Smoothing parameter and model selection for general smooth models

TL;DR

This paper presents a stable, convergent framework for estimating smoothing parameters in general smooth models, allowing for uncertainty quantification and broad applicability across various statistical models.

Contribution

It introduces a unified, numerically stable method for smoothing parameter estimation using penalized likelihood and Laplace approximation, applicable to diverse models including GAMs and Cox models.

Findings

Method enables smoothing parameter uncertainty quantification.

Framework covers a wide range of models including non-exponential family responses.

Implementation simplifies the coding of new model classes.

Abstract

This paper discusses a general framework for smoothing parameter estimation for models with regular likelihoods constructed in terms of unknown smooth functions of covariates. Gaussian random effects and parametric terms may also be present. By construction the method is numerically stable and convergent, and enables smoothing parameter uncertainty to be quantified. The latter enables us to fix a well known problem with AIC for such models. The smooth functions are represented by reduced rank spline like smoothers, with associated quadratic penalties measuring function smoothness. Model estimation is by penalized likelihood maximization, where the smoothing parameters controlling the extent of penalization are estimated by Laplace approximate marginal likelihood. The methods cover, for example, generalized additive models for non-exponential family responses (for example beta, ordered categorical, scaled t distribution, negative binomial and Tweedie distributions), generalized additive models for location scale and shape (for example two stage zero inflation models, and Gaussian location-scale models), Cox proportional hazards models and multivariate additive models. The framework reduces the implementation of new model classes to the coding of some standard derivatives of the log likelihood.