Papers

Question-order effects in human survey data have been reported to approximately satisfy the QQ (quantum question) equality, a parameter-free prediction of the standard projective quantum question-order model. We develop this equality into an audit framework for sequential binary judgments of autoregressive large language models (LLMs). Theoretically, we characterize mechanism families that satisfy QQ robustly, show that classical repetition can reproduce the equality exactly, and combine QQ with the rank-2 Contextuality-by-Default criterion through $|q_{QQ}| \le \mathrm{OSS}$. This separates order sensitivity, QQ imbalance, and residual contextuality rather than treating them as interchangeable signatures. Methodologically, we introduce a committed multi-turn forced-branch protocol that reconstructs order-conditioned joint distributions from next-token log-probabilities under counterbalanced label mappings and pre-specified health gates. A first-signal pilot on an open-weight instruction-tuned model reveals the central measurement problem. Although all pre-specified health gates passed, the binary-conditioned distributions were near-deterministic for 17 of 18 item pairs under the direct-evaluation framing and 7 of 8 under the persona framing. Label assignment materially changed several mapping-specific QQ verdicts, and no item was certified as residually contextual. Thus, under the tested conditions, the observed QQ outcomes did not uniquely identify a response mechanism in the presence of a saturated and label-sensitive measurement interface. The main implication is methodological: next-token probabilities should not be interpreted as survey-response distributions without first establishing adequate dispersion. We therefore argue that saturation screening and label counterbalancing should precede structural interpretation in distribution-level audits of LLM judgments.

In this work we study the Best Policy Identification (BPI) problem in online, tabular Reinforcement Learning. This is an active sequential hypothesis testing problem in which the learner's objective is to identify an optimal policy in a Markov Decision Process (MDP) with high confidence, while minimizing the expected sample complexity to do so. We consider an online setting with deterministic rewards, where the agent must strategically navigate through the MDP in order to effectively explore. Previous works in the literature have provided asymptotically optimal methods for BPI, such as the Navigate and Stop (NaS) algorithm and its variants, however existing analysis remains asymptotic. In this work, we fill that gap by providing the first non-asymptotic sample complexity guarantees for NaS, showing that its sample complexity depends not only on the characteristic time, but also on the connectivity of the underlying MDP, the curvature of the optimal characteristic time, and other instance-dependent quantities. We identify these additional attributes and make explicit their contributions to the overall sample complexity.

The use of ratio variables in convex Data Envelopment Analysis (DEA) models has long been recognized as a methodological issue, as ratios may be incompatible with the convexity assumptions underlying standard DEA. However, in this paper, we prove that if all variables are ratio variables sharing a common denominator, then convex combinations of feasible activities are also feasible. Moreover, we establish a result demonstrating the equivalence between DEA models with common-denominator ratio variables under variable returns to scale and DEA models with volume (non-ratio) variables under constant returns to scale. These significant results enable the development of a framework for evaluating the efficiency of the Organisation for Economic Co-operation and Development (OECD) countries based on the results of the Programme for International Student Assessment (PISA) report, using mean performance scores as outputs. In this framework, we give some methodological extensions, such as the incorporation of the index of economic social and cultural status (ESCS) as an input, thereby enabling fairer comparisons with countries with a lower socio-economic level. Furthermore, we introduce different methods for estimating directions of improvement and calculating targets appropriate to the difficulty of improving each performance score. Finally, we review and introduce several novel contributions to emerging methodologies that can complement classical radial and directional models, such as efficient frontier estimation with adaptive constrained enveloping splines (ACES), stochastic chance-constrained models, and fuzzy models. All these methodologies can be used to analyse data from other PISA or similar reports, allowing non-specialists to implement DEA appropriately.

The randomness, efficiency (power and variability), and desirable allocation proportions are important components for evaluating a response-adaptive design in clinical trials and conflicted demands in applications. The aim of this paper is to provide designs dealing with these dilemmas. We first give a general framework for efficient response-adaptive randomization procedures that attain the Cramér-Rao lower bounds of the allocation variances for any desired allocation proportions. The general framework is flexible for us to define new families of efficient designs with good properties for both two and multiple-treatment clinical trials. We also prove that, among all response-adaptive randomization procedures with the same limit allocation proportions, the selection biases and entropies as measures of the randomness of the designs have their optimal values. Basing on the theory on efficiency and randomness, we propose a new family of doubly adaptive biased coin designs for multi-treatment clinical trials that can target any allocation proportion and are asymptotically best in terms both the randomness and efficiency so that their randomness is asymptotic optimal and asymptotic allocation variance attains the Cramér-Rao lower bound. Theoretical properties, including the strong consistency, the asymptotic normality, and the functional central limit theorem for both the sample allocation proportions and the estimators of the distribution parameters, are developed by using the technique of Gaussian approximation and Gaussian comparing theorems.

In this work, we develop a unified matrix-level asymptotic calculus for plug-in non-parametric maximum likelihood estimators in finite Markov models. Starting from the asymptotic distribution of the estimated transition matrix, the limiting object is kept in its natural matrix form as a Gaussian random matrix, while the corresponding row-wise vector representation remains immediately available. The main point is that the stochastic constraints of the transition matrix need not be removed by a minimal parametrization: they are carried by the tangent directions and by the covariance structure of the limiting Gaussian matrix, whereas the relevant differentials are computed directly in matrix spaces. A single stochastic calculus theorem gives first-order limit distributions, finite-order developments for sufficiently differentiable functionals, and analytic expansions when the functional is analytic. This provides a common source for asymptotic formulas for matrix powers, stationary characteristics, finite-dimensional curves of Markov characteristics, additive-functional variances, entropy-type quantities and reliability indicators. The resulting covariance operators lead directly to confidence intervals, confidence regions, simultaneous finite-dimensional bands and Wald-type tests. Since the derivations are expressed through matrix products and Kronecker representations rather than coordinate-wise calculations, the method also gives substantial simplifications and, in many cases, computational gains. The second-order terms identify curvature corrections of smooth functionals and provide refined approximations whenever higher-order information is useful.

The planted subgraph detection problem asks whether a random graph contains a hidden structured subgraph. In the classical formulation, the entire adjacency matrix is observed and one distinguishes between an Erdős–Rényi random graph and one obtained by planting a copy of a prescribed graph inside an Erdős–Rényi random graph. The statistical and computational limits of this problem under full observation are now well understood, even for arbitrary planted subgraphs. In this paper, we investigate an information-limited version of the problem in which the planted structure is an arbitrary sequence of graphs $\Gamma=(\Gamma_n)_{n\geq1}$, where $\Gamma_n$ is embedded in an ambient graph on $n$ vertices, but the observer does not have access to the full adjacency matrix. Instead, information is acquired through a limited number of non-adaptive edge queries. We study the minimum query complexity required for reliable detection. We derive general information-theoretic lower bounds and complementary algorithmic upper bounds on the query complexity as functions of the query budget and structural properties of the planted graph. The proposed algorithms exploit three distinct structural mechanisms: dense local motifs, high-degree vertices, and global edge density. We establish matching bounds, up to polylogarithmic factors, for several broad families of planted graphs, including clique-like, bounded-cover, and hub-dominated graph classes. Our framework substantially generalizes existing query-complexity results for planted clique and planted dense subgraph models, providing a unified treatment of arbitrary planted subgraphs under restricted graph access.

Validating simulation models against historical output data is essential for their successful deployment in digital-twin environments. We propose a statistical validation framework in which a simulation model is repeatedly initialized from observed system states, and conditional output distributions are obtained by fixing the random primitives from a subset of stochastic input models to their observed realizations while simulating the remaining primitives. These conditional output distributions are then used in goodness-of-fit tests to validate the simulation model with respect to combinations of input models. We also develop diagnostic tools to identify the input models that most contribute to any observed misalignment between a simulation model's outputs and reality. Numerical experiments on an M/M/1 queueing system and a digital-twin-enabled simulation of a tandem queueing system demonstrate that the proposed framework can detect misspecifications in input models that may be missed by existing approaches that validate only the marginal output distribution.

Many multiple-testing procedures control the false discovery rate (FDR) by comparing the two tails of a null distribution. At a fixed cutoff, marginal symmetry makes this natural. Mirror and knockoff+ thresholds select the cutoff from the same data, so the standard finite-sample guarantee uses a stronger property: conditional on magnitudes and nonnull scores, null signs are independent fair coins. Failure can be severe without this property. Models satisfying positive regression dependence on a subset (PRDS) can have exactly uniform null $p$-values and large FDR. Under every fixed positive Gaussian equicorrelation, the all-null FDR converges to one half. Opposing loadings in Gaussian factor models can make FDR and power arbitrarily close to one; near-total failure also occurs for exchangeable, pairwise-uncorrelated scores. At a nominal input level $q<1/2$, no deterministic rule based only on the two tail counts can both reject and control FDR uniformly over our class if more discoveries or fewer controls cannot make rejection harder. We give finite-sample repairs based on joint sign information. Conditional sign odds may be bounded outside an exceptional event or averaged over negative controls; neither route uniformly dominates, and the integrated bounds are sharp. Independent calibration data or a specified Gaussian joint model yield valid adjusted levels. Simulations show that integration retains more power under diffuse Gaussian dependence, whereas exceptional-event calibration is more powerful when very large odds occur only for rare aligned signs; unadjusted FDR exceeds the target in both settings. Covariance alone is insufficient outside a specified joint model. Thus the relevant boundary is not marginal symmetry but joint information that remains valid after adaptive cutoff selection.

Before releasing software to a general population, a developer must weigh a single question: if we ship now, what fraction of users will still hit a defect? This is not a question about how many defects remain, nor whether any particular defect is present – the quantities the reliability literature has long estimated – but about a different and, for a release decision, more consequential one: the probability that a user encounters a defect at all. We give a direct, distribution-free answer. Reading each beta-test report as a draw from the user population and each distinct defect as a class, we show that the fraction of defects reported exactly once, $s/n$, is a conservative upper bound on the probability that a user encounters a defect unseen in testing. This bound is the exact maximum-likelihood estimate of the mass of unseen defects under a general urn construction – the canonical form – into which any population of classes embeds; because that construction charges every singleton to the unseen reservoir, $s/n$ overstates the user's risk rather than understating it, the direction a release decision requires. The estimate needs no operational profile, no assumption on the number or frequency of defects, and no model of the program's internal structure – since a defect's report count already reflects how many users reach it, the estimate is invariant to whether the reachability graph is a tree or a directed acyclic graph, and defects hidden behind other defects are bounded automatically. We validate the estimator against synthetic populations with known ground truth, and discuss the encounter-level data – beta or crash telemetry – under which the user-facing reading holds.

Let $X_1, X_2, \ldots, X_n$ be independent random vectors. For a directed graph $G=(V,E)$ with vertex set $V=\{1,2,\ldots,n\}$ and a collection of bivariate kernels $\{h_e:e\in E\}$, we consider \[ U=\sum_{e=(i,j)\in E} h_e(X_i,X_j). \] This framework generalizes incomplete U-statistics by allowing the random vectors to be non-identically distributed, the kernels to be asymmetric and edge-dependent, and the sampling structure to be specified by an arbitrary graph. We derive several concentration inequalities for $U-\mathbb{E}U$. The main proof strategy exploits edge-coloring results from graph theory and relates the tail behavior of $U$ to the chromatic index of $G$. This approach is elementary, transparent, and readily adaptable to broader settings, including U-statistics of order $m>2$ and statistics involving doubly indexed random vectors.

Public blockchains can make many trading venues simultaneously visible and mechanically reachable, yet an order still has to pay to activate each additional venue: technological connectivity need not translate into economically integrated execution. Automated-market-maker (AMM) pools make this gap directly measurable, because exact pre-trade venue states, transaction-level routing costs, and realized venue use can be reconstructed jointly from public blockchain records. Using 13,768 sampled family-transactions from 29 Ethereum and Base token-pair families and 71 sibling pools, we document a sharp compression from gross to net execution opportunity: the equal-chain lower bound on gross multi-venue gains is 34.85%, while the corresponding upper bound after measured access costs is 6.89%, as access costs eliminate 80.45% of point-identified states with positive gross gains. Realized routing shows a distinct second contrast: actual multi-pool activation occurs in only 1.203% of the opportunity population, yet among 170 eligible realized integrated routes, observed allocation captures 94.8% of aggregate feasible gain. Holding recipient orders and exact venue states fixed, replacing the Ethereum routing-cost regime with Base's lower-cost regime materially expands executable integration under both state populations. For digital-market design, the results imply that blockchain scaling, cross-venue connectivity, and displayed liquidity do not by themselves establish economically integrated execution: integration is an order-specific property that depends on transaction-level access costs.

When applying synthetic control to compositional outcomes (budget, vote shares), researchers commonly minimize Euclidean distances between raw shares. I propose constructing synthetic controls in Aitchison geometry, using centered log-ratio coordinates to select donor weights and form counterfactuals. This approach respects proportional comparisons and is exactly invariant to common multiplicative changes in relative odds. Under a multinomial-choice model, its weights summarize similarity in relative utility indices. Monte Carlo simulations, including a CES allocation model, show that this method performs better when relative incentives determine shares. An application to U.S. school finance demonstrates that this choice can alter reported inference.

Over the past few years, diffusion-based Schrödinger bridge models have been proposed to approximate optimal transport dynamics between two prescribed boundary distributions, with successful applications to generative modeling. More precisely, these methods aim to estimate a path measure whose initial and terminal marginals match the two boundary distributions, while minimizing the Kullback-Leibler divergence with respect to a reference Markov process. In this work, we consider the generalized Schrödinger bridge problem, in which the reference process is a twisted Brownian motion, that is, a Feynman-Kac transform of a Brownian motion induced by a time-dependent differentiable potential. Building on the Iterative Markovian Fitting (IMF) paradigm, and in particular on its special case Diffusion Schrödinger Bridge Matching (DSBM), which corresponds to the zero potential case, we introduce Twisted Schrödinger Bridge Matching (TSBM), a diffusion-based method designed to handle both continuous- and discrete-time potentials. Unlike previous approaches, TSBM provides a rigorous extension of the IMF scheme to the generalized Schrödinger bridge problem. This derivation leads to a new bridge-matching loss that depends explicitly on the gradient of the potential and recovers the DSBM objective when the potential vanishes, yielding improved performance. We further introduce trajectory-based variance-reduction techniques that substantially stabilize optimization and may be useful beyond the present setting. Finally, we empirically demonstrate the benefits of TSBM for trajectory inference across increasingly high-dimensional settings, including crowd navigation and single-cell data. Code available at https://github.com/maxencenoble/twisted-sb-matching.

We propose a unified ridge-regularized Hotelling framework for detecting and locating mean changes in functional time series. A growing basis expansion converts the functional observations into high-dimensional score vectors. Their long-run covariance is estimated by an edge-corrected difference-based procedure. Ridge regularization stabilizes inference under spectral decay. An explicit local-power formula shows that the power-maximizing ridge depends on the unknown spectral orientation of the change. We therefore combine a family of ridge CUSUM statistics by a Cauchy transform and calibrate the aggregate directly from their joint weighted-bridge limit. For multiple changes, we embed local maximum-ridge statistics in a wild binary segmentation procedure, followed by local refinement. Under mild conditions, we establish the validity, local power, consistency, and localization properties of the proposed tests. In the multiple-change setting, the procedure consistently recovers the number of changes and uniformly estimates their locations. The framework accommodates weak dependence and non-Gaussian functional errors. Simulations and two empirical applications demonstrate the favorable finite-sample performance of the proposed methods.

We present a method for calibrating a computer model using non-parametric techniques where the inputs are stochastic but include calibration parameters whose distributions are unknown and control parameters whose distributions are specified. Our solution gives a distributional estimate over the input space that is consistent with observed field data, while also preserving the distribution of the known marginal of the control parameters. This property is desirable since stochastic inputs often include physical processes affecting the experimental conditions, and a scientifically plausible calibration estimate should preserve well-established distributional properties of these inputs. The method builds on recently developed non-parametric computer model calibration techniques based on the disintegration of measure and Bayesian inference.

Estimating entropy from samples is fundamental in information theory and property testing. Shannon entropy measures average uncertainty and can be estimated to constant additive accuracy over a $k$-symbol alphabet using $\Theta(k/\log k)$ samples. Min-entropy depends only on the most likely symbol. Both are special cases of order-$\alpha$ R'{e}nyi entropy, $H_\alpha$. We characterize the sample complexity of estimating min-entropy and R'{e}nyi entropy for $k$ and integer $\alpha>1$; our lower bounds also hold for noninteger $\alpha\ge1.001$. We prove that min-entropy estimation to constant additive accuracy has sample complexity $\Theta(k\log k)$. The upper bound uses the largest empirical frequency and concentration via dyadic grouping. The matching lower bound hides a slightly heavier symbol at a uniformly random location. Thus, min-entropy requires $\Theta(\log^2 k)$ more samples than Shannon entropy and corrects a previously stated $\Theta(k/\log k)$ characterization. For every integer $2\le\alpha\le c_0\log k$, we prove the matching fixed-accuracy bound $\Theta_{c_0}(\alpha k^{1-1/\alpha})$. Previous results gave $\Omega_\alpha(k^{1-1/\alpha})$ for fixed integer $\alpha>1$ and $O_{c_0}(\alpha^2k^{1-1/\alpha})$ for all integer $\alpha>1$. Our upper bound analyzes an unbiased falling-factorial estimator based on $\alpha$-way collisions, while a hidden-heavy-coordinate construction gives the matching lower bound and shows that the factor $\alpha$ is unavoidable. For every real $1.001\le\alpha\le c_0\log k$, we prove the uniform lower bound $\Omega_{c_0}(\alpha k^{1-1/\alpha})$. Finally, since $0\le H_\alpha(p)-H_\infty(p)\le\log k/(\alpha-1)$, min-entropy uniformly approximates $H_\alpha$ when $\alpha$ is a sufficiently large multiple of $\log k$. Combining this reduction with our min-entropy bounds gives $\Theta_\varepsilon(k\log k)$ sample complexity in the high-order regime.

Traditional randomized trials often obscure clinically meaningful heterogeneity in treatment response by focusing on average effects. Leveraging real-world data to emulate clinical trials and estimate heterogeneous treatment effects (HTEs) offers a promising path toward more precise and efficient trial design. In this study, we emulate the DAPA-HF trial using electronic health records from the Mayo Clinic Cloud (MCC) to investigate whether HTE-guided stratification can identify patient subgroups with distinct treatment responses to dapagliflozin versus placebo in patients with heart failure with reduced ejection fraction. All-cause mortality was evaluated using Cox proportional hazards models, with HTEs estimated using a Meta-S learner and subgroups defined using a decision tree-based thresholding approach. In the overall cohort of the emulation, no significant treatment difference was observed (HR, 1.681; 95% CI, 0.828-3.413; p = 0.1507). However, compared with the overall emulated cohort, in which dapagliflozin showed no statistically significant survival benefit, HTE-driven stratification identified subgroups with significant and directionally distinct treatment effects. The beneficial (low-HTE) subgroup showed a significant survival benefit from dapagliflozin (HR = 0.203, 95% CI, 0.087-0.476, p = 0.0002), whereas the harmful (high-HTE) subgroup showed a significant harmful association with markedly increased mortality risk (HR = 6.680, 95% CI, 2.759-16.171, p < 0.0001). These findings indicate that HTE-guided stratification can uncover clinically meaningful beneficial and harmful treatment-effect patterns that are masked in the full-cohort emulation.

We introduce Deep Adaptive Bayesian Screening (DABS), a method for performing adaptive factorial screening in high-dimensional discrete design spaces. DABS learns a policy network offline to sequentially select informative experiments, amortizing Bayesian Optimal Experimental Design. It handles binary designs, incorporates sparsity and interactions via a spike-and-slab prior with strong heredity. The model is trained using a contrastive lower bound on information about factor activity with nuisance effect sizes and noise variance analytically integrated out. Unlike prior amortized Bayesian design approaches, DABS also integrates Gibbs posterior inference at deployment, yielding posterior probabilities of factor activity and credible intervals on effect sizes. We demonstrate DABS on screening problems calibrated to real-world benchmarks and show it achieves superior accuracy and scalability over classical and Bayesian baselines under tight experimental budgets.

Covariance steering provides an efficient framework for designing linear stochastic feedback policies, but its extension to nonlinear systems relies on a Gaussian surrogate obtained through local linearization. Because this surrogate may differ substantially from the true nonlinear state distribution, risk-sensitive quantities such as collision probability and mean-squared error may be inaccurately estimated. This work develops a distributionally robust covariance-steering framework based on the relative entropy, also known as the Kullback-Leibler divergence (KLD), to account for ambiguity in the propagated probability density function. Using a variational representation of exponential integrals, we derive computable upper bounds on risk-sensitive quantities over a KLD ambiguity set. We then formulate an upper bound on the time rate of change of the KLD between the true nonlinear distribution and a Gaussian reference surrogate. Under some assumptions, this bound is controlled by decision variables within a covariance-steering formulation. The resulting constraints are incorporated into a sequential convex programming algorithm to design stochastic guidance policies that keep the true distribution close to its Gaussian surrogate while enforcing bounds on risk-sensitive performance measures. The proposed approach is demonstrated on a challenging nonlinear spacecraft transfer between two near-rectilinear halo orbits.

We discuss Bayesian estimation of the exponentiated Kumaraswamy distribution, based on a location-scale-shape parameterisation. The parameterisation facilitates prior elicitation and interpretation of results, but potentially entails identifiability issues that are addressed through a hierarchical weakly informative prior setting. Our HMC implementation enables off-the-shelf utility for practitioners, and is illustrated on synthetic and real data.

Sparse-support uncertainty is usually quantified by treating the dictionary as known, an assumption that can produce overconfident, label-dependent conclusions when the dictionary is learned from latent sparse mixtures. Near collisions of coherent atoms, a test signal may identify the active physical group even though the training data cannot distinguish the physical rays within it. We develop inference for active physical rays, unit atoms modulo sign, after latent dictionary learning. In a fixed-dimensional Gaussian train-test experiment, we retain all dictionaries compatible with a robust training-moment region, profile the test representation over them, and project surviving configurations onto a permutation-invariant support space. The resulting confidence correspondence can report cross-sheet inconclusiveness, group resolution with child ambiguity, or fine-support resolution. We characterize both its statistical cost and decision-theoretic benefit. Residual block orientation first affects the latent training density at cubic order, yielding information of order $s^6$, where $s$ is the within-block collision scale. The correspondence provides high-probability-over-training conditional test coverage, with resolution governed separately by parent detectability, test-time support separation, and learned-dictionary orientation. In the resolved fixed-shell regime, its projective Hausdorff diameter contracts at the minimax-optimal rate $s \wedge (\sqrt{N}s^2)^{-1}$, up to constants. A restricted-task theorem further determines when coefficient asymmetry allows test replication to supplement training information and when calibration uncertainty remains irreducible. The framework thus yields honest, resolution-adaptive support statements and guides the allocation of training versus test measurements.

Deep survival models are evaluated almost exclusively by the concordance index (C-index), yet they are commonly trained using likelihood objectives such as the Cox partial likelihood, discrete-time negative log-likelihood, and DeepHit likelihood. This mismatch is usually considered acceptable because the C-index can be recomputed on validation data during training. However, for end-to-end training of high-capacity encoders on small, heavily censored oncology cohorts, frequent C-index evaluation is computationally expensive, making the loss value itself an important signal for monitoring, early stopping, and model selection. We show that likelihood losses are unreliable for this purpose and propose a value-monotone concordance loss. We prove that every strictly proper survival likelihood admits directions where the loss decreases while the C-index remains unchanged, causing the loss value to decouple from ranking performance. We then study a sigmoid concordance loss (SCL), whose value approximates one minus the C-index up to a temperature term, ensuring that lower loss corresponds to higher C-index during optimization. The loss is architecture agnostic and reduces to a convex survival ranking support vector machine for linear models. Across eighteen datasets from four modalities using a unified five-fold cross-validation protocol, SCL achieves discrimination comparable to standard likelihood losses and is the best or within one standard deviation of the best C-index. Unlike likelihood losses, SCL maintains a strong correlation between loss value and C-index during training, with rank correlations of 0.96 to 0.99 compared with -0.03 to 0.53 for likelihood losses. Calibration measured by the integrated Brier score is comparable. SCL provides a value-monotone optimization objective whose value can serve as a reliable surrogate for the C-index during expensive end-to-end training.

Simulating dependent Bernoulli outcomes with prescribed means and pairwise Pearson correlations is a common task in risk modeling. A familiar approach is the Gaussian-threshold workflow for binary outcomes, often viewed as a Bernoulli analogue of the Gaussian copula construction. We show that setting latent Gaussian correlations equal to target Bernoulli correlations is generally incorrect after thresholding, and that pairwise tetrachoric calibration is exact only when the calibrated latent matrix is positive semidefinite. We therefore formulate the problem directly over the joint Bernoulli probability mass function. Given target means and pairwise correlations, we impose normalization, nonnegativity, mean constraints, and pairwise cross-moment constraints as a linear program over the $2^N$ atomic probabilities. The resulting PMF formulation either returns an exact law matching the requested first and second moments or certifies infeasibility. A convex-hull characterization further shows that every feasible target admits a law supported on at most $1+N+\binom{N}{2}$ states, while every infeasible target admits a separating quadratic certificate. We then develop a truncated-moment completion scheme that fits a reduced cross-moment table and generates samples by sequential conditioning, together with a sparse-support working-set refinement that can reduce memory usage on structured instances, although the worst-case complexity remains exponential. Together, these constructions provide an exact PMF-based framework for feasibility and simulation at moderate dimension and structured alternatives when the full atomic representation is impractical, while clarifying the limits of Gaussian-threshold constructions.

Background: Mammographic density is a strong marker of breast cancer risk, yet it also reduces mammographic sensitivity through masking. Screening densities are observed at irregular times, requiring methods that accommodate irregular follow-up. Methods: We propose a two-phase framework for irregular mammographic screening data. After regularizing the irregular density histories, the first phase fits a latent Markov model within each BMI group, under the assumption that density does not itself drive risk but the underlying latent disease process does. Extracting the latent-state information leaves density to act only as a detectability factor, and the second phase links the latent process and the final density to cancer diagnosis through a detection–risk model. Detection probabilities are treated as sensitivity parameters to quantify how the estimated risk changes under different masking assumptions. Uncertainty in the inferred latent states is quantified via posterior path sampling and bootstrapping. Results: In 616 patients, higher parity was associated with faster movement toward lower-risk, while a family history of breast cancer was associated with longer persistence in higher-risk. Moving from ignoring masking to accounting for masking with empirically supported detectability scenarios raised the estimated breast cancer risk by 70% for post-menopausal overweight patients and about 40% for obese patients regardless of menopausal status. Conclusions: By attributing risk to latent progression while letting observed density operate through detectability, the framework decouples risk from detectability, quantifies how much masking can distort breast cancer risk, and provides interpretable risk characterization under irregular screening follow-up.

Dropout and Random Gradient Masking (RaM) are two training techniques used to improve performance in deep learning. Both techniques inject randomness into the training dynamics, but in significantly different ways: dropout applies random masks to the activations in the forward pass, whereas RaM leaves the forward pass unchanged and instead masks the gradients. In particular, the noise induced by RaM in the parameter updates is unbiased, so standard explanations for the effectiveness of dropout, such as the penalization effect or the prevention of co-adaptation between neurons, do not apply to RaM. In this work, we show that the difference between the two methods disappears for ResNets in the large depth and width asymptotics: in the complete feature learning regime, they both converge to the same large-scale limiting dynamics. This asymptotic equivalence holds for several variants of dropout and RaM, including layerwise dropout as used in stochastic-depth ResNets, albeit at slower quantitative rates. In fact, we also show that several of these variants collapse to the same limit asymptotically.

The method called 'expected $f_2$' ($\hat{f}_{2,\exp}$), as proposed by Noce et al. (2020) and Xu et al. (2021), has been adopted in two health authority guidelines for dissolution profile comparison when variability precludes the use of the conventional similarity factor $\hat{f}_2$. This position paper, developed by a working group of the European Federation of Statisticians in the Pharmaceutical Industry CMC Statistical Network Europe Special Interest Group (EFSPI CMCSNE SIG), presents a critical evaluation of this method. Fundamental concerns are identified. First, the formula for $\hat{f}_{2,\exp}$ has no traceable origin in the references cited by its proponents. Noce et al. (2020) and Xu et al. (2021) attribute $\hat{f}_{2,\exp}$ to Shah et al. (1998) and Ma et al. (1999, 2000), but neither mentions nor suggests it. Second, no mathematical justification has been provided for the formula. Where Shah et al. (1998) subtract a variance term to reduce the upward bias of $\hat{f}_2$, the $\hat{f}_{2,\exp}$ formula adds this term, thereby increasing rather than correcting the bias. This has also been noted by FDA statisticians Liu et al. (2024). Third, the method exhibits poor statistical properties: for highly variable profiles, the variance term dominates the statistic, resulting in low power even as the true difference between profiles approaches zero. The method can reject equivalence when profiles are identical. Fourth, the formula as published by Noce et al. (2020) contains a notation ambiguity that renders the intended grouping of terms unclear. This ambiguity has propagated into regulatory guidance. A survey of working group members, designed to elicit arguments both for and against the method, found no scientifically meaningful advantage. The EFSPI CMCSNE SIG concludes that $\hat{f}_{2,\exp}$ should not be recommended for dissolution profile comparison.

We develop and analyze neural stochastic residual methods for fully nonlinear parabolic terminal problems under exact diffusion sampling. A quadratic stochastic residual supports global training and backward regression with scalar networks and losses involving only spatial derivatives. For global training, the correction removes a Hessian penalty from the limiting first-order loss. Under a uniform output regularity budget, we bound the continuous residual norm by the square roots of the global loss or normalized backward-regression excess loss, an $O(h^{1/2})$ discretization term for time step $h$, and terminal error. Residual stability then gives value, gradient and Hessian error bounds. Under suitable solution and terminal assumptions, we construct admissible networks with $O(h^{1/2})$ bounds on these approximation errors and the square roots of both loss criteria. In ten-dimensional backward-regression tests using a shared schedule tuned for the quadratic residual, mean relative Hessian errors are $5$–$6\%$ with the quadratic residual and about $15\%$ with the first-order residual. Tuned backward regression achieves Hessian accuracy comparable to physics-informed neural networks across the tested nonlinear coefficients. Translation tests show that the ratio of Hessian error to residual can grow sharply under a fixed sampling law.

Conditional generative modeling remains a challenging problem in semi-supervised settings where labeled data is scarce but unlabeled samples are abundant. To effectively leverage structural information embedded within the unlabeled dataset and compensate for sparse conditioning signals, we propose a semi-supervised framework combining conditional stochastic interpolation with low-dimensional latent representations. RepG decomposes generation into two stages: label-dependent latent sampling and high-dimensional reconstruction. This isolates the supervised learning of conditional dependencies to a low-dimensional space, requiring few labels while utilizing the abundant unlabeled data purely for reconstruction. Theoretically, we establish an error decomposition showing that the Kullback-Leibler divergence of RepG comprises stage-wise estimation errors and a structural bias quantified by conditional mutual information. For deep neural network estimators, we derive non-asymptotic convergence rates proving that RepG significantly improves sample complexity. By confining the supervised estimation burden to the low intrinsic dimension of the latent representation, RepG achieves a strictly faster convergence rate. Complemented by a minimax lower bound, our theoretical results demonstrate that this method effectively mitigates the curse of dimensionality inherent in direct ambient-space generative modeling.

This paper proposes a causal independence principle for value – the value Causal Markov Condition (v-CMC) – and develops the conceptual and mathematical foundations of a "causal value theory" linking causality and utility. After motivating a local formulation of the v-CMC, we introduce a probability-value duality that translates standard causal-inference results into the value setting. In particular, we formulate local, global, and decomposition versions of the v-CMC and prove their equivalence. We also define v-separation and show that it is sound and complete for conditional value independence. Furthermore, we derive a Bellman-type recursion as a special case of the v-CMC, thereby generalizing standard Bellman recursion from linear chains to causal DAGs. Finally, we show how the v-CMC supports modular transfer and updating of utility information across causal contexts and develop algorithms for causally structured utility elicitation and canonical influence-diagram construction.

Bayesian hierarchical models with high-dimensional latent structure require scalable posterior approximations that preserve key dependencies while remaining computationally tractable. Mean-field variational inference (MFVI) is efficient, but can be unreliable when local variables are strongly correlated or tightly coupled to global variables. We propose KNorm-RVB, a generalized reparametrized variational Bayes framework for latent Gaussian and latent non-Gaussian models with sparse local precision matrices. KNorm-RVB maps the conditional posterior of local variables toward a standard Gaussian via normalization followed by skewness reduction, enabled by a novel K-component skew-symmetric density representation. This reparametrization centers the transformed conditional local posterior at an optimized reflection point and decorrelates local and global variables, making MFVI much more effective. Under symmetry conditions, we show that MFVI recovers the local posterior mean and correlation matrix exactly, motivating KNorm-RVB's normalization and symmetrization of the conditional local posterior before applying MFVI. We combine a Gaussian variational family for reparametrized local variables with a flexible closed skew normal family for the remaining variables. Across generalized linear mixed models, mixed multinomial logit models, spatial autoregressive models, and stochastic volatility models, KNorm-RVB improves posterior approximation accuracy over existing methods.