Papers

Portfolio optimization is a critical task in investment. Most existing portfolio optimization methods require information on the distribution of returns of the assets that make up the portfolio. However, such distribution information is usually unknown to investors. Various methods have been proposed to estimate distribution information, but their accuracy greatly depends on the uncertainty of the financial markets. Due to this uncertainty, a model that could well predict the distribution information at one point in time may perform less accurately compared to another model at a different time. To solve this problem, we investigate a method for portfolio optimization based on Bayesian predictive synthesis (BPS), one of the Bayesian ensemble methods for meta-learning. We assume that investors have access to multiple asset return prediction models. By using BPS with dynamic linear models to combine these predictions, we can obtain a Bayesian predictive posterior about the mean rewards of assets that accommodate the uncertainty of the financial markets. In this study, we examine how to construct mean-variance portfolios and quantile-based portfolios based on the predicted distribution information.

Participants in clinical trials are often viewed as a unique, finite population. Yet, statistical analyses often assume that participants were randomly sampled from a larger population. Under Complete Randomization, Randomization-Based Inference (RBI; a finite population inference) and Analysis of Variance (ANOVA; a random sampling inference) provide asymptotically equivalent difference-in-means tests. However, sequentially-enrolling trials typically employ restricted randomization schemes, such as block or Maximum Tolerable Imbalance (MTI) designs, to reduce the chance of chronological treatment imbalances. The impact of these restrictions on RBI and ANOVA concordance is not well understood. With real-world frames of reference, such as rare and ultra-rare diseases, we review full versus random sampling of finite populations and empirically evaluate finite population Type I error when using ANOVA following randomization restrictions. Randomization restrictions strongly impacted ANOVA Type I error, even for trials with 1,000 participants. Properly adjusting for restrictions corrected Type I error. We corrected for block randomization, yet leave open how to correct for MTI designs. More directly, RBI accounts for randomization restrictions while ensuring correct finite population Type I error. Novel contributions are: 1) deepening the understanding and correction of RBI and ANOVA concordance under block and MTI restrictions and 2) using finite populations to estimate the convergence of Type I error to a nominal rate. We discuss the challenge of specifying an estimand's population and reconciling with sampled trial participants.

This paper studies slow, fast and opportunistic fluid antenna multiple access (FAMA) under the effect of Nakagami-m fading channels, considering the new and realistic spatial blockcorrelation model. Expressions for the outage probability (OP), based on the signal-to-interference ratio (SIR), are derived for slow FAMA. Interestingly, we provide mathematical relationships that allow the expressions of fast FAMA to be obtained from slow FAMA. Multiplexing gains for an opportunistic FAMA (OFAMA) network are presented for both slow and fast FAMA scenarios. Our analytical results are validated through Monte Carlo simulations, under various channel and system parameters. All expressions derived in this work are original.

Clustered Federated Learning (CFL) improves performance under non-IID client heterogeneity by clustering clients and training one model per cluster, thereby balancing between a global model and fully personalized models. However, most CFL methods require the number of clusters K to be fixed a priori, which is impractical when the latent structure is unknown. We propose DPMM-CFL, a CFL algorithm that places a Dirichlet Process (DP) prior over the distribution of cluster parameters. This enables nonparametric Bayesian inference to jointly infer both the number of clusters and client assignments, while optimizing per-cluster federated objectives. This results in a method where, at each round, federated updates and cluster inferences are coupled, as presented in this paper. The algorithm is validated on benchmark datasets under Dirichlet and class-split non-IID partitions.

Conventional joint modeling approaches generally characterize the relationship between longitudinal biomarkers and discrete event occurrences within terminal, recurring or competing risk settings, thereby offering a limited representation of complex, multi-state trajectories. We propose a general multi-state joint modeling framework that unifies longitudinal biomarker dynamics with multi-state time-to-event processes defined on arbitrary directed graphs. The proposed framework also accomodates nonlinear longitudinal submodels and scalable inference via stochastic gradient descent. This formulation encompasses both Markovian and semi-Markovian transition structures, allowing recurrent cycles and terminal absorptions to be naturally represented. The longitudinal and event processes are linked through shared latent structures within nonlinear mixed-effects models, extending classical joint modeling formulations. We derive the complete likelihood, model selection criteria, and develop scalable inference procedures based on stochastic gradient descent to enable high-dimensional and large-scale applications. In addition, we formulate a dynamic prediction framework that provides individualized state-transition probabilities and personalized risk assessments along complex event trajectories. Through simulation and application to the PAQUID cohort, we demonstrate accurate parameter recovery and individualized prediction.

In assessing Overall Survival (OS) in oncology studies, it is essential for the efficacy measure to be Logic-respecting, for otherwise patients may be incorrectly targeted. This paper explains, while Time Ratio (TR) is Logic-respecting, Hazard Ratio (HR) is not Logic-respecting. With Time Ratio (TR) being recommended, a smooth transitioning strategy is suggested. The conclusion states: Logicality requires, and Subgroup Mixable Estimation (SME) delivers, an efficacy assessment for the overall population within the range of minimum and maximum efficacy in the subgroups, no matter how outcome is measured, whichever logic-respecting efficacy measure is chosen, the same efficacy assessment regardless of how subgroups are stratified.

In the ever-changing and intricate landscape of financial markets, portfolio optimisation remains a formidable challenge for investors and asset managers. Conventional methods often struggle to capture the complex dynamics of market behaviour and align with diverse investor preferences. To address this, we propose an innovative framework, termed Diffusion-Augmented Reinforcement Learning (DARL), which synergistically integrates Denoising Diffusion Probabilistic Models (DDPMs) with Deep Reinforcement Learning (DRL) for portfolio management. By leveraging DDPMs to generate synthetic market crash scenarios conditioned on varying stress intensities, our approach significantly enhances the robustness of training data. Empirical evaluations demonstrate that DARL outperforms traditional baselines, delivering superior risk-adjusted returns and resilience against unforeseen crises, such as the 2025 Tariff Crisis. This work offers a robust and practical methodology to bolster stress resilience in DRL-driven financial applications.

Conformal prediction provides prediction sets with coverage guarantees. The informativeness of conformal prediction depends on its efficiency, typically quantified by the expected size of the prediction set. Prior work on the efficiency of conformalized regression commonly treats the miscoverage level $\alpha$ as a fixed constant. In this work, we establish non-asymptotic bounds on the deviation of the prediction set length from the oracle interval length for conformalized quantile and median regression trained via SGD, under mild assumptions on the data distribution. Our bounds of order $\mathcal{O}(1/\sqrt{n} + 1/(\alpha^2 n) + 1/\sqrt{m} + \exp(-\alpha^2 m))$ capture the joint dependence of efficiency on the proper training set size $n$, the calibration set size $m$, and the miscoverage level $\alpha$. The results identify phase transitions in convergence rates across different regimes of $\alpha$, offering guidance for allocating data to control excess prediction set length. Empirical results are consistent with our theoretical findings.

Fourier analysis on the Boolean hypercube is fundamentally defined as the orthogonal decomposition of the space of pseudo-Boolean functions with respect to the uniform probability measure. In this work, we propose an ANOVA-based generalization of the Fourier decomposition on the Boolean hypercube endowed with any arbitrary probability measure. We provide an explicit decomposition basis which generalizes the Walsh-Hadamard (or parity functions) basis under any arbitrary probability measure on the Boolean hypercube. We formulate the computation of the entire functional decomposition as a least squares problem and also provide a method to address the classical curse of dimensionality challenge. We provide a comprehensive generalization of Fourier analysis on the Boolean hypercube, enabling the handling of non-uniform configuration spaces inherent to real-world machine learning tasks, e.g. when dealing with one-hot encoded features. Finally, we demonstrate its practical impact in the field of explainable AI, by conducting comparative studies with feature attribution methods such as SHAP or TreeHFD.

In epidemiology, some have argued that multiple comparison corrections are not necessary as there is rarely interest in the universal null hypothesis. From a parameter estimation perspective, epidemiologists may still be interested in multiple parameters. In this context, standard confidence intervals are not guaranteed to provide simultaneous coverage of more than one parameter. In other words, use of confidence intervals in these cases will understate the uncertainty due to random error. To address this challenge, one can use confidence bands, an extension of confidence intervals to parameter vectors. We illustrate the use of confidence bands in three case studies: estimation of multiple causal effects, effect measure modification by a binary variable, and effect measure modification by a continuous variable. Each example uses publicly available data is accompanied by SAS, R, and Python code. The type of confidence region reported by epidemiologists should depend on whether scientific interest is in a single parameter or a set of parameters. For sets of parameters, like in cases where multiple actions or outcomes, effect measure modification, dose-response, or other functions are of interest, sup-t confidence bands are preferred due to their statistical properties, computational simplicity, and ease of presentation.

We study the propagation of outliers in cyclic causal graphs with linear structural equations, tracing them back to one or several "root cause" nodes. We show that it is possible to identify a short list of potential root causes provided that the perturbation is sufficiently strong and propagates according to the same structural equations as in the normal mode. This shortlist consists of the true root causes together with those of its parents lying on a cycle with the root cause. Notably, our method does not require prior knowledge of the causal graph and yields encouraging results on simulated data and real data from biology and cloud computing.

Reinforcement learning (RL) aims to learn and evaluate a sequential decision rule, often referred to as a "policy", that maximizes the population-level benefit in an environment across possibly infinitely many time steps. However, the sequential decisions made by an RL algorithm, while optimized to maximize overall population benefits, may disadvantage certain individuals who are in minority or socioeconomically disadvantaged groups. To address this problem, we introduce PyCFRL, a Python library for ensuring counterfactual fairness in offline RL. PyCFRL implements a novel data preprocessing algorithm for learning counterfactually fair RL policies from offline datasets and provides tools to evaluate the values and counterfactual unfairness levels of RL policies. We describe the high-level functionalities of PyCFRL and demonstrate one of its major use cases through a data example. The library is publicly available on PyPI and Github (https://github.com/JianhanZhang/PyCFRL), and detailed tutorials can be found in the PyCFRL documentation (https://pycfrl-documentation.netlify.app).

We present a method that models the evolution of an unbounded number of time series clusters by switching among an unknown number of regimes with linear dynamics. We develop a Bayesian non-parametric approach using a hierarchical Dirichlet process as a prior on the parameters of a Switching Linear Dynamical System and a Gaussian process prior to model the statistical variations in amplitude and temporal alignment within each cluster. By modeling the evolution of time series patterns, the method avoids unnecessary proliferation of clusters in a principled manner. We perform inference by formulating a variational lower bound for off-line and on-line scenarios, enabling efficient learning through optimization. We illustrate the versatility and effectiveness of the approach through several case studies of electrocardiogram analysis using publicly available databases.

We prove that the norm of a $d$-dimensional Lévy process possesses a finite second moment if and only if the convex distance between an appropriately rescaled process at time $t$ and a standard Gaussian vector is integrable in time with respect to the scale-invariant measure $t^{-1} dt$ on $[1,\infty)$. We further prove that under the standard $\sqrt{t}$-scaling, the corresponding convex distance is integrable if and only if the norm of the Lévy process has a finite $(2+\log)$-moment. Both equivalences also hold for the integrability with respect to $t^{-1} dt$ of the multivariate Kolmogorov distance. Our results imply: (I) polynomial Berry-Esseen bounds on the rate of convergence in the convex distance in the CLT for Lévy processes cannot hold without finiteness of $(2+\delta)$-moments for some $\delta>0$ and (II) integrability of the convex distance with respect to $t^{-1} dt$ in the domain of non-normal attraction cannot occur for any scaling function.

We introduce an offline nonparametric estimator for concave multi-asset propagator models based on a dataset of correlated price trajectories and metaorders. Compared to parametric models, our framework avoids parameter explosion in the multi-asset case and yields confidence bounds for the estimator. We implement the estimator using both proprietary metaorder data from Capital Fund Management (CFM) and publicly available S&P order flow data, where we augment the former dataset using a metaorder proxy. In particular, we provide unbiased evidence that self-impact is concave and exhibits a shifted power-law decay, and show that the metaorder proxy stabilizes the calibration. Moreover, we find that introducing cross-impact provides a significant gain in explanatory power, with concave specifications outperforming linear ones, suggesting that the square-root law extends to cross-impact. We also measure asymmetric cross-impact between assets driven by relative liquidity differences. Finally, we demonstrate that a shape-constrained projection of the nonparametric kernel not only ensures interpretability but also slightly outperforms established parametric models in terms of predictive accuracy.

The pseudo-observation method is regularly applied to time-to-event data. However, to date such analyses have relied on not formally verified statements or ad-hoc methods regarding covariance estimation. This paper strives to close this gap in the literature. To begin with, we demonstrate that the usual Huber-White estimator is not consistent for the limiting covariance of parameter estimates in pseudo-observation regression approaches. By confirming that a plug-in estimator can be used instead, we obtain asymptotically exact and consistent tests for general linear hypotheses in the parameters of the model. Additionally, we confirm that naive bootstrapping can not be used for covariance estimation in the pseudo-observation model either. However, it can be used for hypothesis testing by applying a suitable studentization. Simulations illustrate the good performance of our proposed methods in many scenarios. Finally, we obtain a general uniform law of large numbers for U- and V-statistics, as such statistics are central in the mathematical analysis of the inference procedures developed in this work.

This paper investigates the rank aggregation problem through the lens of multi-way comparison data derived from rater scores. Departing from traditional parametric frameworks, such as the Bradley-Terry and Plackett-Luce models, we propose a model-free method that accommodates highly heterogeneous preference distributions across raters and encompasses weak stochastic transitivity in pairwise comparisons as a special case. We establish the theoretical foundations of the proposed estimator by proving its consistency, demonstrating that the proportion of discordant pairs (Kendall tau) converges to zero in probability as the number of raters diverges. Furthermore, we derive upper and lower bounds for a performance metric based on Kendall's tau. In certain asymptotic regimes, these bounds coincide up to logarithmic factors, so the estimator is nearly minimax optimal. These results are obtained by analyzing the convergence behavior of a U-empirical process; the novel technical results developed for this analysis may be of independent theoretical interest. The practical utility of our method is validated through extensive simulations and applications to sports player rankings and survey preference aggregation.

Population dynamics models play an important role in a number of fields, such as actuarial science, demography, and ecology, as they help explain past fluctuations and predict future population. The accuracy of these models is often influenced by the uncertainty introduced by sampling error. Statistical inference for these models can be difficult when, in addition to the process' inherent stochasticity, one also needs to account for sampling error. Ignoring the latter can lead to biases in the estimation, which in turn can produce erroneous conclusions about the system's behavior. The Gompertz model is widely used to infer population size dynamics, but a full likelihood approach can be computationally prohibitive when sampling error is accounted for. We close this gap by developing efficient computational tools for statistical inference in the Gompertz model with Poisson sampling error based on the full likelihood. The approach is illustrated in both the Bayesian and frequentist paradigms. Performance is illustrated with simulations and data analysis.

We present a novel approach to modeling market dynamics using ordinary differential equations that explicitly incorporates product competitiveness and consumer behavior. Our framework treats market segments as interacting populations in a dynamical system analogous to predator-prey models, where competitive advantages drive market share transitions through mechanistic modeling of market flows including new product adoption, refresh cycles, and obsolescence dynamics.

The comparison of a parameter in $k$ populations is a classical problem in statistics. Testing for the equality of means or variances are typical examples. Most procedures designed to deal with this problem assume that $k$ is fixed and that samples with increasing sample sizes are available from each population. This paper introduces and studies a test for the comparison of an estimable parameter across $k$ populations, when $k$ is large and the sample sizes from each population are small when compared with $k$. The proposed test statistic is asymptotically distribution-free under the null hypothesis of parameter homogeneity, enabling asymptotically exact inference without parametric assumptions. Additionally, the behaviour of the proposal is studied under alternatives. Simulations are conducted to evaluate its finite-sample performance, and a linear bootstrap method is implemented to improve its behaviour for small $k$. Finally, an application to a real dataset is presented.

Most capture-recapture models assume that individuals either do not emigrate or emigrate permanently from the sampling area during the sampling period. This assumption is violated when individuals temporarily leave the sampling area and return during later capture occasions, which can result in biased or less precise inferences under normal capture-recapture models. Existing temporary emigration models require that individuals are uniquely and correctly identified. To our knowledge, no studies to date have addressed temporary emigration in the presence of latent individual identification, which can arise in many scenarios such as misidentification, data integration, and batch marking. In this paper, we propose a new latent multinomial temporary emigration modelling framework for analysing capture-recapture data with latent identification. The framework is applicable to both closed- and open-population problems, accommodates data with or without individual identification, and flexibly incorporates different emigration processes, including the completely random and Markovian emigration. Through simulations, we demonstrate that model parameters can be reliably estimated in various emigration scenarios. We apply the proposed framework to a real dataset on golden mantella collected using batch marks under Pollock's robust design. The results show that accounting for temporary emigration provides a better fit to the data compared to the previous model without temporary emigration.

Geodesic slice sampling, introduced in Durmus et al., 2024, is a slice sampling based Markov chain Monte Carlo method for approximate sampling from distributions on Riemannian manifolds. We prove that it is uniformly ergodic for distributions with compact support that have a bounded density with respect to the Riemannian measure. The constants in our convergence bound are available explicitly, and we investigate their dependence on the hyperparameters of the geodesic slice sampler, the target distribution and the underlying domain.

Bayesian causal discovery benefits from prior information elicited from domain experts, and in heterogeneous domains any prior knowledge would be badly needed. However, so far prior elicitation approaches have assumed a single causal graph and hence are not suited to heterogeneous domains. We propose a causal elicitation strategy for heterogeneous settings, based on Bayesian experimental design (BED) principles, and a variational mixture structure learning (VaMSL) method – extending the earlier differentiable Bayesian structure learning (DiBS) method – to iteratively infer mixtures of causal Bayesian networks (CBNs). We construct an informative graph prior incorporating elicited expert feedback in the inference of mixtures of CBNs. Our proposed method successfully produces a set of alternative causal models (mixture components or clusters), and achieves an improved structure learning performance on heterogeneous synthetic data when informed by a simulated expert. Finally, we demonstrate that our approach is capable of capturing complex distributions in a breast cancer database.

Self-attention layers have become fundamental building blocks of modern deep neural networks, yet their theoretical understanding remains limited, particularly from the perspective of random matrix theory. In this work, we provide a rigorous analysis of the singular value spectrum of the attention matrix and establish the first Gaussian equivalence result for attention. In a natural regime where the inverse temperature remains of constant order, we show that the singular value distribution of the attention matrix is asymptotically characterized by a tractable linear model. We further demonstrate that the distribution of squared singular values deviates from the Marchenko-Pastur law, which has been believed in previous work. Our proof relies on two key ingredients: precise control of fluctuations in the normalization term and a refined linearization that leverages favorable Taylor expansions of the exponential. This analysis also identifies a threshold for linearization and elucidates why attention, despite not being an entrywise operation, admits a rigorous Gaussian equivalence in this regime.

Transformer has become the dominant architecture for sequence modeling, yet a detailed understanding of how its structural parameters influence expressive power remains limited. In this work, we study the approximation properties of transformers, with particular emphasis on the role of the number of attention heads. Our analysis begins with the introduction of a generalized $D$-retrieval task, which we prove to be dense in the space of continuous functions, thereby providing the basis for our theoretical framework. We then establish both upper and lower bounds on the parameter complexity required for $\epsilon$-approximation. Specifically, we show that transformers with sufficiently many heads admit efficient approximation, whereas with too few heads, the number of parameters must scale at least as $O(1/\epsilon^{cT})$, for some constant $c$ and sequence length $T$. To the best of our knowledge, this constitutes the first rigorous lower bound of this type in a nonlinear and practically relevant setting. We further examine the single-head case and demonstrate that an embedding dimension of order $O(T)$ allows complete memorization of the input, where approximation is entirely achieved by the feed-forward block. Finally, we validate our theoretical findings with experiments on both synthetic data and real-world tasks, illustrating the practical relevance of our results.

We study fine-grained gap-dependent regret bounds for model-free reinforcement learning in episodic tabular Markov Decision Processes. Existing model-free algorithms achieve minimax worst-case regret, but their gap-dependent bounds remain coarse and fail to fully capture the structure of suboptimality gaps. We address this limitation by establishing fine-grained gap-dependent regret bounds for both UCB-based and non-UCB-based algorithms. In the UCB-based setting, we develop a novel analytical framework that explicitly separates the analysis of optimal and suboptimal state-action pairs, yielding the first fine-grained regret upper bound for UCB-Hoeffding (Jin et al., 2018). To highlight the generality of this framework, we introduce ULCB-Hoeffding, a new UCB-based algorithm inspired by AMB (Xu et al.,2021) but with a simplified structure, which enjoys fine-grained regret guarantees and empirically outperforms AMB. In the non-UCB-based setting, we revisit the only known algorithm AMB, and identify two key issues in its algorithm design and analysis: improper truncation in the $Q$-updates and violation of the martingale difference condition in its concentration argument. We propose a refined version of AMB that addresses these issues, establishing the first rigorous fine-grained gap-dependent regret for a non-UCB-based method, with experiments demonstrating improved performance over AMB.

The continuous nature of belief states in POMDPs presents significant computational challenges in learning the optimal policy. In this paper, we consider an approach that solves a Partially Observable Reinforcement Learning (PORL) problem by approximating the corresponding POMDP model into a finite-state Markov Decision Process (MDP) (called Superstate MDP). We first derive theoretical guarantees that improve upon prior work that relate the optimal value function of the transformed Superstate MDP to the optimal value function of the original POMDP. Next, we propose a policy-based learning approach with linear function approximation to learn the optimal policy for the Superstate MDP. Consequently, our approach shows that a POMDP can be approximately solved using TD-learning followed by Policy Optimization by treating it as an MDP, where the MDP state corresponds to a finite history. We show that the approximation error decreases exponentially with the length of this history. To the best of our knowledge, our finite-time bounds are the first to explicitly quantify the error introduced when applying standard TD learning to a setting where the true dynamics are not Markovian.

We establish that randomly initialized neural networks, with large width and a natural choice of hyperparameters, have nearly independent outputs exactly when their activation function is nonlinear with zero mean under the Gaussian measure: $\mathbb{E}_{z \sim \mathcal{N}(0,1)}[\sigma(z)]=0$. For example, this includes ReLU and GeLU with an additive shift, as well as tanh, but not ReLU or GeLU by themselves. Because of their nearly independent outputs, we propose neural networks with zero-mean activation functions as a promising candidate for the Alignment Research Center's computational no-coincidence conjecture – a conjecture that aims to measure the limits of AI interpretability.

Recent work in dynamic causal inference introduced a class of discrete-time stochastic processes that generalize martingale difference sequences and arrays as follows: the random variates in each sequence have expectation zero given certain lagged filtrations but not given the natural filtration. We formalize this class of stochastic processes and prove stable central limit theorems (CLTs) via martingale-coboundary decomposition, leveraging the classical martingale CLT. We develop a variety of sufficient conditions, including conditions under which the limiting variance has a simple form that depends on variances and covariances of neighboring variates. We demonstrate the application of these results to inference for time-averaged treatment effects in switchback designs and present a simulation study supporting their validity. The CLTs enable various extensions to existing methodology for design-based approaches to dynamic causal inference, including time-lagged effects, random limiting variances, cross-unit dependence, and vector-valued estimands.

Online matching problems arise in many complex systems, from cloud services and online marketplaces to organ exchange networks, where timely, principled decisions are critical for maintaining high system performance. Traditional heuristics in these settings are simple and interpretable but typically tailored to specific operating regimes, which can lead to inefficiencies when conditions change. We propose a reinforcement learning (RL) approach that learns to orchestrate a set of such expert policies, leveraging their complementary strengths in a data-driven, adaptive manner. Building on the Adv2 framework (Jonckheere et al., 2024), our method combines expert decisions through advantage-based weight updates and extends naturally to settings where only estimated value functions are available. We establish both expectation and high-probability regret guarantees and derive a novel finite-time bias bound for temporal-difference learning, enabling reliable advantage estimation even under constant step size and non-stationary dynamics. To support scalability, we introduce a neural actor-critic architecture that generalizes across large state spaces while preserving interpretability. Simulations on stochastic matching models, including an organ exchange scenario, show that the orchestrated policy converges faster and yields higher system level efficiency than both individual experts and conventional RL baselines. Our results highlight how structured, adaptive learning can improve the modeling and management of complex resource allocation and decision-making processes.