FLARE: Analytical uncertainty quantification and linear-nonlinear decomposition for interpretable clinical risk prediction

In clinical risk prediction, accurate estimation of predictive uncertainty and transparent interpretation of model behaviour are as important as discrimination performance. While logistic regression supports statistical inference, its ability to capture nonlinear relationships is limited. Kernel-based, tree-based, and deep learning methods can accommodate nonlinear effects but generally do not provide closed-form uncertainty quantification, parametric inference, or analytical decomposition of variable effects into linear and nonlinear components. Moreover, these methods fall short in constructing uncertainty profiles at the individual patient and risk group levels. Fourier–Laplace Analytical Risk Estimation (FLARE) is presented as a unified framework for nonlinear clinical risk modelling and analytical uncertainty quantification. FLARE combines Orthogonal Random Fourier Features with a Laplace approximation to estimate uncertainty arising from model parameters, kernel hyperparameters, and their interaction. The framework provides (i) closed-form prediction uncertainty at the patient level, (ii) Wald tests, confidence intervals, and odds ratios through Jacobian-based projection, (iii) decomposition of variable effects into linear and nonlinear components via path-dependent gradient analysis, and (iv) analytical Hessian-based patient-specific interaction analysis within a single model. FLARE was evaluated on four real-world clinical datasets and two synthetic datasets. Under linear data-generating conditions, FLARE produced results functionally equivalent to logistic regression, while yielding improved predictive performance in nonlinear settings. Analytical confidence intervals achieved coverage close to nominal levels, and attribution patterns showed strong agreement with SHAP-based explanations. In the FLChain cohort, interactions undetectable at the population level were found to be substantially amplified in specific high-risk patients, with certain covariate pairs exhibiting markedly stronger effects locally than at the population mean. FLARE integrates the inferential transparency of classical statistical models with the representational power of nonlinear methods within a single analytical framework. The method provides closed-form uncertainty estimates, interpretable linear–nonlinear effect decomposition, and patient-level interaction assessment, facilitating a more comprehensive understanding of model behaviour in clinical prediction settings.