Statistical Models

This manuscript formalizes the most popular model validation tools used in general insurance actuarial modeling. These include graphical tools like calibration plots, actual-vs-expected plots, lift charts, Murphy diagrams, as well as classical statistical tools such as Bregman losses, deviance losses, elementary losses, Murphy's decomposition and Gini scores. Particular emphasis is placed on whether calibration and discrimination are studied under a policy-weighted or an exposure-weighted population measure. This distinction is crucial in ensuring that premium schemes are calibrated on the correct scale.

When providing forecasted probabilities with a predictive model, the ideal model offers perfect calibration: the true probability of the outcome (i.e., the probability that $Y=1$) exactly matches the forecasted probability $f(X)$. In practice, models inevitably exhibit calibration error, and it is therefore important to be able to measure this miscalibration to assess a model's reliability. The Expected Calibration Error (ECE) is the most widely used measure of miscalibration, but is known to be impossible to estimate the ECE with guaranteed accuracy in an assumption-free setting. In this work, we propose an alternative measure, the rankECE, that is based on comparing points with neighboring values of the predicted probability $f(X)$. Our theoretical guarantees and empirical results establish that rankECE provides a better proxy for ECE as compared to binned approximations to ECE, which are the most commonly-used approximations in practice.

We provide methods for calculating the robustness of the predictions of two types of generative classifiers whose underlying distribution is a Probabilistic Graphical Model (PGM): naive Bayes classifiers and generative forests (a probabilistic extension of random forests). Following the paradigm of robustness quantification, we define the robustness of a prediction as the extent to which the distribution of the classifier can be perturbed without changing this prediction. We consider perturbations obtained by varying the local models of the PGMs within general neighborhoods and focus in particular on epsilon-contamination, total variation distance and chi-squared divergence balls. We test our methods on benchmark datasets, demonstrate that the robustness value of a prediction serves as an indicator for its trustworthiness and compare our approach with other such indicators.

Many of the series that generative time-series models are benchmarked on place a large probability mass on a single value --- it does not rain, no ride is requested, no part is ordered. We report what happens when such data is evaluated carefully. First, the standard rolling-origin protocol can score a model on a window whose atom structure bears no resemblance to the dataset: on one benchmark the dataset is $42\%$ zeros and the evaluation windows are $13\%$, on another $47\%$ against $5\%$. This is not a cosmetic problem --- it reversed one of our own conclusions, turning the strongest occurrence model in our study into what looked like a cautionary tale. Second, we give a control in which CRPS is invariant \emph{by construction} while the temporal coupling is destroyed, which measures exactly how much that coupling contributes to a chosen statistic. Third, benchmarking seven models on a matched protocol over five seeds, an autoregressive hurdle beats a conditional flow on five of six datasets, by up to a factor of $153$, while the flow's own occurrence statistics vary by up to $62\%$ across training seeds and every baseline is deterministic. Finally, the model ordering is not the same under five different occurrence statistics, and the two that do not share a construction agree with each other least.

This paper develops procedures for nonparametric goodness-of-fit testing under covariate shift, where labelled data are drawn from a source population but goodness-of-fit is evaluated for a target population. The distribution mismatch is quantified by either a bounded moment condition or a sub-exponential tail condition on the target-to-source density ratio. Our method combines truncated importance-weighting kernel ridge regression with a multiplier bootstrap to construct confidence sets for the regression function. The truncation stabilizes the importance- weighting kernel ridge regression as well as the bootstrap calibration, making our approach applicable even when the density ratio has heavy tails. We prove nonasymptotic validity and sharpness of the resulting confidence sets under suitable operator compatibility conditions, and establish explicit error rates for coverage probability under specific conditions on the target- to-source density ratio and on the spectral decay of the kernel integral operator. Numerical experiments corroborate our theoretical findings.

Given an algebraic statistical model, a challenging problem is classifying the data according to the number of positive critical points of the likelihood function. The positive critical points are the positive solutions to an algebraic system, say likelihood equations. So, identifying the number of positive critical points is a real root classification problem for the likelihood equations. A discriminant variety of a likelihood-equation system geometrically describes the data for which the number of real solutions becomes unusual. As an essential component of the discriminant variety, the nonproperness set collects the data such that the likelihood-equation system has a solution at infinity. So, the number of real solutions varies when the data passes the nonproperness set, and identifying the nonproperness set plays a crucial role in the real root classification. In this work, we develop a novel method for computing nonproperness sets of likelihood-equation systems. We prove the correctness of this method. We show experimentally that it is far more efficient than the known methods in the literature.

Standard model comparison is global, aggregating losses across the covariate space to declare a single winner. This can obscure heterogeneous performance, where different models are preferable in different regions. We introduce conformalized local model comparison, a split-sample framework for constructing calibrated local best-model maps. Given a model comparison score, such as the difference between two squared losses, the method uses three disjoint splits to fit competing models, estimate local centers and scales from out-of-sample scores, and conformally calibrate residual uncertainty. At a target point, the procedure declares a local winner only when a one-sided conformal bound excludes a tie, with the score's sign determining the favored model. We prove finite-sample marginal control for one-sided erroneous declarations on the realized future comparison score, establish pointwise consistency of the localized mean-score estimator away from tie boundaries, show that aggregate comparison can disagree sharply with the prevalence of local superiority, and derive a squared-loss bias--variance decomposition that clarifies how model structure affects local wins. Synthetic and real-data experiments show that the method recovers heterogeneous winner regions, abstains under uncertainty, and yields higher conditional gain than global selection.

AI benchmarks increasingly leverage item-level statistical models, particularly item response theory (IRT), to estimate model capabilities, rank systems, select informative examples, and diagnose benchmark quality. However, AI benchmark data often departs from the data regime of human testing, for which standard IRT estimation tools were originally developed: benchmarks typically involve fewer evaluated models, far more items, and capability distributions that may be skewed, clustered, or multimodal. We examine how these regime mismatches challenge the reliability of IRT modeling for AI evaluation. Using item parameters and capability distributions derived from six widely used LLM benchmarks, we simulate response matrices under three common IRT models and compare four estimation tools used in recent benchmark studies: marginal maximum likelihood, Markov chain Monte Carlo, variational inference, and a neural pseudo-Siamese estimator. Across 18,000 simulation conditions, we systematically evaluate computational feasibility, scalability, and the reliability of IRT inferences about model rankings, predicted performance, and item characteristics. Results show that classical estimators can become infeasible in large benchmark settings, whereas scalable estimators can produce unreliable item-level and ranking inferences with small or nonnormally distributed model sets. This study identifies when latent trait models reliably support or risk distorting AI benchmarking claims, and what sample sizes and diagnostics are needed for trustworthy use.

In software analytics, rerunning the same analysis twice often yields different models and conclusions. This reduces trust in the model and limits its use. We find that model instability is a major problem. Across 127 multi-objective SE optimization problems (12,700 test cases), repeated runs of a state-of-the-art optimizer agree on only 13.7% of test cases, even under improved settings. We argue that this instability is not merely noise to tolerate, but a property that can be measured and managed. By adjusting how labels are spent, how complex the models become, and how splits are scored, we obtain models that agree 4.8 times as often as the default configuration. The standard deviation of optimization error falls by 22% on average (mean std 17.4 to 13.6), while recommendation quality improves rather than degrades. In terms of quality, the refined settings are statistically top-ranked on 119 of 127 datasets, compared to 74 for the defaults. We then test causal and data-locality interventions and find that they help only partially, suggesting a residual stability floor. Our evidence suggests there are fundamental limits to stability set by the data itself (noise, scarce labels, proxy objectives, and the many near-equivalent models a dataset admits). We conclude that instability should be treated as a standard evaluation axis in SE optimization, which should be routinely measured, reported alongside performance, and used to calibrate trust in any single run. The methods in this paper provide a baseline against which future efforts to reduce SBSE instability can be judged. To support open science, we offer the following reproduction package: https://tinyurl.com/Model-Instability

This paper explores the "Granularity Paradox" in time-series forecasting, wherein finer temporal disaggregation (e.g., Monthly to Weekly/Daily) improves in-sample diagnostics and dataset size (N), but degrades out-of-sample accuracy due to recursive error compounding over longer horizons (H). Conversely, coarse aggregation (Annual) eliminates recursive error propagation but reduces data available to estimators. We formalize this trade-off and benchmark 10 models - spanning naïve, statistical, machine learning, and deep learning architectures - across six granularities using a 13-year public procurement dataset. The empirical results reveal a non-monotonic threshold structure: recursive autoregressive and seasonal models degrade substantially under high-frequency forecasting (e.g., Holt-Winters reaches a Test R-squared of -151 and TPFE of 425.85% at the Daily grain), while the LSTM traces a U-shaped error curve, worsening from Monthly (19.66%) through Bi-Weekly (35.94%) before overcoming the error propagation penalty at Daily (TPFE of 4.35%, R-squared of 0.66). Linear Regression remains stable across all granularities (16.3-17.0% TPFE), confirming that the paradox is driven by recursive feedback topology, not model complexity. The results demonstrate that standard pointwise metrics (RMSE, MAE) systematically mask cumulative error propagation, and that evaluating forecasts without goal-dependent cumulative metrics produces misleading assessments of model adequacy. We introduce a consensus-dissensus diagnostic comparing the directional behaviour of pointwise metrics against cumulative TPFE across granularities, enabling the identification of models whose standard diagnostics mask systematic error propagation.

A central challenge in statistical modeling is identifying the subset of features that belong in the true regression model. The classical best subset selection problem, recently made tractable via mixed-integer optimization (MIO), finds the globally optimal sparse solution. It does not, however, make use of any information beyond the observed data. In many applied settings, domain experts can meaningfully rank or score the relevance of candidate predictors, yet no existing framework integrates such probabilistic expert assessments directly into the best-subsets objective. This paper presents Expert-Implied Bayesian Best Subsets (EBBS), a method that incorporates domain-expert probability estimates of feature relevance into the MIO best-subsets problem through a maximum a posteriori (MAP) framework. Expert views from multiple respondents are aggregated into a single prior probability per feature using the Poisson binomial distribution for marginal probability estimates, the pairwise win rate for pairwise comparisons, or the normalized mean rank for ordinal rankings. This probability enters the objective function as a log-odds penalty term that smoothly encourages or discourages the selection of each feature consistent with the expert consensus. This paper provides analytic derivations of the MAP formulation and characterizes its theoretical properties. The proposed model reduces to Best Subsets when experts all have no views. Empirical results on synthetic and real datasets are forthcoming.

We study the problem of model selection among probabilistic forecasting models evaluated on datasets of multiple time series. The performance of a model on a single time series is quantified by the average value (score) of a proper scoring rule over a test set, but extending model selection to data sets of time series requires aggregating these scores. Common approaches either rely on scaling scores and averaging them (mean scaled score) or avoid scaling by using alternative statistics such as mean ranks or win rates. However, these approaches can yield conflicting conclusions. We show that such discrepancies arise from the skewness of the distribution of the scores, which is particularly pronounced when test sets are short. The skewness can cause non-mean criteria (e.g., mean rank, median, win rate) to select misspecified models. In contrast, the mean score is immune from this problem. We further show that, as the size of the test sets increases, all aggregation criteria converge to the same model selection decision, mitigating these discrepancies. Our experiments on intermittent demand time series, including data from the M5 competition, highlight the importance of sufficiently large test sets; the mean scaled score appears to be the more reliable approach, also because empirically we found its decision to remain consistent when different scaling factors are adopted.

Generative surveying -- where collections of LLM-based personas provide feedback on messages -- has emerged as a cheap and scalable alternative to traditional market research. However, LLMs are sensitive to small variations in prompt design and conclusions drawn from generative surveys may depend on arbitrary phrasing choices. Controlling for this sensitivity requires including semantically equivalent perturbations in the analysis. In this paper, we show that standard hypothesis tests, including the sign test and Wilcoxon signed-rank test, are invalid under a statistical model for generative surveying that includes realistic perturbation structure. We propose a permutation test that is valid under this model and formally characterize the conditions under which standard tests fail. Applying our framework to a simple generative surveying problem, we estimate relevant parameters, characterize the power of the permutation test under realistic conditions, and provide practical guidance on budget allocation across personas, perturbations, and replicates. Finally, we show that both the magnitude and direction of the estimated effect are sensitive to the choice of model, even within the same model family.

Kernel Stein discrepancy (KSD) is among the most popular goodness-of-fit (GoF) measures on general domains with a large number of successful deployments. One of the main applications of KSD is in constructing powerful GoF tests. However, tests relying on the classical U-/V-statistic-based KSD estimators have two major drawbacks. (i) Their runtime scales quadratically in the number of samples. (ii) Their asymptotic null distribution is computationally intractable in most cases, typically handled by bootstrapping. While it is known that the Nyström method permits accelerating KSD estimation with no loss of statistical accuracy under mild conditions, to the best of our knowledge, the fundamental question of its impact on bootstrap-based GoF testing is open; resolving this question is the focus of the current paper. In particular, we prove that the key properties of the quadratic-time bootstrapped KSD-based GoF test (asymptotic level and local consistency) are preserved by its Nyström acceleration. We numerically demonstrate the efficiency of the accelerated KSD estimator and bootstrap in the context of GoF testing of spherical and functional data. Our numerical results show that the Nyström-accelerated method performs statistically on-par with the quadratic-time approach, while requiring substantially smaller runtime.

Comprehensive evaluation of machine learning models is the key to make sure that they perform as robustly and consistently as desired. In order to summarize the experimental results and pick a winner, Critical Difference (CD) diagrams are used. Standard CD diagrams rely on discrete ranks, discarding the magnitude of performance gaps between models, raising an issue which we call magnitude-blindness. In order to address this issue, we propose Magnitude-Aware Rank Statistics (MARS) that incorporates a relative margin coefficient as a weight for the discrete ranks. This coefficient scales ranks based on the distance between the best and worst performers, with a dynamic projection to handle boundary cases. Followed by the calculation of a CD value, MARS results in a more realistic statistical representation of differences of model performances and more insights on how methods actually perform in vast and extensive experimental settings.

Deep generative models offer a natural foundation for out-of-distribution (OOD) detection, yet prior work has shown that their assigned likelihoods are notoriously unreliable indicators for in- vs out-of-distribution data. In this paper, we address this problem by leveraging the diffeomorphic and mass-preserving properties of continuous normalising flows. Our analysis shows that OOD samples are mapped to noise samples that are highly atypical under the noise prior in ways not captured by the likelihood. Based on this observation, we propose a new method -- Signal in the Noise (SITN) -- for OOD detection on the single-sample level. SITN requires no access to OOD data, incurs minimal computational overhead, and provides strict control of false positive rates. Comprehensive evaluations through standard benchmarks and synthetic perturbations highlight the method's effectiveness and the absence of the complexity bias inherent to likelihood-based methods.

Agent-based models (ABMs) are increasingly used in macroeconomics, but their analysis still often relies on ad hoc Monte Carlo campaigns with heterogeneous statistical effort across parameter settings. We show how statistical model checking (SMC), implemented through MultiVeStA, can provide a principled analysis layer for a realistic macroeconomic ABM without rewriting the simulator in a dedicated formalism. Our case study is the heuristic-switching Keynes+Schumpeter(K+S) model, analysed hrough a transient sensitivity campaign over one-parameter sweeps, two macro observables (unemployment and GDP growth), and one auxiliary micro-level probe (market share) on the post-warmup phase of a 600-step horizon. The analysis is driven by reusable temporal queries, observable-specific precision targets, and confidence-based stopping rules that automatically determine the simulation effort required by each configuration. Results show a clear contrast across parameter families: macro-financial and structural sweeps produce the strongest transient effects, whereas several heuristic-rule sweeps remain much weaker under the same precision policy. More broadly, the paper shows that SMC can support reproducible and informative quantitative analysis of substantively rich economic ABMs, while making uncertainty estimates and simulation cost explicit parts of the reported results.

We define susceptibilities as a measure of the response of an observable quantity of a parameterized statistical model to a perturbation of the data for a general class of observables. We define estimators for these susceptibilities as statistics in a sequence of n data-points and prove that these estimators are consistent and asymptotically unbiased in the large n regime.

We introduce Mira, a sample-based score for assessing the accuracy of a candidate conditional distribution using only joint samples from the true data-generating process. Relying on the principle that distributions coincide if they assign equal probability mass to all regions, we derive an analytic expression for the Mira statistic, whose average defines the Mira score. This formulation further allows us to compute theoretical reference values and uncertainty estimates when the candidate distribution matches the true one. This framework enables model comparison by quantifying the alignment between the conditional distribution of a candidate model and the true data generating process. Consequently, Mira enables Bayesian model comparison through direct posterior validation, bypassing the challenging evidence computation. We demonstrate its effectiveness across several toy problems and Bayesian inference tasks.

We study a classification problem with three key challenges: pervasive informative missingness, the integration of partial prior expert knowledge into the learning process, and the need for interpretable decision rules. We propose a framework that encodes prior knowledge through an expert-guided class-conditional model for one or more classes, and use this model to construct a small set of interpretable goodness-of-fit features. The features quantify how well the observed data agree with the expert model, isolating the contributions of different aspects of the data, including both observed and missing components. These features are combined with a few transparent auxiliary summaries in a simple discriminative classifier, resulting in a decision rule that is easy to inspect and justify. We develop and apply the framework in the context of seismic monitoring used to assess compliance with the Comprehensive Nuclear-Test-Ban Treaty. We show that the method has strong potential as a transparent screening tool, reducing workload for expert analysts. A simulation designed to isolate the contribution of the proposed framework shows that this interpretable expert-guided method can even outperform strong standard machine-learning classifiers, particularly when training samples are small.

In differentially private (DP) tabular data synthesis, the consensus is that statistical models are better than neural network (NN)-based methods. However, we argue that this conclusion is incomplete and overlooks the challenge of densely correlated datasets, where intricate dependencies can overwhelm statistical models. In such complex scenarios, neural networks are more suitable due to their capacity to fit complex distributions by learning directly from samples. Despite this potential, existing NN-based algorithms still suffer from significant limitations. We therefore propose MargNet, incorporating successful algorithmic designs of statistical models into neural networks. MargNet applies an adaptive marginal selection strategy and trains the neural networks to generate data that conforms to the selected marginals. On sparsely correlated datasets, our approach achieves utility close to the best statistical method while offering an average 7$\times$ speedup over it. More importantly, on densely correlated datasets, MargNet establishes a new state-of-the-art, reducing fidelity error by up to 26% compared to the previous best. We release our code on GitHub.\footnote{https://github.com/KaiChen9909/margnet}

Conditional independence testing is a critical component of feature screening, invariant statistical models, and causal discovery. Many of these algorithms rely on the sequential application of conditional independence tests, and their stability hinges on how their outcomes interact. We study this meta-dependence'' between conditional independence properties using the following geometric intuition: satisfying each conditional independence property constrains the space of possible joint distributions to a manifold. The meta-dependence'' of multiple conditional independences in a probability distribution is informed by its position relative to these manifolds. We provide a simple-to-compute measure of this meta-dependence using moment projections, with a closed-form expression for multivariate Gaussian distributions, and consolidate our findings empirically using both synthetic and real-world data. Our measure of meta-dependence does not rely on graphical properties of the distribution and can be computed directly from summary statistics such as a covariance matrix, allowing for various applications. We demonstrate one use case of meta-dependence, using a simple redundancy metric to tune significance thresholds and improve causal discovery.

Most statistical models for pairwise comparisons, including the Bradley-Terry (BT) and Thurstone models and many extensions, make a relatively strong assumption of stochastic transitivity. This assumption imposes the existence of an unobserved global ranking among all the players/teams/items and monotone constraints on the comparison probabilities implied by the global ranking. However, the stochastic transitivity assumption does not hold in many real-world scenarios of pairwise comparisons, especially games involving multiple skills or strategies. As a result, models relying on this assumption can have suboptimal predictive performance. In this paper, we propose a general family of statistical models for pairwise comparison data without a stochastic transitivity assumption, substantially extending the BT and Thurstone models. In this model, the pairwise probabilities are determined by a (approximately) low-dimensional skew-symmetric matrix. Likelihood-based estimation methods and computational algorithms are developed, which allow for sparse data with only a small proportion of observed pairs. Theoretical analysis shows that the proposed estimator achieves minimax-rate optimality, which adapts effectively to the sparsity level of the data. The spectral theory for skew-symmetric matrices plays a crucial role in the implementation and theoretical analysis. The proposed method's superiority against the BT model, along with its broad applicability across diverse scenarios, is further supported by simulations and real data analysis.