor, and those predictors may include both standard smooth terms and nested transformed-covariate terms (Collarin et al., 24 Nov 2025). If the nested terms are removed, this becomes a standard multi-parameter GAM (Collarin et al., 24 Nov 2025).

Within this literature, the same underlying idea is described in several ways. “GAMLSS” emphasizes location, scale, and shape parameters for arbitrary univariate distributions (Marra et al., 2016). “Multiple GAMs” emphasizes several parameters of a probability distribution depending on predictors through smooth functions with $L_2$ regularization (El-Bachir et al., 2018). “Structured additive distributional regression” emphasizes additive predictor decompositions, basis-penalty representations, and Bayesian penalization for all parameters of a univariate or multivariate conditional distribution (Kock et al., 2023). The terminological variation is substantial, but the common principle is parameter-wise additive modeling.

A standard GAM is therefore a special case in which only one parameter is modeled, usually through a single additive predictor (Marra et al., 2016). This suggests that multi-parameter GAMs are best understood not as a separate modeling family, but as a strict extension of additive regression from conditional means to conditional distributions.

2. Predictor architecture and effect types

The generic additive predictor used in the bivariate copula model is

$\eta_i=\beta_0+\sum_{k=1}^K s_k(z_{ki}), \qquad i=1,\dots,n,$

with basis expansion

$s_k(z_{ki})=\sum_{j_k=1}^{J_k}\beta_{k j_k}\, b_{k j_k}(z_{ki}),$

and matrix form

$\eta = Z\beta$

(Marra et al., 2016). In the multi-parameter setting, this construction is replicated for each parameter-specific equation. For example, in the bivariate location-scale-shape copula model, separate predictors may be assigned to

$(\eta_{\mu_1},\eta_{\mu_2},\eta_{\sigma_1},\eta_{\sigma_2},\eta_{\nu_1},\eta_{\nu_2},\eta_\theta),$

all estimated jointly from a single penalized likelihood (Marra et al., 2016).

The multivariate structured additive framework gives the same architecture in a slightly different notation: $\eta_k(\mathbf{x})=\beta_{0k}+f_{1k}(\mathbf{x})+\dots+f_{S_k k}(\mathbf{x}),$ with each effect represented through a basis expansion

$Y_i \sim F_i(y_i;\theta_i), \qquad \theta_i=(\theta_i^{(1)},\ldots,\theta_i^{(D)})\in\mathbb{R}^D,$ 0

(Kock et al., 2023).

Across the cited work, the permitted effect classes are broad. The bivariate copula model explicitly allows ordinary linear terms, nonlinear smooth functions of continuous covariates, random effects, spatial effects, varying coefficient terms, and multivariate smooths (Marra et al., 2016). For spatial effects over discrete regions, Markov random field smoothers are used, with adjacency-based penalties equivalent to a Gaussian Markov random field prior in stochastic terms (Marra et al., 2016). The multivariate distributional-regression framework similarly permits constant terms with flat priors, linear effects with ridge-type priors, nonlinear effects of continuous covariates using cubic B-splines with second-order difference penalties, random effects with iid priors, and discrete spatial effects using adjacency-based design matrices and Markov random field priors (Kock et al., 2023).

Link functions are used to enforce parameter-space constraints. In the bivariate copula model, positive parameters are modeled through logarithmic links such as

$Y_i \sim F_i(y_i;\theta_i), \qquad \theta_i=(\theta_i^{(1)},\ldots,\theta_i^{(D)})\in\mathbb{R}^D,$ 1

while the copula dependence parameter uses transforms adapted to its support, including $Y_i \sim F_i(y_i;\theta_i), \qquad \theta_i=(\theta_i^{(1)},\ldots,\theta_i^{(D)})\in\mathbb{R}^D,$ 2 for the Gumbel copula and $Y_i \sim F_i(y_i;\theta_i), \qquad \theta_i=(\theta_i^{(1)},\ldots,\theta_i^{(D)})\in\mathbb{R}^D,$ 3 for copulae with $Y_i \sim F_i(y_i;\theta_i), \qquad \theta_i=(\theta_i^{(1)},\ldots,\theta_i^{(D)})\in\mathbb{R}^D,$ 4 such as Gaussian or AMH (Marra et al., 2016). In the multiple-GAM framework, the same principle appears through transformed parameterizations such as $Y_i \sim F_i(y_i;\theta_i), \qquad \theta_i=(\theta_i^{(1)},\ldots,\theta_i^{(D)})\in\mathbb{R}^D,$ 5 in generalized extreme-value regression (El-Bachir et al., 2018).

A recurring theme is that identifiability constraints are integral to the predictor construction. Smooth terms in the bivariate copula model are centered using Wood’s parsimonious approach (Marra et al., 2016). In the nested-transformation framework, the outer smooth is constrained by

$Y_i \sim F_i(y_i;\theta_i), \qquad \theta_i=(\theta_i^{(1)},\ldots,\theta_i^{(D)})\in\mathbb{R}^D,$ 6

and boundary conditions are imposed to ensure sufficient smoothness at the edges of the fixed support interval (Collarin et al., 24 Nov 2025). In the multivariate Bayesian framework, identifiability is enforced through constraints of the form

$Y_i \sim F_i(y_i;\theta_i), \qquad \theta_i=(\theta_i^{(1)},\ldots,\theta_i^{(D)})\in\mathbb{R}^D,$ 7

(Kock et al., 2023).

3. Penalization, smoothing, and estimation

Quadratic penalization is fundamental to multi-parameter GAMs. In the multiple-GAM formulation, smoothness is controlled by

$Y_i \sim F_i(y_i;\theta_i), \qquad \theta_i=(\theta_i^{(1)},\ldots,\theta_i^{(D)})\in\mathbb{R}^D,$ 8

with a block-diagonal smoothing matrix because different smooth terms and different distributional parameters have different smoothing parameters (El-Bachir et al., 2018). For fixed $Y_i \sim F_i(y_i;\theta_i), \qquad \theta_i=(\theta_i^{(1)},\ldots,\theta_i^{(D)})\in\mathbb{R}^D,$ 9, coefficients are estimated by maximizing the penalized log-likelihood

(El-Bachir et al., 2018).

The bivariate copula additive model uses an analogous penalized likelihood,

where penalization is assembled blockwise over all parameter-specific predictors (Marra et al., 2016). Estimation is carried out by penalized maximum likelihood using a trust region algorithm with integrated automatic multiple smoothing parameter selection (Marra et al., 2016). The preference for trust region optimization is explicitly motivated by the fact that the objective can be non-concave and may contain flat regions; trust region optimization is more stable than line-search Newton methods in such settings (Marra et al., 2016).

The paper on “Fast Automatic Smoothing for Generalized Additive Models” develops an empirical Bayes approach for automatically learning the optimal degree of $g_j(\theta_{ji}) = \eta_{ji} = \left(\mathbf Z^0_{ji}\right)\boldsymbol\gamma_{j0} + \sum_{k=1}^{K_j} f_{jk}\!\left(\mathbf x_i^{S_{jk}}\right) + \sum_{u=1}^{U_j} s_{ju}\!\left(\tilde s_{ju}\!\left(\mathbf x_i^{\tilde S_{ju}}\right)\right), \qquad j=1,\dots,m,$ 2 regularization for multiple GAMs by maximization of a marginal likelihood through an approximate expectation-maximization algorithm that involves a double Laplace approximation at the E-step and leads to an efficient M-step (El-Bachir et al., 2018). Its key closed-form smoothing update is

and the paper emphasizes three claimed improvements: speed, stability, and accuracy (El-Bachir et al., 2018).

The covariate-transformation framework instead estimates regression coefficients and transformation parameters jointly by maximizing a posterior log-density

using Newton’s method (Collarin et al., 24 Nov 2025). Smoothing parameters are selected in an empirical Bayes framework by maximizing a Laplace approximate marginal likelihood,

(Collarin et al., 24 Nov 2025). The practical optimization strategy is BFGS on the LAML, nested around Newton updates for $g_j(\theta_{ji}) = \eta_{ji} = \left(\mathbf Z^0_{ji}\right)\boldsymbol\gamma_{j0} + \sum_{k=1}^{K_j} f_{jk}\!\left(\mathbf x_i^{S_{jk}}\right) + \sum_{u=1}^{U_j} s_{ju}\!\left(\tilde s_{ju}\!\left(\mathbf x_i^{\tilde S_{ju}}\right)\right), \qquad j=1,\dots,m,$ 6 (Collarin et al., 24 Nov 2025).

The cited literature therefore presents several estimation paradigms within the same broader class: penalized maximum likelihood with trust-region optimization and integrated smoothing selection (Marra et al., 2016), empirical Bayes smoothing via approximate EM and double Laplace approximation (El-Bachir et al., 2018), joint MAP estimation with LAML-based smoothing selection (Collarin et al., 24 Nov 2025), and Bayesian MCMC with shrinkage priors in multivariate settings (Kock et al., 2023). A plausible implication is that the class “multi-parameter GAM” is methodologically unified at the modeling level but not tied to a single inferential technology.

4. Distributional regression beyond the univariate case

The move from ordinary GAMs to multi-parameter GAMs is already a move from mean regression to distributional regression. The bivariate copula additive model makes this explicit by extending the scope of GAMLSS from one response to a bivariate response with continuous margins, using a copula so that the two margins and the dependence parameter can all be modeled simultaneously and each parameter can itself be a structured additive function of covariates (Marra et al., 2016).

Its conditional joint CDF is constructed through Sklar’s theorem as

where $g_j(\theta_{ji}) = \eta_{ji} = \left(\mathbf Z^0_{ji}\right)\boldsymbol\gamma_{j0} + \sum_{k=1}^{K_j} f_{jk}\!\left(\mathbf x_i^{S_{jk}}\right) + \sum_{u=1}^{U_j} s_{ju}\!\left(\tilde s_{ju}\!\left(\mathbf x_i^{\tilde S_{ju}}\right)\right), \qquad j=1,\dots,m,$ 8 and $g_j(\theta_{ji}) = \eta_{ji} = \left(\mathbf Z^0_{ji}\right)\boldsymbol\gamma_{j0} + \sum_{k=1}^{K_j} f_{jk}\!\left(\mathbf x_i^{S_{jk}}\right) + \sum_{u=1}^{U_j} s_{ju}\!\left(\tilde s_{ju}\!\left(\mathbf x_i^{\tilde S_{ju}}\right)\right), \qquad j=1,\dots,m,$ 9 are continuous marginal CDFs and $m$ 0 is a bivariate copula with association parameter $m$ 1 (Marra et al., 2016). The joint density factorizes into a copula density times the two marginal densities, but the copula dependence and all marginal distribution parameters are estimated simultaneously, not in stages (Marra et al., 2016). This joint estimation is one of the paper’s important methodological points.

The multivariate extension goes further. “Truly Multivariate Structured Additive Distributional Regression” constructs a $m$ 2-dimensional joint model from arbitrary parametric marginals and a Gaussian copula, with additive predictors for all marginal and dependence parameters (Kock et al., 2023). For component $m$ 3,

$m$ 4

and the total number of distributional parameters in the joint model is

$m$ 5

(Kock et al., 2023). The dependence structure is modeled through a covariate-dependent Gaussian copula correlation matrix $m$ 6, parameterized via a modified Cholesky-type representation so that unconstrained lower-triangular entries can be linked directly to additive predictors (Kock et al., 2023).

This multivariate framework is intended to fill the gap between multivariate Gaussian GAMLSS-type models, which allow additive predictors but are distributionally restrictive, and copula regression approaches, which allow flexible margins but often have fixed dependence, low-dimensional restriction, or limited additive modeling (Kock et al., 2023). The model therefore preserves the central GAMLSS principle—distributional parameter regression via structured additive predictors—in a genuinely multivariate setting (Kock et al., 2023).

The copula constructions also clarify an important misconception. Multi-parameter GAMs are not limited to “location, scale and shape” in a narrow univariate sense. In the bivariate copula model, the dependence parameter is an additional distributional parameter governing association between $m$ 7 and $m$ 8 (Marra et al., 2016). In the truly multivariate model, the dependence structure contributes $m$ 9 parameters before accounting for covariate effects (Kock et al., 2023). This suggests that, in the broadest modern usage, a multi-parameter GAM is a model for the full conditional distribution, including dependence parameters when the response is multivariate.

5. Embedded covariate transformations and model-based feature engineering

A substantial recent extension embeds interpretable, parameterized covariate transformations directly inside multi-parameter GAM predictors and estimates them jointly with regression coefficients and smoothing parameters (Collarin et al., 24 Nov 2025). The key extension is the inclusion of nested smooth effects

$L_2$ 0

where $L_2$ 1 is an inner transformation and $L_2$ 2 is an outer smooth (Collarin et al., 24 Nov 2025).

The inner transformation must be scalar-valued and sufficiently differentiable; in the paper’s implementation it is fourth-order differentiable with respect to its own parameters (Collarin et al., 24 Nov 2025). The transformed scalar is not treated as a preprocessing feature. Instead, it appears inside the model as the argument of a learned smooth, and the transformation parameters are learned within the likelihood/penalty optimization, with joint uncertainty propagation (Collarin et al., 24 Nov 2025).

Three main transformation classes are developed. The first is adaptive exponential smoothing,

$L_2$ 3

which permits interpretable time-varying smoothing regimes (Collarin et al., 24 Nov 2025). The second is multivariate kernel smoothing,

$L_2$ 4

which yields interpretable local indices such as neighboring house-price levels (Collarin et al., 24 Nov 2025). The third is linear combinations or single-index effects,

$L_2$ 5

covering single-index effects and distributed lag summaries (Collarin et al., 24 Nov 2025).

A novel issue arises because the argument of the outer spline basis depends on unknown transformation parameters. The paper’s practical solution is a penalty-based scaling construction,

$L_2$ 6

with variance penalty

$L_2$ 7

(Collarin et al., 24 Nov 2025). In practice the paper uses $L_2$ 8 and $L_2$ 9 for the fixed outer-basis support construction (Collarin et al., 24 Nov 2025).

The framework is explicitly positioned as broader than conventional feature engineering because the transformation is inside the model, estimated jointly with all coefficients, penalized and uncertainty-quantified in the same Bayesian-GAM framework, and incorporated into smoothing parameter selection through the marginal likelihood (Collarin et al., 24 Nov 2025). This suggests a shift in the role of preprocessing: instead of fixing transformed covariates before model fitting, one can encode the transformation class as part of the distributional additive model itself.

6. Bayesian and scalable perspectives

Not all cited work addresses multi-parameter GAMs directly. “Scalable GAM using sparse variational Gaussian processes” develops a scalable Bayesian generalized additive model by representing each additive component with a Gaussian process and performing sparse variational inference with a structured, non-mean-field Gaussian posterior over inducing variables (Adam et al., 2018). In the paper’s own framing, this is not a new multi-parameter GAM in the distributional-regression sense, but a scalable inference scheme for additive predictors built from multiple latent GPs (Adam et al., 2018).

Its likelihood depends on a single additive predictor

$\eta_i=\beta_0+\sum_{k=1}^K s_k(z_{ki}), \qquad i=1,\dots,n,$ 0

and the framework treats only one additive predictor or one latent scalar quantity entering the likelihood (Adam et al., 2018). It is therefore a single-parameter GAM framework in the narrow distributional-regression sense (Adam et al., 2018). Nevertheless, the paper’s relevance to multi-parameter GAMs lies in its inference architecture. The prior structure gives each additive term an independent GP prior, while posterior dependence between components is captured through a structured precision parameterization

$\eta_i=\beta_0+\sum_{k=1}^K s_k(z_{ki}), \qquad i=1,\dots,n,$ 1

(Adam et al., 2018).

The paper’s core reusable idea is “Gaussian a posteriori coupling between the components”: allowing the variational posterior over inducing variables to have nonzero cross-covariance even though the prior is independent across components (Adam et al., 2018). The proposed method keeps storage and KL cost at the mean-field level while paying an extra factor of $\eta_i=\beta_0+\sum_{k=1}^K s_k(z_{ki}), \qquad i=1,\dots,n,$ 2 in the expected log-likelihood term to retain posterior coupling (Adam et al., 2018). For readers interested in multi-parameter GAMs, the paper is best understood as an inference template rather than a complete solution (Adam et al., 2018).

The multivariate structured additive distributional-regression paper adopts a fully Bayesian route, using Gaussian smoothing priors with inverse-gamma hyperpriors on smoothing variances,

$\eta_i=\beta_0+\sum_{k=1}^K s_k(z_{ki}), \qquad i=1,\dots,n,$ 3

and a blockwise Gibbs-within-Metropolis–Hastings MCMC algorithm with IWLS proposals for coefficient blocks (Kock et al., 2023). The authors emphasize that the model is highly parameterized and that shrinkage priors are what make the model estimable in practice (Kock et al., 2023).

Taken together, these papers show two distinct but complementary Bayesian directions in the field: scalable approximate inference for additive latent-function models (Adam et al., 2018), and high-dimensional posterior sampling with shrinkage priors for multivariate distributional additive models (Kock et al., 2023). A plausible implication is that future multi-parameter GAM methodology will continue to depend heavily on inference innovations, not only on new predictor structures.

7. Applications, empirical findings, and practical limitations

The empirical literature summarized here shows why parameter-wise additive modeling matters. In the bivariate copula case, an electricity application allows both margins and the copula dependence to vary with time and raw material prices, showing that not only means but also variances and association can vary over covariates; the estimated dependence fluctuates over time, demonstrating that a constant-dependence model would be inadequate (Marra et al., 2016). In the North Carolina birth example, the model jointly analyzes birth weight and gestational age with county-level spatial effects in the margins and the copula parameter; joint probabilities of low birth weight and premature delivery were underestimated under an independence model (Marra et al., 2016).

The multiple-GAM smoothing paper uses the generalized extreme-value model as a flagship example. It supports Gaussian mean and standard deviation, Poisson rate, Exponential rate, Gamma shape and scale, Binomial probability, and generalized extreme-value location, scale, and shape (El-Bachir et al., 2018). Its simulation study generates $\eta_i=\beta_0+\sum_{k=1}^K s_k(z_{ki}), \qquad i=1,\dots,n,$ 4 observations across six models and reports that multgam is the only method supporting all the classical models considered in the study; for the GEV model, multgam failed on 17/100 replicates, whereas mgcv::gam failed on 46/100 replicates (El-Bachir et al., 2018). In the application to monthly maxima of daily Central England Temperature from 1772 to 2016, the shape parameter shows significant seasonal variation with narrow confidence intervals, and remains negative throughout the year (El-Bachir et al., 2018).

The embedded-transformation framework presents two large applied examples. For Great Britain electricity net-demand forecasting, the proposed nested model attains the best CRPS, RMSE, MAE, and AIC among the compared models, with the table reporting CRPS $\eta_i=\beta_0+\sum_{k=1}^K s_k(z_{ki}), \qquad i=1,\dots,n,$ 5, RMSE $\eta_i=\beta_0+\sum_{k=1}^K s_k(z_{ki}), \qquad i=1,\dots,n,$ 6, MAE $\eta_i=\beta_0+\sum_{k=1}^K s_k(z_{ki}), \qquad i=1,\dots,n,$ 7, and AIC $\eta_i=\beta_0+\sum_{k=1}^K s_k(z_{ki}), \qquad i=1,\dots,n,$ 8 for the proposed model (Collarin et al., 24 Nov 2025). For London house-price modeling, the nested model with 400 spatial basis functions outperforms standard models with up to 2000 basis functions; the nested model reports Log-score $\eta_i=\beta_0+\sum_{k=1}^K s_k(z_{ki}), \qquad i=1,\dots,n,$ 9, CRPS $s_k(z_{ki})=\sum_{j_k=1}^{J_k}\beta_{k j_k}\, b_{k j_k}(z_{ki}),$ 0, RMSE $s_k(z_{ki})=\sum_{j_k=1}^{J_k}\beta_{k j_k}\, b_{k j_k}(z_{ki}),$ 1, MAE $s_k(z_{ki})=\sum_{j_k=1}^{J_k}\beta_{k j_k}\, b_{k j_k}(z_{ki}),$ 2, and AIC $s_k(z_{ki})=\sum_{j_k=1}^{J_k}\beta_{k j_k}\, b_{k j_k}(z_{ki}),$ 3, and the paper states that it was about 18 times faster than the standard model with 2000 basis functions (Collarin et al., 24 Nov 2025).

The truly multivariate paper contributes two further applications. In childhood malnutrition data from Nigeria, the model jointly analyzes stunting, wasting, and underweight, and the Student $s_k(z_{ki})=\sum_{j_k=1}^{J_k}\beta_{k j_k}\, b_{k j_k}(z_{ki}),$ 4 margin specification fits better than Gaussian margins because the Gaussian margins showed deviations from normality, especially in the right tail (Kock et al., 2023). In traffic detection data from Berlin, the model combines negative binomial margins for counts and Student $s_k(z_{ki})=\sum_{j_k=1}^{J_k}\beta_{k j_k}\, b_{k j_k}(z_{ki}),$ 5 margins for log-speeds in a 4-dimensional mixed discrete-continuous response, demonstrating a setting that cannot be naturally handled by multivariate Gaussian GAMLSS (Kock et al., 2023).

Several limitations are repeatedly emphasized. The bivariate copula framework is for continuous margins and currently focuses on one-parameter copulas; extension to two-parameter and non-exchangeable copulas is possible in principle but may create identifiability and information-content issues when combined with flexible additive predictors (Marra et al., 2016). The nested-transformation framework requires scalar-valued and sufficiently differentiable transformations, specially chosen bases for nested outer smooths, and does not fully propagate smoothing parameter uncertainty (Collarin et al., 24 Nov 2025). The multiple-GAM smoothing method is empirical Bayes and requires third derivatives of the log-likelihood (El-Bachir et al., 2018). The truly multivariate copula model is mathematically valid for arbitrary $s_k(z_{ki})=\sum_{j_k=1}^{J_k}\beta_{k j_k}\, b_{k j_k}(z_{ki}),$ 6, but the proposed MCMC algorithm does not scale well to very large $s_k(z_{ki})=\sum_{j_k=1}^{J_k}\beta_{k j_k}\, b_{k j_k}(z_{ki}),$ 7, and the Gaussian copula may be restrictive if asymmetric tail dependence matters (Kock et al., 2023).

These limitations help clarify a second common misconception: multi-parameter GAMs are not merely “more flexible GAMs.” They are distributional models whose flexibility comes with nontrivial demands in identifiability, smoothing selection, constrained parameterization, derivative computation, and computational scalability (Marra et al., 2016). The literature suggests that their practical success depends as much on stable estimation and regularization strategies as on predictor specification itself.