Papers

The Hierarchical Kernel Transformer (HKT) is a multi-scale attention mechanism that processes sequences at L resolution levels via trainable causal downsampling, combining level-specific score matrices through learned convex weights. The total computational cost is bounded by 4/3 times that of standard attention, reaching 1.3125x for L = 3. Four theoretical results are established. (i) The hierarchical score matrix defines a positive semidefinite kernel under a sufficient condition on the symmetrised bilinear form (Proposition 3.1). (ii) The asymmetric score matrix decomposes uniquely into a symmetric part controlling reciprocal attention and an antisymmetric part controlling directional attention; HKT provides L independent such pairs across scales, one per resolution level (Propositions 3.5-3.6). (iii) The approximation error decomposes into three interpretable components with an explicit non-Gaussian correction and a geometric decay bound in L (Theorem 4.3, Proposition 4.4). (iv) HKT strictly subsumes single-head standard attention and causal convolution (Proposition 3.4). Experiments over 3 random seeds show consistent gains over retrained standard attention baselines: +4.77pp on synthetic ListOps (55.10+-0.29% vs 50.33+-0.12%, T = 512), +1.44pp on sequential CIFAR-10 (35.45+-0.09% vs 34.01+-0.19%, T = 1,024), and +7.47pp on IMDB character-level sentiment (70.19+-0.57% vs 62.72+-0.40%, T = 1,024), all at 1.31x overhead.

Neurophysiologists are nowadays able to record from a large number of extracellular electrodes and to extract, from the raw data, the sequences of action potentials or spikes generated by many neurons. Unfortunately these ”many neurons” still represent only a tiny fraction of the neuronal population that constitutes the network. Using association statistics such as the estimation of the cross-correlation functions, they are trying to infer the structure of the network formed by the recorded neurons. But this inference is compromised by the tremendous under-sampling of the neuronal population. We propose to focus instead on simple spike train statistics, like the empirical spikes frequency, or the interspike interval distribution. Their sampling distributions can be estimated by simulations, and, given a few observed spike train statistics, they provide enough information to infer the structure of the underlying network. We show that, on a ”toy model”, our method gives significantly better results than the sub-network reconstruction method with regards to the inference of the connection probability of the original network.

Bathymetry reconstruction is an important problem in various fields, including oceanography and environmental monitoring. This paper presents a Bayesian inference approach to reconstructing bathymetries from point measurements of the water height. We test the method for parameterized and discretized bathymetries with synthetic data to evaluate its performance and limitations. Our results indicate that the Bayesian framework provides a robust approach to bathymetry reconstruction. Finally, we use the framework to reconstruct a real-world bathymetry in a wave flume from experimental measurements and compare its performance to an adjoint optimization method. The Bayesian approach improves the normalized root mean squared error (NRMSE) of the reconstruction and provides better qualitative features, while also quantifying uncertainty.

We study statistical parameter estimation in the setting of data markets when providers differ in both provision costs and estimation-relevant data quality. We define a cost-per-information score that summarizes each provider's provision cost per unit of information about the buyer's estimation objective. When quality is known ex ante, we describe a second-score procurement mechanism that ranks providers by this score, and endogenously chooses both a provider and a sample size while making truthful cost reports optimal. We then turn to the more realistic setting where data quality is private, and can only be assessed noisily via the delivered data. In this setting, we propose a simple mechanism that augments the second-score rule with a lenient ex post statistical test of the reported quality. We prove that, under mild conditions, the mechanism admits near-truthful reports that become approximately optimal and individually rational as the procured sample size grows. Under these near-truthful reports, the buyer asymptotically recovers the performance of the corresponding known-quality second-score mechanism. Our analysis highlights how the choice of verification test and the buyer's accuracy-cost tradeoff jointly shape participation and misreporting incentives in data markets.

SVG generation is typically evaluated by comparing rendered outputs to reference images, which captures visual similarity but not the structural properties that make SVG editable, decomposable, and reusable. Inspired by the classical jackknife, we introduce element-level leave-one-out (LOO) analysis. The procedure renders the SVG with and without each element, which yields element-level signals for quality assessment and structural analysis. From this single mechanism, we derive (i) per-element quality scores that enable zero-shot artifact detection; (ii) element-concept attribution via LOO footprints crossed with VLM-grounded concept heatmaps; and (iii) four structural metrics: purity, coverage, compactness, and locality, which quantify SVG modularity from complementary angles. These metrics extend SVG evaluation from image similarity to code structure, enabling element-level diagnosis and comparison of how visual concepts are represented, partitioned, and organized within SVG code. Their practical relevance is validated on over 19,000 edits (5 types) across 5 generation systems and 3 complexity tiers.

Digital firms routinely run many online experiments on shared user populations. When product decisions are compositional, such as combinations of interface elements, flows, messages, or incentives, the number of feasible interventions grows combinatorially, while available traffic remains limited. Overlapping experiments can therefore generate interaction effects that are poorly handled by decentralized A/B testing. We study how to design large-scale factorial experiments when the objective is not to estimate every treatment effect, but to identify a high-performing policy under a fixed experimentation budget. We propose a two-stage design that centralizes overlapping experiments into a single factorial problem and models expected outcomes as a low-rank tensor. In the first stage, the platform samples a subset of intervention combinations, uses tensor completion to infer performance on untested combinations, and eliminates weak factor levels using estimated marginal contributions. In the second stage, it applies sequential halving to the surviving combinations to select a final policy. We establish gap-independent simple-regret bounds and gap-dependent identification guarantees showing that the relevant complexity scales with the degrees of freedom of the low-rank tensor and the separation structure across factor levels, rather than the full factorial size. In an offline evaluation based on a product-bundling problem constructed from 100 million Taobao interactions, the proposed method substantially outperforms one-shot tensor completion and unstructured best-arm benchmarks, especially in low-budget and high-noise settings. These results show how centralized, policy-aware experimentation can make combinatorial product design operationally feasible at platform scale.

We study identification of a structural group effect when the group indicator $G\in\{0,1\}$ is unobserved but the analyst observes a calibrated probability score $p$ satisfying $E[G\mid p,X]=p$. Under a constant-coefficient structural mean model, the latent-group coefficient $\tau$ is point-identified by a closed-form ratio of observable moments whose denominator is the residual score variance $V^{*}=E[(p-E[p\mid X])^2]$. Identification fails exactly when the score is a deterministic function of $X$; we construct an explicit continuum of observationally equivalent models showing the failure is genuine. The marginal latent mean gap decomposes as $\tau$ plus a compositional term that is itself identified in closed form, and we characterise when the two coincide. The oracle estimator is $\sqrt{n}$-consistent and asymptotically normal with a closed-form sandwich variance. Under calibration error bounded by $\delta$, the bias obeys a sharp bound proportional to $\delta/V^{*}$, and hard-threshold classification attenuates the estimated gap. Monte Carlo experiments confirm the theory, including the variance-weighted estimand under heterogeneous effects.

We leverage historical outage data to quantify the resilience benefits of undergrounding a circuit. The historical performance of the overhead circuit is compared to the performance if the circuit had been undergrounded in the past. The number of outages, customers affected, outage duration, and customer hours lost are used as metrics to quantify the benefits of undergrounding. Results show 75% and 78% reductions in customer hours lost per year for two selected circuits, as well as a significant reduction in the average number of outages and customers affected per year, highlighting the advantages of undergrounding. The benefits of investments that result in 10% faster outage restoration are also calculated by rerunning history with the faster restoration included.

Computational models support high-stakes decisions across engineering and science, and practitioners increasingly seek probabilistic predictions to quantify uncertainty in such models. Existing approaches generate predictions either by sampling input parameter distributions or by augmenting deterministic outputs with uncertainty representations, including distribution-free and distributional methods. However, sampling-based methods are often computationally prohibitive for real-time applications, and many existing uncertainty representations either ignore input dependence or rely on restrictive Gaussian assumptions that fail to capture asymmetry and heavy-tailed behavior. Therefore, we extend the ACCurate and Reliable Uncertainty Estimate (ACCRUE) framework to learn input-dependent, non-Gaussian uncertainty distributions, specifically two-piece Gaussian and asymmetric Laplace forms, using a neural network trained with a loss function that balances predictive accuracy and reliability. Through synthetic and real-world experiments, we show that the proposed approach captures an input-dependent uncertainty structure and improves probabilistic forecasts relative to existing methods, while maintaining flexibility to model skewed and non-Gaussian errors.

Hypergraph data are often projected onto a weighted graph by constructing an adjacency matrix whose $(i,j)$ entry counts the number of hyperedges containing both nodes $i$ and $j$. This reduction is computationally convenient, but it can lose information: distinct hypergraphs may induce the same matrix, and the matrix entries are generally dependent because each hyperedge contributes to multiple pairs. We study the planted clique problem under this matrix-only observation model. For detection, we show that a spectral norm test is asymptotically powerful at the $\sqrt{n}$ scale, with explicit dependence on the background hyperedge probability $p$. For recovery, we analyze a polynomial-time spectral method based on the leading eigenvector and prove exact recovery at the canonical $\sqrt{n}$ scale, again with explicit dependence on $p$. We also extend both results to sparse regimes in which the hyperedge probability may depend on \(n\). Our analysis adapts a leave–one–out eigenvector framework to this setting. These results provide rigorous detection and recovery guarantees when only the adjacency matrix is observed.

How can we conduct causal inference when the outcome of interest is latent and imperfectly measured by multiple indicators? We develop a nonparametric framework for identifying and estimating average treatment effects on latent outcomes in randomized experiments. Researchers commonly aggregate multiple outcome indicators into a summary measure before estimating treatment effects. We identify a fundamental but often overlooked comparability problem arising from this practice: adding, removing, or replacing indicators can alter the causal estimand. We formalize this problem and show that commonly used procedures generally fail to preserve comparability across measurement sets. To address this problem, we propose a design-based approach that uses nonparametric bridge functions. We further characterize the semiparametric efficiency bound, construct an efficient estimator, and establish necessary and sufficient conditions under which additional measurements yield strict efficiency gains. We illustrate our approach through an application to the Hajj visa lottery. Our framework provides practical guidance for measurement design, identification, and efficient causal estimation with latent outcomes.

Time-series stationarity is a property that statistical characteristics such as trend, variance, seasonality remain constant over time. It is considered fundamental to many forecasting and analysis methods. Different tests detect different types of non-stationarity: structural breaks or deterministic trends, clustered or time-dependent variance, stochastic or deterministic seasonality. A series might pass one test while failing another; single-test approaches seldom distinguish between conceptually different types of non-stationarity that require different types of tests and transformations. `StationarityToolkit` addresses this by providing a comprehensive Python library that runs 10 statistical tests across three categories: trend (4 tests), variance (4 tests), and seasonality (2 tests). Rather than a binary stationary/non-stationary verdict, users receive detailed diagnostics with actionable notes for each detection. The toolkit automatically infers the frequency of the data provided (requires datetime index), provides clear interpretations with test statistics and p-values, and supports an iterative test-transform-retest workflow essential for real-world data sets.

Network segmentation is a foundational enterprise security control. Despite its recognized benefits, segmentation initiatives frequently fail in practice, and the field lacks a systematic empirical explanation for why these projects do not achieve their intended outcomes. This paper presents an empirical study of failed segmentation projects based on a survey of 400 U.S.-based\ network security practitioners. The survey was grounded in a two-part failure framework that separately measures general IT project failure factors and segmentation-specific technical and operational barriers. Clustering analysis of the responses reveals four distinct failure archetypes, ranging from projects in which every factor contributed simultaneously to projects where governance was comparatively better but specific segmentation challenges proved decisive. The archetypes correspond to real differences in how segmentation was attempted: projects that included campus networks were more likely to experience the broadest or most technically intense failures. On the other hand, the archetypes did not differ significantly by workload type.

Benign overfitting, the phenomenon that interpolating noisy data can be statistically harmless, is usually characterized through effective-rank conditions on the covariance spectrum. These conditions are fragile: a perturbation of vanishing relative operator and Frobenius norm can turn a benign covariance into one whose minimum-norm interpolator has non-vanishing excess risk. We argue that the natural geometry of benign overfitting is optimal transport of the log-eigenvalues, measured in units of the sample size $n$. In log-spectral coordinates, the deterministic equivalent of the risk becomes a functional of two measures smoothed by a logistic kernel centred at the interpolation threshold $\log\kappa$. We compute its exact first variation: the threshold moves by a sensitivity-weighted average of the log-eigenvalue displacements, and the influence of each eigenvalue decays exponentially with its log-distance to the threshold. We then derive global, dimension-free stability bounds. The variance functional is $8/(27n)$-Lipschitz in the 1-Wasserstein distance between log-spectra, with a sharp constant, and is controlled more generally by a threshold-localized transport discrepancy; an analogous bound controls the bias. Consequently, any rank-$k$ update of the covariance, of arbitrary norm, changes the variance functional by at most $3k/n$, and benign overfitting survives every perturbation whose localized transport cost is $o(n)$. The results extend to ridge regression and to infinite-dimensional trace-class covariances. Matching lower bounds show that the $n$-scaling is necessary and cannot be replaced by operator norms, Frobenius norms, or dimension-normalized spectral distances. Simulations, one-step feature learning in two-layer networks, and random features of real images support the theory and delineate its limits.

Large language models (LLMs) can struggle to memorize factual knowledge in their parameters, often leading to hallucinations and poor performance on knowledge-intensive tasks. In this paper, we formalize fact memorization from an information-theoretic perspective and study how training data distributions affect fact accuracy. We show that fact accuracy is suboptimal (below the capacity limit) whenever the amount of information contained in the training data facts exceeds model capacity. This is further exacerbated when the fact frequency distribution is skewed (e.g. a power law). We propose data selection schemes based on the training loss alone that aim to limit the number of facts in the training data and flatten their frequency distribution. On semi-synthetic datasets containing high-entropy facts, our selection method effectively boosts fact accuracy to the capacity limit. When pretraining language models from scratch on an annotated Wikipedia corpus, our selection method enables a GPT2-Small model (110m parameters) to memorize 1.3X more entity facts compared to standard training, matching the performance of a 10X larger model (1.3B parameters) pretrained on the full dataset.

Recent advances in single-cell technologies have advanced our understanding of gene regulation and cellular heterogeneity at single-cell resolution. Single-cell data contain both gene expression levels and the proportion of expressing cells, which makes them structurally different from bulk data. Currently, methodological work on causal mediation analysis for single-cell data remains limited and often requires specific distributional assumptions. To address this challenge, we present QuasiMed, a mediation framework specialized for single-cell data. Our proposed method comprises three steps, including (i) screening mediator candidates through penalized regression and marginal models (similar to sure independence screening), (ii) estimation of indirect effects through the average expression and the proportion of expressing cells, (iii) and hypothesis testing with multiplicity control. The key benefit of QuasiMed is that it specifies only the mean functions of the mediation models through a quasi-regression framework, thereby relaxing strict distributional assumptions. The method performance was evaluated through the real-data-inspired simulations, and demonstrated high power, false discovery rate control, and computational efficiency. Lastly, we applied QuasiMed to ROSMAP single-cell data to illustrate its potential to identify mediating causal pathways. R package is freely available on GitHub repository at https://github.com/sjahnn/QuasiMed.

We initiate the study of language generation in the limit, a model recently introduced by Kleinberg and Mullainathan [KM24], under the constraint of differential privacy. We consider the continual release model, where a generator must eventually output a stream of valid strings while protecting the privacy of the entire input sequence. Our first main result is that for countable collections of languages, privacy comes at no qualitative cost: we provide an $\varepsilon$-differentially-private algorithm that generates in the limit from any countable collection. This stands in contrast to many learning settings where privacy renders learnability impossible. However, privacy does impose a quantitative cost: there are finite collections of size $k$ for which uniform private generation requires $\Omega(k/\varepsilon)$ samples, whereas just one sample suffices non-privately. We then turn to the harder problem of language identification in the limit. Here, we show that privacy creates fundamental barriers. We prove that no $\varepsilon$-DP algorithm can identify a collection containing two languages with an infinite intersection and a finite set difference, a condition far stronger than the classical non-private characterization of identification. Next, we turn to the stochastic setting where the sample strings are sampled i.i.d. from a distribution (instead of being generated by an adversary). Here, we show that private identification is possible if and only if the collection is identifiable in the adversarial model. Together, our results establish new dimensions along which generation and identification differ and, for identification, a separation between adversarial and stochastic settings induced by privacy constraints.

We propose a flexible Bayesian approach for estimating the joint density of a multivariate random variable of interest in the presence of categorical covariates. Leveraging a Gaussian copula framework, our method captures the dependence structure across different coordinates of the multivariate response. The conditional (on covariates) marginal (across outcomes) distributions are modeled as flexible mixtures with shared atoms across coordinates. The mixture weights are allowed to vary with covariates through a novel Tucker tensor factorization-based structure which enables the identification of coordinate-specific subsets of influential covariates. In particular, we replace the traditional mode matrices with coordinate-specific random partition models on the covariate levels, offering a flexible mechanism to aggregate covariate levels that exhibit similar effects on the response. We introduce a Markov chain Monte Carlo algorithm that scales with the number of aggregated levels rather than the original levels, significantly reducing memory requirements and improving computational efficiency. We demonstrate the method's numerical performance through simulation experiments and its practical applicability through the analysis of NHANES dietary data.

What are the limits of controlling language models via synthetic training data? We develop a reinforcement learning (RL) primitive, the Dataset Policy Gradient (DPG), which can precisely optimize synthetic data generators to produce a dataset of targeted examples. When used for supervised fine-tuning (SFT) of a target model, these examples cause the target model to do well on a differentiable metric of our choice. Our approach achieves this by taking exact data attribution via higher-order gradients and using those scores as policy gradient rewards. We prove that this procedure closely approximates the true, intractable gradient for the synthetic data generator. To illustrate the potential of DPG, we show that, using only SFT on generated examples, we can cause the target model's LM head weights to (1) embed a QR code, (2) embed the pattern $\texttt{67}$, and (3) have lower $\ell^2$ norm. We additionally show that we can cause the generator to (4) rephrase inputs in a new language and (5) produce a specific UUID, even though neither of these objectives is conveyed in the generator's input prompts. These findings suggest that DPG is a powerful and flexible technique for shaping model properties using only synthetic training examples.

When designing and evaluating an experiment or observational study, it is useful to have a realistic hypothesis regarding the average treatment effect. We present an approach to conceptualizing this average by first considering a distribution of effects. We demonstrate with examples in medicine, economics, and psychology.

Out-of-distribution (OoD) generalization occurs when representation learning encounters a distribution shift. This occurs frequently in practice when training and testing data come from different environments. Covariate shift is a type of distribution shift that occurs only in the input data, while the concept distribution stays invariant. We propose RIA - Regularization for Invariance with Adversarial training, a new method for OoD generalization under convariate shift. Motivated by an analogy to $Q$-learning, it performs an adversarial exploration for counterfactual data environments. These new environments are induced by adversarial label invariant data augmentations that prevent a collapse to an in-distribution trained learner. It works with many existing OoD generalization methods for covariate shift that can be formulated as constrained optimization problems. We develop an alternating gradient descent-ascent algorithm to solve the problem in the context of causally generated graph data, and perform extensive experiments on OoD graph classification for various kinds of synthetic and natural distribution shifts. We demonstrate that our method can achieve high accuracy compared with OoD baselines.

Systematic investment strategies are exposed to a subtle but pervasive vulnerability: the progressive erosion of their effectiveness as market regimes change. Traditional risk measures, designed to capture volatility or drawdowns, overlook this form of structural fragility. This article introduces a quantitative framework for assessing the durability of systematic strategies through minimum regime performance (MRP), defined as the lowest realized risk-adjusted return across distinct historical regimes. MRP serves as a lower bound on a strategy's robustness, capturing how performance deteriorates when underlying relationships weaken or competitive pressures compress alpha. Applied to a broad universe of established factor strategies, the measure reveals a consistent trade-off between efficiency and resilience – strategies with higher long-term Sharpe ratios do not always exhibit higher MRPs. By translating the persistence of investment efficacy into a measurable quantity, the framework provides investors with a practical diagnostic for identifying and managing strategy-decay risk, a novel dimension of portfolio fragility that complements traditional measures of market and liquidity risk.

Background: Mediation analysis is widely used to investigate how treatments and programs exert their effects, but standard ordinary least squares (OLS) inference can be unreliable when regression errors are non-Gaussian. In medical and public-health studies, this can affect whether indirect and direct effects are judged clinically or scientifically meaningful. Methods: We developed a semiparametric causal mediation framework for linear models allowing possibly non-Gaussian errors, covering both standard models and models with treatment–mediator interaction. The method combines semiparametric efficient regression estimation, a reproducible multi-start fitting algorithm for numerical stability, and stacked estimating equations for confidence-interval construction without requiring Gaussian error assumptions. Results: Across Gaussian, skewed, and mixture-error simulations, the semiparametric estimator reduced root mean squared error and confidence-interval length relative to OLS, with the largest gains under non-Gaussian errors. In a near-boundary power design, the OLS confidence interval achieved 18.3% empirical power, whereas the semiparametric confidence interval identified significant effects in all replications. In the uis drug-treatment data, it yielded sharper treatment-specific effect estimates under clear treatment–mediator interaction. In the jobs social-program data, the semiparametric analysis produced shorter confidence intervals for mediated effects and detected nonzero mediation where OLS did not. Conclusions: Semiparametric mediation analysis can improve the precision and reliability of effect decomposition in studies with non-Gaussian outcomes, offering a practical alternative to OLS when indirect and direct effects may inform clinical or policy decision-making.

mmid (Multi-Modal Integration and Downstream analyses for healthcare analytics) is a Python package that offers multi-modal fusion and imputation, classification, time-to-event prediction and clustering functionalities under a single interface, filling the gap of sequential data integration and downstream analyses for healthcare applications in a structured and flexible environment. mmid wraps in a unique package several algorithms for multi-modal decomposition, prediction and clustering, which can be combined smoothly with a single command and proper configuration files, thus facilitating reproducibility and transferability of studies involving heterogeneous health data sources. A showcase on personalised cardiovascular risk prediction is used to highlight the relevance of a composite pipeline enabling proper treatment and analysis of complex multi-modal data. We thus employed mmid in an example real application scenario involving cardiac magnetic resonance imaging, electrocardiogram, and polygenic risk scores data from the UK Biobank. We proved that the three modalities captured joint and individual information that was used to (1) early identify cardiovascular disease before clinical manifestations with cardiological relevance, and (2) do it better than single data sources alone. Moreover, mmid allowed to impute partially observable data modalities without considerable performance losses in downstream disease prediction, thus proving its relevance for real-world health analytics applications (which are often characterised by the presence of missing data).

We propose a mathematical formalisation of the “wave model” originally developed in historical linguistics but with further applications in human sciences. This model assumes new traits appear in a population and spread to nearby populations depending on their closeness. It is mostly used to describe joint evolution of closely related populations, for example of several dialects. These situations of permanent contact are not accurately represented by its competitors based on tree structures. We built a fully Bayesian generative model where innovation spread along a fixed graph and disappear according to a death process. We then develop a Metropolis-Hastings within Gibbs sampler to sample from the posterior distribution on the graph. We test our method on simulated datasets as well as on several real dataset.

We consider a linear contextual bandit model where contexts and rewards are governed by a finite hidden Markov chain. We first revisit the simplified model by Nelson et al. (2022), in which rewards are linear functions of the posterior probabilities over the hidden states given the observed contexts (called beliefs), rather than functions of the hidden states themselves. This simplified model may be handled through a direct reduction to standard linear contextual bandits. We extend the theoretical analysis of this reduction to take into account the estimation of the parameters of the hidden Markov model [HMM] in the regret bound and to provide high-probability bounds not depending anymore on the reward functions and only depending on the model through the estimation of the HMM parameters. Second, and most importantly, we instead study the more natural and more complex model incorporating direct dependencies in the hidden states (on top of dependencies on the observed contexts, as is natural for contextual bandits). Under a classic HMM forgetting condition, the main algorithmic tool introduced to cope with the various statistical dependencies that the reward structure introduces is to only periodically update reward-model parameters.

We propose a novel entropy flow on weighted graphs, which provides a principled framework that characterizes the evolution of probability distributions over graph structures while sharing geometric intuition with discrete Ricci flow. We provide its rigorous formulation, establish its fundamental theoretical properties, and prove the long-time existence and convergence of its solutions. To demonstrate its applicability, we employ entropy flow for community detection in real-world networks. Empirically, it achieves detection accuracy fully comparable to that of discrete Ricci flow. Crucially, by avoiding computations of optimal transport distances and shortest paths, our approach overcomes the fundamental computational bottleneck of Ollivier and Lin-Lu-Yau Ricci flows. As a result, entropy flow requires only $1.61\%$-$3.20\%$ of the computation time of Ricci flow. These results indicate that entropy flow provides a theoretically rigorous and computationally efficient framework for large-scale graph analysis.

In the last decades, energy-based models (EBMs) have become an important class of probabilistic models in which a component of the likelihood is intractable and therefore cannot be evaluated explicitly. Consequently, parameter estimation in EBMs is challenging for conventional inference methods. In this work, we provide a unified framework that connects noise contrastive estimation (NCE), reverse logistic regression (RLR), multiple importance sampling (MIS), and bridge sampling within the context of EBMs. We further show that these methods are equivalent under specific conditions. This unified perspective clarifies relationships among existing methods and enables the development of new estimators, with the potential to improve statistical and computational efficiency. Furthermore, this study helps elucidate the success of NCE in terms of its flexibility and robustness, while also identifying scenarios in which its performance can be further improved. Hence, rather than being a purely descriptive review, this work offers a unifying perspective and additional methodological contributions. The MATLAB code used in the numerical experiments is also made freely available to support the reproducibility of the results.

Background: Composite endpoints in cardiovascular trials combine heterogeneous outcomes-mortality, nonfatal events, hospitalizations, and biomarkers-yet conventional analytical methods sacrifice information by targeting a single dimension. Cox time-to-first-event ignores post-first-event data; Win Ratio discards tied pairs; negative binomial regression treats death as noninformative censoring. Methods: We propose CWOT-CE: a Choquet integral-based composite endpoint analysis that encodes K = 6 outcome dimensions-survival, event-free time, AUC recurrent burden, last event time, biomarker, and alive status-and aggregates them through a non-additive fuzzy measure with pairwise interaction terms. The recurrent event process is represented as two complementary scalar summaries: the area under the cumulative count curve (AUC burden) and the last event time. Inference is via permutation test with exact finite-sample Type I error control and dual confidence interval by inversion. We conducted a simulation study comparing CWOT-CE against Cox TTFE, Win Ratio (WRrec), and WLW across 20 clinically motivated scenarios (1,000-5,000 replications). Results: Under the sharp null (5,000 replications), all methods maintained nominal Type I error (CWOT-CE: 4.8%, MCSE 0.3%). Across 17 non-null scenarios, CWOT-CE outperformed Cox TTFE in 15 (mean +28.8 pp), WLW in 14 (mean +27.2 pp), and Win Ratio in 10, with 5 ties and only 2 narrow losses (mean +5.6 pp). CWOT-CE showed particular advantages in high-correlation settings (+35.4 pp vs. WR), mortality-driven effects (+10.7 pp), and balanced multi-component effects (+10.1 pp). Shapley decomposition correctly identified effect-bearing components across all calibration scenarios. Conclusions: CWOT-CE with block recurrent encoding is broadly effective across clinically relevant scenarios while offering unique interpretive advantages through component attribution.

In this work, we analyze 126 publicly available IAM climate scenarios modeled by six leading teams in climate science. We define a simple numerical metric that measures the decarbonization speed implied by each IAM scenario. With this metric, the narrative based, high-dimensional time series scenario datasets can be ranked and compared in a transparent way. We find that the ranking of IAM scenarios according to the decarbonization speed is consistent with their representative concentration pathway assumptions, showing that the decarbonization metric is a useful summary of a scenario's mitigation policy. We further construct an empirical distribution and a fitted parametric distribution of the decarbonization speed estimates. Key statistics such as mean, median and their confidence intervals by the bootstrap resample technique are also reported.