Papers

The $p_1$ model plays a fundamental role in modeling directed networks, where the reciprocal effect parameter $\rho$ is of special interest in practice. However, due to nonlinear factors in this model, how to estimate $\rho$ efficiently is a long-standing open problem. We tackle the problem by the cycle count approach. The challenge is, due to the nonlinear factors in the model, for any given type of generalized cycles, the expected count is a complicated function of many parameters in the model, so it is unclear how to use cycle counts to estimate $\rho$. However, somewhat surprisingly, we discover that, among many types of generalized cycles with the same length, we can carefully pick a pair of them such that in the ratio between the expected cycle counts of the two types, the non-linear factors cancel out nicely with each other, and as a result, the ratio equals to $\mathrm{exp}(\rho)$ exactly. Therefore, though the expected count of cycles of any type is not tractable, the ratio between the expected cycle counts of a (carefully chosen) pair of generalized cycles may have an utterly simple form. We study to what extent such pairs exist, and use our discovery to derive both an estimate for $\rho$ and a testing procedure for testing $\rho = \rho_0$. In a setting where we allow a wide range of reciprocal effects and a wide variety of network sparsity and degree heterogeneity, we show that our estimator achieves the optimal rate and our test achieves the optimal phase transition. Technically, first, motivated by what we observe on real networks, we do not want to impose strong conditions on reciprocal effects, network sparsity, and degree heterogeneity. Second, our proposed statistic is a type of $U$-statistic, the analysis of which involves complex combinatorics and is error-prone. For these reasons, our analysis is long and delicate.

Handling missing data in time series is a complex problem due to the presence of temporal dependence. General-purpose imputation methods, while widely used, often distort key statistical properties of the data, such as variance and dependence structure, leading to biased estimation and misleading inference. These issues become more pronounced in models that explicitly rely on capturing serial dependence, as standard imputation techniques fail to preserve the underlying dynamics. This paper proposes a novel multiple imputation method specifically designed for parameter estimation in observation-driven models (ODM). The approach takes advantage of the iterative nature of the systematic component in ODM to propagate the dependence structure through missing data, minimizing its impact on estimation. Unlike traditional imputation techniques, the proposed method accommodates continuous, discrete, and mixed-type data while preserving key distributional and dependence properties. We evaluate its performance through Monte Carlo simulations in the context of GARMA models, considering time series with up to 70% missing data. An application to the proportion of stocked energy stored in South Brazil further demonstrates its practical utility.

This paper is motivated by evaluating the benefits of patients receiving mechanical circulatory support (MCS) devices in end-stage heart failure management inference, in which hypothesis testing for a treatment effect on the risk of recurrent events is challenged in the presence of terminal events. Existing methods based on cumulative frequency unreasonably disadvantage longer survivors as they tend to experience more recurrent events. The While-Alive-based (WA) test has provided a solution to address this survival-length-bias problem, and it performs well when the recurrent event rate holds constant over time. However, if such a constant-rate assumption is violated, the WA test can exhibit an inflated type I error and inaccurate estimation of treatment effects. To fill this methodological gap, we propose a Proportional Rate Marginal Structural Model-assisted Test (PR-MSMaT) in the causal inference framework of separable treatment effects for recurrent and terminal events. Using the simulation study, we demonstrate that our PR-MSMaT can properly control type I error while gaining power comparable to the WA test under time-varying recurrent event rates. We employ PR-MSMaT to compare different MCS devices with the postoperative risk of gastrointestinal bleeding among patients enrolled in the Interagency Registry of Mechanically Assisted Circulatory Support program.

Bayesian model selection commonly relies on Laplace approximation or the Bayesian Information Criterion (BIC), which assume that the effective model dimension equals the number of parameters. Singular learning theory replaces this assumption with the real log canonical threshold (RLCT), an effective dimension that can be strictly smaller in overparameterized or rank-deficient models. We study linear-Gaussian rank models and linear subspace (dictionary) models in which the exact marginal likelihood is available in closed form and the RLCT is analytically tractable. In this setting, we show theoretically and empirically that the error of Laplace/BIC grows linearly with (d/2 minus lambda) times log n, where d is the ambient parameter dimension and lambda is the RLCT. An RLCT-aware correction recovers the correct evidence slope and is invariant to overcomplete reparameterizations that represent the same data subspace. Our results provide a concrete finite-sample characterization of Laplace failure in singular models and demonstrate that evidence slopes can be used as a practical estimator of effective dimension in simple linear settings.

Clinical decision-making demands uncertainty quantification that provides both distribution-free coverage guarantees and risk-adaptive precision, requirements that existing methods fail to jointly satisfy. We present a hybrid Bayesian-conformal framework that addresses this fundamental limitation in healthcare predictions. Our approach integrates Bayesian hierarchical random forests with group-aware conformal calibration, using posterior uncertainties to weight conformity scores while maintaining rigorous coverage validity. Evaluated on 61,538 admissions across 3,793 U.S. hospitals and 4 regions, our method achieves target coverage (94.3% vs 95% target) with adaptive precision: 21% narrower intervals for low-uncertainty cases while appropriately widening for high-risk predictions. Critically, we demonstrate that well-calibrated Bayesian uncertainties alone severely under-cover (14.1%), highlighting the necessity of our hybrid approach. This framework enables risk-stratified clinical protocols, efficient resource planning for high-confidence predictions, and conservative allocation with enhanced oversight for uncertain cases, providing uncertainty-aware decision support across diverse healthcare settings.

We introduce an operator-theoretic framework for analyzing directional dependence in multivariate time series based on order-constrained spectral non-invariance. Directional influence is defined as the sensitivity of second-order dependence operators to admissible, order-preserving temporal deformations of a designated source component, summarized through orthogonally invariant spectral functionals. We show that the resulting supremum–infimum dispersion functional is the unique diagnostic within this class satisfying order consistency, orthogonal invariance, Loewner monotonicity, second-order sufficiency, and continuity, and that classical Granger causality, directed coherence, and Geweke frequency-domain causality arise as special cases under appropriate restrictions. An information-theoretic impossibility result establishes that entrywise-stable edge-based tests require quadratic sample size scaling in distributed (non-sparse) regimes, whereas spectral tests detect at the optimal linear scale. We establish uniform consistency and valid shift-based randomization inference under weak dependence. Simulations confirm correct size and strong power across distributed and nonlinear alternatives, and an empirical application illustrates system-level directional causal structure in financial markets.

In stochastic message passing, an edge's sampled role changes the node states used to compute adaptive weights at later layers. Weight averaging therefore depends on whether edge roles persist across layers or are resampled at each layer. We study this dependence in Persistent Tri-State Message Passing (P3MP), which combines persistent additive, subtractive, and inactive roles with weights computed from each sample's endpoint states. For two layers, we separate target-state feedback from weight-source covariance and derive the condition under which local and shared weights reverse their preactivation ordering. Shared weights retain a $1/K$ fraction of the feedback for $K$ samples. The interaction is zero for a source-only scorer. Permuting complete weight vectors between samples quantifies the same role-state dependence while preserving their empirical distribution. Exact enumeration and feature-only checkpoint measurements agree with the analytical predictions. Across independently trained feature-only models, local weights yield higher accuracy and lower negative log-likelihood under persistence, while shared weights yield higher accuracy and lower loss under layerwise resampling. The same accuracy ordering holds on deduplicated Chameleon and Squirrel.

In this work, we propose a modified Bayesian Information Criterion (BIC) specifically designed for mixture models and hierarchical structures. This criterion incorporates the determinant of the Hessian matrix of the log-likelihood function, thereby refining the classical Bayes Factor by accounting for the curvature of the likelihood surface. Such geometric information introduces a more nuanced penalization for model complexity. The proposed approach improves model selection, particularly under small-sample conditions or in the presence of noise variables. Through theoretical derivations and extensive simulation studies-including both linear and linear mixed models-we show that our criterion consistently outperforms traditional methods such as BIC, Akaike Information Criterion (AIC), and related variants. The results suggest that integrating curvature-based information from the likelihood landscape leads to more robust and accurate model discrimination in complex data environments.

We consider a system of $N$ Hawkes processes and observe the actions of a subpopulation of size $K \le N$ up to time $t$, where $K$ is large. The influence relationships between each pair of individuals are modeled by i.i.d.Bernoulli($p$) random variables, where $p \in [0,1]$ is an unknown parameter. Each individual acts at a baseline rate $\mu > 0$ and, additionally, at an excitation rate of the form $N^{-1} \sum_{j=1}^{N} \theta_{ij} \int_{0}^{t} \phi(t-s)\,dZ_s^{j,N}$, which depends on the past actions of all individuals that influence it, scaled by $N^{-1}$ (i.e. the mean-field type), with the influence of older actions discounted through a memory kernel $\phi \colon \mathbb{R}{+} \to \mathbb{R}{+}$. Here, $\mu$ and $\phi$ are treated as nuisance parameters. The aim of this paper is to establish a central limit theorem for the estimator of $p$ proposed in \cite{D}, under the subcritical condition $\Lambda p < 1$.

We propose a new estimation method for the Stable Trait, Auto Regressive Trait, and State (STARTS) model, which is well known for its frequent occurrence of improper solutions. The proposed approach is implemented through a two-stage estimation procedure that combines matrix decomposition factor analysis (MDFA) based on eigenvalue decomposition with conventional SEM estimation principles. By reformulating the STARTS model within a factor-analytic framework, this study presents a novel way of applying MDFA in the context of structural equation modeling (SEM). Through a simulation study and an empirical application to ToKyo Teen Cohort data, the proposed method was shown to entail a substantially lower risk of improper solutions than commonly used maximum likelihood, conditional ML, and (unweighted) least squares estimators, while tending to yield solutions similar to those obtained by ML. Compared with Bayesian estimation, the proposed method does not require the specification of appropriate (weakly informative) prior distributions and may effectively mitigate bias issues that arise when the number of time points is small. Applying the proposed method, as well as conducting sensitivity analyses informed by it, will enable researchers to more effectively delineate the range of plausible conclusions from data in estimating the STARTS model and other SEMs.

Conformal Test Martingales (CTMs) are a standard method within the Conformal Prediction framework for testing the crucial assumption of data exchangeability by monitoring deviations from uniformity in the p-value sequence. Although exchangeability implies uniform p-values, the converse does not hold. This raises the question of whether a significant break in exchangeability can occur, such that the p-values remain uniform, rendering CTMs blind. We answer this affirmatively, demonstrating the phenomenon of conformal blindness. Through explicit construction, for the theoretically ideal “predictive oracle” conformity measure (given by the true conditional density), we demonstrate the possibility of an \emph{$A$-cryptic change-point} (where $A$ refers to the conformity measure). Using bivariate Gaussian distributions, we identify a line along which a change in the marginal means does not alter the distribution of the conformity scores, thereby producing perfectly uniform p-values. Simulations confirm that even a massive distribution shift can be perfectly cryptic to the CTM, highlighting a fundamental limitation and emphasising the critical role of the alignment of the conformity measure with potential shifts. By contrasting the predictive oracle with recent results on detection-optimal scores, we emphasise that validity monitoring in safety-critical systems requires careful separation of predictive and diagnostic goals.

This paper, introducing a novel method in philomatics, draws on Wittgenstein's concept of family resemblance from analytic philosophy to develop a clustering algorithm for machine learning. According to Wittgenstein's Philosophical Investigations (1953), family resemblance holds that members of a concept or category are connected by overlapping similarities rather than a single defining property. Consequently, a family of entities forms a chain of items sharing overlapping traits. This philosophical idea naturally lends itself to a graph-based approach in machine learning. Accordingly, we propose the Wittgenstein's Family Resemblance (WFR) clustering algorithm and its kernel variant, kernel WFR. This algorithm computes resemblance scores between neighboring data instances, and after thresholding these scores, a resemblance graph is constructed. The connected components of this graph define the resulting clusters. Simulations on benchmark datasets demonstrate that WFR is an effective nonlinear clustering algorithm that does not require prior knowledge of the number of clusters or assumptions about their shapes.

Understanding how educators interact and form peer networks in online professional development contexts has become increasingly important as MOOCs for educators (MOOC-Eds) proliferate. This study examines peer discussion network formation and evolution in 'The Digital Learning Transition in K-12 Schools', a MOOC-Ed offered to U.S. and international educators in Spring 2013. Using cross-sectional and temporal exponential random graph models (ERGMs and TERGMs), the study analyzes two network subsamples: the largest connected component (N = 363) and active participants with three or more interactions (N = 227). Results reveal strong reciprocity and transitive closure effects across both networks, with participants six to nine times more likely to reciprocate interactions and over twice as likely to form ties with peers sharing common discussion partners. Assigned discussion group homophily emerged as the strongest predictor of tie formation, while regional homophily and willingness to connect also significantly influenced network structure. Temporal analysis showed discussion activity peaked mid-course before declining sharply, with network structure evolving from broadly distributed participation to concentrated interaction among a tightly connected core. These findings illuminate the mechanisms driving peer-supported learning in online professional development contexts and suggest design implications for fostering sustained educator engagement in MOOC-based learning environments.

The increasing deployment of artificial intelligence (AI) in clinical settings challenges foundational assumptions underlying traditional frameworks of medical evidence. Classical statistical approaches, centered on randomized controlled trials, frequentist hypothesis testing, and static confidence intervals, were designed for fixed interventions evaluated under stable conditions. In contrast, AI-driven clinical systems learn continuously, adapt their behavior over time, and operate in non-stationary environments shaped by evolving populations, practices, and feedback effects. In such systems, clinical harm arises less from average error rates than from calibration drift, rare but severe failures, and the accumulation of suboptimal decisions over time. In this perspective, we argue that prevailing notions of statistical significance are insufficient for characterizing evidence and safety in learning health systems. Drawing on risk-theoretic concepts from quantitative finance and online decision theory, we propose reframing medical evidence for adaptive AI systems in terms of time-indexed calibration stability, bounded downside risk, and controlled cumulative regret. We emphasize that this approach does not replace randomized trials or causal inference, but complements them by addressing dimensions of risk and uncertainty that emerge only after deployment. This framework provides a principled mathematical language for evaluating AI-driven clinical systems under continual learning and offers implications for clinical practice, research design, and regulatory oversight.

Recent works have demonstrated promising performances of neural networks on hyperbolic spaces and symmetric positive definite (SPD) manifolds. These spaces belong to a family of Riemannian manifolds referred to as symmetric spaces of noncompact type. In this paper, we propose a novel approach for developing neural networks on such spaces. Our approach relies on a unified formulation of the distance from a point to a hyperplane on the considered spaces. We show that some existing formulations of the point-to-hyperplane distance can be recovered by our approach under specific settings. Furthermore, we derive a closed-form expression for the point-to-hyperplane distance in higher-rank symmetric spaces of noncompact type equipped with G-invariant Riemannian metrics. The derived distance then serves as a tool to design fully-connected (FC) layers and an attention mechanism for neural networks on the considered spaces. Our approach is validated on challenging benchmarks for image classification, electroencephalogram (EEG) signal classification, image generation, and natural language inference.

Non-stationary parametric bandits have attracted much attention recently. There are three principled ways to deal with non-stationarity, including sliding-window, weighted, and restart strategies. As many non-stationary environments exhibit gradual drifting patterns, the weighted strategy is commonly adopted in real-world applications. However, previous theoretical studies show that its analysis is more involved and the algorithms are either computationally less efficient or statistically suboptimal. This paper revisits the weighted strategy for non-stationary parametric bandits. In linear bandits (LB), we discover that this undesirable feature is due to an inadequate regret analysis, which results in an overly complex algorithm design. We propose a refined analysis framework, which simplifies the derivation and, importantly, produces a simpler weight-based algorithm that is as efficient as window/restart-based algorithms while retaining the same regret as previous studies. Furthermore, our new framework can be used to improve regret bounds of other parametric bandits, including Generalized Linear Bandits (GLB) and Self-Concordant Bandits (SCB). For example, we develop a simple weighted GLB algorithm with an $\tilde{O}(k_\mu^{5/4} c_\mu^{-3/4} d^{3/4} P_T^{1/4}T^{3/4})$ regret, improving the $\tilde{O}(k_\mu^{2} c_\mu^{-1}d^{9/10} P_T^{1/5}T^{4/5})$ bound in prior work, where $k_\mu$ and $c_\mu$ characterize the reward model's nonlinearity, $P_T$ measures the non-stationarity, $d$ and $T$ denote the dimension and time horizon. Moreover, we extend our framework to non-stationary Markov Decision Processes (MDPs) with function approximation, focusing on Linear Mixture MDP and Multinomial Logit (MNL) Mixture MDP. For both classes, we propose algorithms based on the weighted strategy and establish dynamic regret guarantees using our analysis framework.

This paper develops the algorithmic and dynamical foundations of recursive ensemble learning driven by Fibonacci-type update flows. In contrast with classical boosting Freund and Schapire (1997); Friedman (2001), where the ensemble evolves through first-order additive updates, we study second-order recursive architectures in which each predictor depends on its two immediate predecessors. These Fibonacci flows induce a learning dynamic with memory, allowing ensembles to integrate past structure while adapting to new residual information. We introduce a general family of recursive weight-update algorithms encompassing Fibonacci, tribonacci, and higher-order recursions, together with continuous-time limits that yield systems of differential equations governing ensemble evolution. We establish global convergence conditions, spectral stability criteria, and non-asymptotic generalization bounds under Rademacher Bartlett and Mendelson (2002) and algorithmic stability analyses. The resulting theory unifies recursive ensembles, structured weighting, and dynamical systems viewpoints in statistical learning. Experiments with kernel ridge regression Rasmussen and Williams (2006), spline smoothers Wahba (1990), and random Fourier feature models Rahimi and Recht (2007) demonstrate that recursive flows consistently improve approximation and generalization beyond static weighting. These results complete the trilogy begun in Papers I and II: from Fibonacci weighting, through geometric weighting theory, to fully dynamical recursive ensemble learning systems.

The growing frequency and intensity of wildfires pose serious threats to communities in wildland-urban interface regions. Understanding evacuation behavior is critical for effective emergency planning. This study analyzes evacuation during the 2025 Palisades and Eaton Fires using high-resolution Facebook data. We propose a comprehensive framework to derive wildfire evacuation-related metrics, including compliance rate, departure timing, delay, origin-destination flows, travel distance, and destination types. A new metric, Damage-Evacuation Disparity Index (DEDI), identifies areas with severe structural damage but low evacuation compliance. Results reveal spatiotemporal heterogeneity: residents closer to the fire evacuated earlier, whereas late or nighttime orders led to lower compliance and longer delays. Contrasting patterns between East and West Altadena further illustrate this disparity. DEDI-identified communities exhibited higher social vulnerability and fire risk. Most evacuations concluded in residential areas, while longer trips concentrated in hotels and public facilities. These findings showcase the Facebook data's potential for data-driven wildfire evacuation planning.

We study the expectation-maximization (EM) algorithm for general latent-variable models under (i) distributional misspecification and (ii) nonidentifiability induced by a group action. We formulate EM on the quotient parameter space and measure error using an arbitrary integral probability metric (IPM). Our main results give (a) a sharp local linear convergence rate for population EM governed by the spectral radius of the linearization on a local slice, and (b) tight finite-sample bounds for sample EM obtained via perturbed contraction inequalities and generic chaining/entropy control of EM-induced empirical processes.

This paper develops a practical framework for using observational data to audit the consumer surplus effects of AI-driven decisions, specifically in targeted pricing and algorithmic lending. Traditional approaches first estimate demand functions and then integrate to compute consumer surplus, but these methods can be challenging to implement in practice due to model misspecification in parametric demand forms and the large data requirements and slow convergence of flexible nonparametric or machine learning approaches. Instead, we exploit the randomness inherent in modern algorithmic pricing, arising from the need to balance exploration and exploitation, and introduce an estimator that avoids explicit estimation and numerical integration of the demand function. Each observed purchase outcome at a randomized price is an unbiased estimate of demand and by carefully reweighting purchase outcomes using novel cumulative propensity weights (CPW), we are able to reconstruct the integral. Building on this idea, we introduce a doubly robust variant named the augmented cumulative propensity weighting (ACPW) estimator that only requires one of either the demand model or the historical pricing policy distribution to be correctly specified. Furthermore, this approach facilitates the use of flexible machine learning methods for estimating consumer surplus, since it achieves fast convergence rates by incorporating an estimate of demand, even when the machine learning estimate has slower convergence rates. Neither of these estimators is a standard application of off-policy evaluation techniques as the target estimand, consumer surplus, is unobserved. To address fairness, we extend this framework to an inequality-aware surplus measure, allowing regulators and firms to quantify the profit-equity trade-off. Finally, we validate our methods through comprehensive numerical studies.

We provide an overview of high dimensional dynamical systems driven by random matrices, focusing on applications to simple models of learning and generalization in machine learning theory. Using both cavity method arguments and path integrals, we review how the behavior of a coupled infinite dimensional system can be characterized as a stochastic process for each single site of the system. We provide a pedagogical treatment of dynamical mean field theory (DMFT), a framework that can be flexibly applied to these settings. The DMFT single site stochastic process is fully characterized by a set of (two-time) correlation and response functions. For linear time-invariant systems, we illustrate connections between random matrix resolvents and the DMFT response. We demonstrate applications of these ideas to machine learning models such as gradient flow, stochastic gradient descent on random feature models and deep linear networks in the feature learning regime trained on random data. We demonstrate how bias and variance decompositions (analysis of ensembling/bagging etc) can be computed by averaging over subsets of the DMFT noise variables. From our formalism we also investigate how linear systems driven with random non-Hermitian matrices (such as random feature models) can exhibit non-monotonic loss curves with training time, while Hermitian matrices with the matching spectra do not, highlighting a different mechanism for non-monotonicity than small eigenvalues causing instability to label noise. Lastly, we provide asymptotic descriptions of the training and test loss dynamics for randomly initialized deep linear neural networks trained in the feature learning regime with high-dimensional random data. In this case, the time translation invariance structure is lost and the hidden layer weights are characterized as spiked random matrices.

Transfer learning aims to improve performance on a target task by leveraging information from related source tasks. We propose a nonparametric regression transfer learning framework that explicitly models heterogeneity in the source-target relationship. Our approach relies on a local transfer assumption: the covariate space is partitioned into finitely many cells such that, within each cell, the target regression function can be expressed as a low-complexity transformation of the source regression function. This localized structure enables effective transfer where similarity is present while limiting negative transfer elsewhere. We introduce estimators that jointly learn the local transfer functions and the target regression, together with fully data-driven procedures that adapt to unknown partition structure and transfer strength. We establish sharp minimax rates for target regression estimation, showing that local transfer can mitigate the curse of dimensionality by exploiting reduced functional complexity. Our theoretical guarantees take the form of oracle inequalities that decompose excess risk into estimation and approximation terms, ensuring robustness to model misspecification. Numerical experiments illustrate the benefits of the proposed approach.

Functional magnetic resonance imaging (fMRI) provides an indirect measurement of neuronal activity via hemodynamic responses that vary across brain regions and individuals. Ignoring this hemodynamic variability can bias downstream connectivity estimates. Furthermore, the hemodynamic parameters themselves may serve as important imaging biomarkers. Estimating spatially varying hemodynamics from resting-state fMRI (rsfMRI) is therefore an important but challenging blind inverse problem, since both the latent neural activity and the hemodynamic coupling are unknown. In this work, we propose a methodology for inferring hemodynamic coupling on the cortical surface from rsfMRI. Our approach avoids the highly unstable joint recovery of neural activity and hemodynamics by marginalizing out the latent neural signal and basing inference on the resulting marginal likelihood. To enable scalable, high-resolution estimation, we employ a deep neural network combined with conditional normalizing flows to accurately approximate this intractable marginal likelihood, while enforcing spatial coherence through priors defined on the cortical surface that admit sparse representations. Uncertainty in the hemodynamic estimates is quantified via a double-bootstrap procedure. The proposed approach is extensively validated using synthetic data and real fMRI datasets, demonstrating clear improvements over current methods for hemodynamic estimation and downstream connectivity reproducibility.

Learning models with categorical variables requires optimizing expectations over discrete distributions, a setting in which stochastic gradient-based optimization is challenging due to the non-differentiability of categorical sampling. A common workaround is to replace the discrete distribution with a continuous relaxation, yielding a smooth surrogate that admits reparameterized gradient estimates via the reparameterization trick. Building on this idea, we introduce ReDGE, a novel and efficient diffusion-based soft reparameterization method for categorical distributions. Our approach defines a flexible class of gradient estimators that includes the Straight-Through estimator as a special case. Experiments spanning latent variable models and inference-time reward guidance in discrete diffusion models demonstrate that ReDGE consistently matches or outperforms existing gradient-based methods. The code will be made available at https://github.com/samsongourevitch/redge.

Variable importance in regression analyses is of considerable interest in a variety of fields. There is no unique method for assessing variable importance. However, a substantial share of the available literature employs Shapley values, either explicitly or implicitly, to decompose a suitable goodness-of-fit measure, in the linear regression model typically the classical $R^2$. Beyond linear regression, there is no generally accepted goodness-of-fit measure, only a variety of pseudo-$R^2$s. We formulate and discuss the desirable properties of goodness-of-fit measures that enable Shapley values to be interpreted in terms of relative, and even absolute, importance. We suggest to use a pseudo-$R^2$ based on the Kullback-Leibler divergence, the Kullback-Leibler $R^2$, which has a convenient form for generalized linear models and permits to unify and extend previous work on variable importance for linear and nonlinear models. Several examples are presented, using data from public health and insurance.

Bayesian inference in complex generative models is often obstructed by the absence of tractable likelihoods and the infeasibility of computing gradients of high-dimensional simulators. Existing likelihood-free methods for generalized Bayesian inference typically rely on gradient-based optimization or reparameterization, which can be computationally expensive and often inapplicable to black-box simulators. To overcome these limitations, we introduce a gradient-free ensemble transform Langevin dynamics method for generalized Bayesian inference using the maximum mean discrepancy. By relying on ensemble-based covariance structures rather than simulator derivatives, the proposed method enables robust posterior approximation without requiring access to gradients of the forward model, making it applicable to a broader class of likelihood-free models. The method is affine invariant, computationally efficient, and robust to model misspecification. Through numerical experiments on well-specified chaotic dynamical systems, and misspecified generative models with contaminated data, we demonstrate that the proposed method achieves comparable or improved accuracy relative to existing gradient-based methods, while substantially reducing computational cost.

Accurate hindcasting of sea state events is a cornerstone of coastal engineering, risk assessment, and climate-related studies, yet it remains limited by uncertainties in physical parameterizations and model structure. This study introduces an automated calibration framework based on Bayesian Optimization (BO) using the Tree-structured Parzen Estimator (TPE) to constrain key dissipative processes in the ANEMOC-3 hindcast wave model, including bottom-friction losses, depth-induced wave breaking, and dissipation driven by wave strong opposing currents. The methodology enables the joint optimization of continuous physical parameters and discrete model structure choices within a unified probabilistic search space, significantly reducing model-observation misfit. Calibration is conducted over the high energy storm conditions of February 2014, while transferability is assessed both temporally and spatially, through independent validation on January 2014 and January 2018 events and across a network of offshore and coastal buoy observations. The optimized configurations retain skill beyond the calibration period and across observation sites, yielding systematically improved agreement with buoy measurements in terms of bias, root mean square error, and scatter index relative to the reference configuration. These results highlight the potential of Bayesian Optimization as a scalable and robust framework for automating the calibration of complex wave hindcast systems. Future developments will address multi-objective optimization, uncertainty quantification, and the integration of complementary observational datasets.

Modern optimization problems in scientific and engineering domains often rely on expensive black-box evaluations, such as those arising in physical simulations or deep learning pipelines, where gradient information is unavailable or unreliable. In these settings, conventional optimization methods quickly become impractical due to prohibitive computational costs and poor scalability. We propose ALMAB-DC, a unified and modular framework for scalable black-box optimization that integrates active learning, multi-armed bandits, and distributed computing, with optional GPU acceleration. The framework leverages surrogate modeling and information-theoretic acquisition functions to guide informative sample selection, while bandit-based controllers dynamically allocate computational resources across candidate evaluations in a statistically principled manner. These decisions are executed asynchronously within a distributed multi-agent system, enabling high-throughput parallel evaluation. We establish theoretical regret bounds for both UCB-based and Thompson-sampling-based variants and develop a scalability analysis grounded in Amdahl's and Gustafson's laws. Empirical results across synthetic benchmarks, reinforcement learning tasks, and scientific simulation problems demonstrate that ALMAB-DC consistently outperforms state-of-the-art black-box optimizers. By design, ALMAB-DC is modular, uncertainty-aware, and extensible, making it particularly well suited for high-dimensional, resource-intensive optimization challenges.

Machine learning is central to empirical asset pricing, but portfolio construction still relies on point predictions and largely ignores asset-specific estimation uncertainty. We propose a simple change: sort assets using uncertainty-adjusted prediction bounds instead of point predictions alone. Across a broad set of ML models and a U.S. equity panel, this approach improves portfolio performance relative to point-prediction sorting. These gains persist even when bounds are built from partial or misspecified uncertainty information. They arise mainly from reduced volatility and are strongest for flexible machine learning models. Identification and robustness exercises show that these improvements are driven by asset-level rather than time or aggregate predictive uncertainty.

This paper develops a novel methodology for testing the goodness-of-fit of sparse parametric regression models based on projected empirical processes and p-value combination, where the covariate dimension may substantially exceed the sample size. In such ultra-high dimensional settings, traditional empirical process-based tests often fail due to the curse of dimensionality or their reliance on the asymptotic linearity and normality of parameter estimators–properties that may not hold under ultra-high dimensional scenarios. To overcome these challenges, we first extend the classic martingale transformation to ultra-high dimensional settings under mild conditions and construct a Cramer-von Mises type test based on a martingale-transformed, projected residual-marked empirical process for any projection on the unit sphere. The martingale transformation renders this projected test asymptotically distribution-free and enables us to derive its limiting distribution using only standard convergence rates of parameter estimators. While the projected test is consistent for almost all projections on the unit sphere under mild conditions, it may still suffer from power loss for specific projections. Therefore, we further employ powerful p-value combination procedures, such as the Cauchy combination, to aggregate p-values across multiple projections, thereby enhancing overall robustness. Furthermore, recognizing that empirical process-based tests excel at detecting low-frequency signals while local smoothing tests are generally superior for high-frequency alternatives, we propose a novel hybrid test that aggregates both approaches using Cauchy combination. The resulting hybrid test is powerful against both low-frequency and high-frequency alternatives. $\cdots$