Semi-structured multi-state delinquency model for mortgage default

Credit risk models increasingly need to combine predictive accuracy with transparent explanations and auditable fairness constraints. Logistic regression remains attractive because its coefficients are easy to interpret, but it can miss nonlinear structure. Flexible models can improve prediction, but their explanations are often post-hoc and may not describe the decision rule itself. We introduce $\texttt{findr}$, short for flexible, interpretable deep regression, a semi-structured framework for binary credit risk modelling that decomposes the logit into an interpretable structured component and an orthogonal neural residual. The orthogonalisation separates coefficient-based effects from residual nonlinear variation, while an in-processing Wasserstein penalty mitigates group disparities by comparing score distributions during training. The framework also includes diagnostics that measure the structured component's contribution to logit variation, decision agreement, and local directional consistency. We evaluate $\texttt{findr}$ in a simulation study and on eight public credit datasets using score-level accuracy-fairness frontiers. The results show that $\texttt{findr}$ behaves close to logistic regression when the signal is approximately linear, while recovering much of the predictive gain of neural models when nonlinear structure is relevant. The diagnostics identify when coefficient-based explanations remain close to the full fitted model and when residual variation must also be examined. These findings support semi-structured modelling as a practical way to make performance, fairness, and interpretability trade-offs explicit in credit risk decisions.

We develop a semi-parametric state-space model for time-series data with latent regime transitions. Classical Markov-switching models use fixed parametric transition functions, such as logistic or probit links, which restrict flexibility when transitions depend on nonlinear and context-dependent effects. We replace this assumption with learned functions $f_0, f_1 \in \calH$, where $\calH$ is either a reproducing kernel Hilbert space or a spline approximation space, and define transition probabilities as $p_{jk,t} = \sigmoid(f(\bx_{t-1}))$. The transition functions are estimated jointly with emission parameters using a generalized Expectation-Maximization algorithm. The E-step uses the standard forward-backward recursion, while the M-step reduces to a penalized regression problem with weights from smoothed occupation measures. We establish identifiability conditions and provide a consistency argument for the resulting estimators. Experiments on synthetic data show improved recovery of nonlinear transition dynamics compared to parametric baselines. An empirical study on financial time series demonstrates improved regime classification and earlier detection of transition events.

Accurate estimation of long-term care transition probabilities is central to disability insurance pricing, reserving, and solvency assessment. Classical actuarial multi-state models commonly rely on Markov, semi-Markov, or proportional-hazard specifications, which provide a direct connection to cohort projection but may be restrictive for irregular longitudinal health data with nonlinear aging patterns and heterogeneous covariate histories. This paper develops a well-calibrated estimator of multi-state transition probabilities for irregular longitudinal health data. The model learns from individual health history, incorporates the time elapsed between observations, and conditions transition probabilities on demographic and socioeconomic attributes. It produces a valid probability distribution over the next observed health state, with four possible states: healthy, mild disability, severe disability, and death. Individual probabilities are aggregated by age group and origin state to form transition matrices compatible with actuarial cohort projection. Using longitudinal data from the Health and Retirement Study, we compare the proposed estimator with logistic regression, gradient-boosted trees, a recurrent neural network, and a last-state persistence benchmark. The evaluation considers probabilistic accuracy, endpoint discrimination and calibration for severe disability and death, risk concentration, and transition matrix error after aggregation. The proposed estimator improves severe disability discrimination relative to logistic regression and gradient-boosted tree benchmarks, maintains strong calibration, and yields the lowest transition matrix error among the evaluated models in the held-out test analysis. Results show that a structured machine learning estimator can support long-term care transition modeling when judged by calibration and projection fidelity, beyond discrimination.