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The adaptive Iterative Soft-Thresholding Algorithm (ISTA) has been a popular algorithm for finding a desirable solution to the LASSO problem without explicitly tuning the regularization parameter $λ$. Despite that the adaptive ISTA is a successful practical algorithm, few theoretical results exist. In this paper, we present the theoretical analysis on the adaptive ISTA with the thresholding strategy of estimating noise level by median absolute deviation. We show properties of the fixed points of the algorithm, including scale equivariance, non-uniqueness, and local stability, prove the local linear convergence guarantee, and show its global convergence behavior.

Bayesian adaptive clinical trials offer a flexible and efficient alternative to traditional fixed-design trials, but their implementation is often hindered by the complexity of Bayesian computations and the need for advanced statistical programming expertise. The authors introduce a custom fine-tuned LLM designed to assist with this and lower barriers to adoption of Bayesian methods for adaptive clinical trials. This paper describes the development and fine-tuning of BACTA-GPT, a Large Language Model (LLM)-based tool designed to assist in the implementation of Bayesian Adaptive Clinical Trials. This engine uses GPT-3.5 as the underlying model and takes in Natural Language input from the Statistician or the Trialist. The fine-tuned model demonstrates a viable proof-of-concept in its objectives. Test case evaluations show that the model is capable of generating a fit-for-purpose Bayesian model for an adaptive trial and evaluate its operating characteristics via simulations using R and JAGS. The integration of AI code generation has significant potential to lower technical barriers for the design and implementation of Bayesian Adaptive trials. But they also require attention to important considerations regarding validation and quality control.

Motivated by the need for efficient estimation of conditional expectations, we consider a least-squares function approximation problem with heavily polluted data. Existing methods that are powerful in the small noise regime are suboptimal when large noise is present. We propose a hybrid approach that combines Christoffel sampling with certain types of optimal experimental design to address this issue. We show that the proposed algorithm enjoys appropriate optimality properties for both sample point generation and noise mollification, leading to improved computational efficiency and sample complexity compared to existing methods. We also extend the algorithm to convex-constrained settings with similar theoretical guarantees. When the target function is defined as the expectation of a random field, we extend our approach to leverage adaptive random subspaces and establish results on the approximation capacity of the adaptive procedure. Our theoretical findings are supported by numerical studies on both synthetic data and on a more challenging stochastic simulation problem in computational finance.

In the US, `black box' studies are increasingly being used to estimate the error rate of forensic disciplines. A sample of forensic examiner participants are asked to evaluate a set of items whose source is known to the researchers but not to the participants. Participants are asked to make a source determination (typically an identification, exclusion, or some kind of inconclusive). We study inconclusives in two black box studies, one on fingerprints and one on bullets. Rather than treating all inconclusive responses as functionally correct (as is the practice in reported error rates in the two studies we address), irrelevant to reported error rates (as some would do), or treating them all as potential errors (as others would do), we propose that the overall pattern of inconclusives in a particular black box study can shed light on the proportion of inconclusives that are due to examiner variability. Raw item and examiner variances are computed, and compared with the results of a logistic regression model that takes account of which items were addressed by which examiner. The error rates reported in black box studies are substantially smaller than ``failure rate" analyses that take inconclusives into account. The magnitude of this difference is highly dependent on the particular study at hand.

In this paper, we explore the knowledge transfer under the setting of matrix completion, which aims to enhance the estimation of a low-rank target matrix with auxiliary data available. We propose a transfer learning procedure given prior information on which source datasets are favorable. We study its convergence rates and prove its minimax optimality. Our analysis reveals that with the source matrices close enough to the target matrix, out method outperforms the traditional method using the single target data. In particular, we leverage the advanced sharp concentration inequalities introduced in \cite{brailovskaya2024universality} to eliminate a logarithmic factor in the convergence rate, which is crucial for proving the minimax optimality. When the relevance of source datasets is unknown, we develop an efficient detection procedure to identify informative sources and establish its selection consistency. Simulations and real data analysis are conducted to support the validity of our methodology.

Structure-agnostic causal inference studies how well one can estimate a treatment effect given black-box machine learning estimates of nuisance functions (like the impact of confounders on treatment and outcomes). Here, we find that the answer depends in a surprising way on the distribution of the treatment noise. Focusing on the partially linear model of \citet{robinson1988root}, we first show that the widely adopted double machine learning (DML) estimator is minimax rate-optimal for Gaussian treatment noise, resolving an open problem of \citet{mackey2018orthogonal}. Meanwhile, for independent non-Gaussian treatment noise, we show that DML is always suboptimal by constructing new practical procedures with higher-order robustness to nuisance errors. These \emph{ACE} procedures use structure-agnostic cumulant estimators to achieve $r$-th order insensitivity to nuisance errors whenever the $(r+1)$-st treatment cumulant is non-zero. We complement these core results with novel minimax guarantees for binary treatments in the partially linear model. Finally, using synthetic demand estimation experiments, we demonstrate the practical benefits of our higher-order robust estimators.

The fractional order generalization of Shannon entropy proposed by Ubriaco has been studied for discrete distributions. In the current paper, we conduct a detailed study of the continuous analogue of this entropy termed as fractional differential entropy and find some interesting properties which makes it stand out among the existing entropies in literature. The studied entropy measure is evaluated analytically and numerically for some well-known continuous distributions, which will be quite useful in reliability analysis works and other statistical studies of complex systems. Further, it has been used to model the one-dimensional vertical velocity profile of turbulent flows in wide open channels. A one-parametric spatial distribution function is utilized for better estimation of the velocity distribution. The validity of the model has been established using experimental and field data through regression analysis. A comparative study is also presented to show the superiority of the proposed model over the existing entropy-based models.

Inducing-point-based sparse variational Gaussian processes have become the standard workhorse for scaling up GP models. Recent advances show that these methods can be improved by introducing a diagonal scaling matrix to the conditional posterior density given the inducing points. This paper first considers an extension that employs a block-diagonal structure for the scaling matrix, provably tightening the variational lower bound. We then revisit the unifying framework of sparse GPs based on Power Expectation Propagation (PEP) and show that it can leverage and benefit from the new structured approximate posteriors. Through extensive regression experiments, we show that the proposed block-diagonal approximation consistently performs similarly to or better than existing diagonal approximations while maintaining comparable computational costs. Furthermore, the new PEP framework with structured posteriors provides competitive performance across various power hyperparameter settings, offering practitioners flexible alternatives to standard variational approaches.

A penalized maximum likelihood estimation approach is proposed for discrete-time hidden Markov models where covariates affect the observed responses and serial dependence is considered. The proposed penalized maximum likelihood method addresses the issue of latent state separation that typically occurs when this model is applied to binary and categorical response variables with a limited number of categories, resulting in extremely large estimates of the support points of the latent variable assumed with a discrete, left unspecified distribution. We also propose a cross-validation approach for jointly selecting the number of hidden states and the roughness of the penalty term. The proposal is illustrated through a simulation study comparing parameter estimation accuracy and computational efficiency across different estimation procedures. We also demonstrate the potential of this class of models through the analysis of longitudinal data collected during spinal anesthesia to monitor the occurrence of hypotension in patients, and we compare the results with those obtained from other standard models.

Graphical models have been popularly used for capturing conditional independence structure in multivariate data, which are often built upon independent and identically distributed observations, limiting their applicability to complex datasets such as network-linked data. This paper proposes a nonparametric graphical model that addresses these limitations by accommodating heterogeneous graph structures without imposing any specific distributional assumptions. The proposed estimation method effectively integrates network embedding with nonparametric graphical model estimation. It further transforms the graph learning task into solving a finite-dimensional linear equation system by leveraging the properties of vector-valued reproducing kernel Hilbert space. Moreover, theoretical guarantees are established for the proposed method in terms of the estimation consistency and exact recovery of the heterogeneous graph structures. Its effectiveness is also demonstrated through a variety of simulated examples and a real application to the statistician coauthorship dataset.

The use of metrics underpins the quantification, communication and, ultimately, the functioning of a wide range of disciplines as diverse as labour recruitment, institutional management, economics and science. For application of metrics, customised scores are widely employed to optimise progress monitoring towards a goal, to contribute to decision-making, and to quantify situations under evaluation. However, the development of such metrics in complex and rigorous settings intrinsically relies on mathematical processes which are not always readily accessible. Here, we propose a framework for construction of metrics suitable for a wide range of disciplines, following a specified workflow that combines existing decision analysis and utility theory concepts to create a customisable performance metric (with corresponding uncertainty) that can be used to quantitatively evaluate goal achievement. It involves dividing criteria into two groups (root and additional) to utilise a newly proposed alternative form of utility function designed to build such customised metrics. Once the metrics are produced by this approach, these metrics can be used on a varied set of contexts, including their use in subsequent statistical analysis with the metric values as a response variable, or informing a decision-making process. The flexibility of the metric construction makes it suitable for a wide range of fields and applications, and could provide a valuable first step for monitoring and comparison in many different settings.

The goal of this paper is to estimate an optimal combination of biomarkers for individuals with Duchenne muscular dystrophy (DMD), which provides the most sensitive combinations of biomarkers to assess disease progression (in this case, optimal with respect to standardized response mean (SRM) for 4 muscle biomarkers). The biomarker data is an incomplete (missing and irregular) multivariate longitudinal data. We propose a normal model with structured covariance designed for our setting. To sample from the posterior distribution of parameters, we develop a Markov Chain Monte Carlo (MCMC) algorithm to address the positive definiteness constraint on the structured correlation matrix. In particular, we propose a novel approach to compute the support of the parameters in the structured correlation matrix; we modify the approach from \cite{Barnard} on the set of largest possible submatrices of the correlation matrix, where the correlation parameter is a unique element. For each posterior sample, we compute the optimal weights of our construct. We conduct data analysis and simulation studies to evaluate the algorithm and the frequentist properties of the posteriors of correlations and weights. We found that the lower extremities are the most responsive muscles at the early and late ambulatory disease stages and the biceps brachii is the most responsive at the non-ambulatory disease stage.

We present D-BIRD, a Bayesian dynamic item response model for estimating student ability from sparse, longitudinal assessments. By decomposing ability into a cohort trend and individual trajectory, D-BIRD supports interpretable modeling of learning over time. We evaluate parameter recovery in simulation and demonstrate the model using real-world personalized learning data.

We compare the integration error of Monte Carlo (MC) and quasi-Monte Carlo (QMC) methods for approximating the normalizing constant of posterior distributions and certain marginal likelihoods. In doing so, we characterize the dependency of the relative and absolute integration errors on the number of data points ($n$), the number of grid points ($m$) and the dimension of the integral ($p$). We find that if the dimension of the integral remains fixed as $n$ and $m$ tend to infinity, the scaling rate of the relative error of MC integration includes an additional $n^{1/2}\log(n)^{p/2}$ data-dependent factor, while for QMC this factor is $\log(n)^{p/2}$. In this scenario, QMC will outperform MC if $\log(m)^{p - 1/2}/\sqrt{mn\log(n)} < 1$, which differs from the usual result that QMC will outperform MC if $\log(m)^p/m^{1/2} < 1$.The accuracies of MC and QMC methods are also examined in the high-dimensional setting as $p \rightarrow \infty$, where MC gives more optimistic results as the scaling in dimension is slower than that of QMC when the Halton sequence is used to construct the low discrepancy grid; however both methods display poor dimensional scaling as expected. An additional contribution of this work is a bound on the high-dimensional scaling of the star discrepancy for the Halton sequence.

Authors Yi-Cheng Chen, Ping-En Lu, Cheng-Shang Chang, and Tzu-Hsuan Liu use the Finite Impulse Response (FIR) linear system filtering method to track and predict the number of people infected and recovered from COVID-19, in a pandemic context in which there was still no vaccine and the only way to avoid contagion was isolation. To estimate the coefficients of these FIR filters, Chen et al. used machine learning methods through a classical optimization problem with regularization (ridge regression). These estimated coefficients are called ridge coefficients. The epidemic mathematical model adopted by these researchers to formulate the FIR filters is the time-dependent discrete SIR. In this paper, we propose a small modification to the algorithm of Chen et al. to obtain the ridge coefficients. We then used this modified algorithm to track and predict the number of people infected and recovered from COVID-19 in the state of Minas Gerais/Brazil, within a prediction window, during the initial period of the pandemic. We also compare the predicted data with the respective real data to check how good the approximation is. In the modified algorithm, we set values for the FIR filter orders and for the regularization parameters, both different from the respective values defined by Chen et al. in their algorithm. In this context, the numerical results obtained by the modified algorithm in some simulations present better approximation errors compared to the respective approximation errors presented by the algorithm of Chen et al.

Denoising Diffusion Probabilistic Models represent an entirely new class of generative AI methods that have yet to be fully explored. Critical damping has been successfully introduced in Critically-Damped Langevin Dynamics (CLD) and Critically-Damped Third-Order Langevin Dynamics (TOLD++), but has not yet been applied to dynamics of arbitrary order. The proposed line of work generalizes Higher-Order Langevin Dynamics (HOLD), a recent state-of-the-art diffusion method, by introducing the concept of critical damping from systems analysis.

Diffusion models have demonstrated exceptional capabilities in generating high-fidelity images but typically suffer from inefficient sampling. Many solver designs and noise scheduling strategies have been proposed to dramatically improve sampling speeds. In this paper, we introduce a new sampling method that is up to $186\%$ faster than the current state of the art solver for comparative FID on ImageNet512. This new sampling method is training-free and uses an ordinary differential equation (ODE) solver. The key to our method resides in using higher-dimensional initial noise, allowing to produce more detailed samples with less function evaluations from existing pretrained diffusion models. In addition, by design our solver allows to control the level of detail through a simple hyper-parameter at no extra computational cost. We present how our approach leverages momentum dynamics by establishing a fundamental equivalence between momentum diffusion models and conventional diffusion models with respect to their training paradigms. Moreover, we observe the use of higher-dimensional noise naturally exhibits characteristics similar to stochastic differential equations (SDEs). Finally, we demonstrate strong performances on a set of representative pretrained diffusion models, including EDM, EDM2, and Stable-Diffusion 3, which cover models in both pixel and latent spaces, as well as class and text conditional settings. The code is available at https://github.com/apple/ml-tada.

Parameter estimation for ordinary differential equations (ODEs) plays a fundamental role in the analysis of dynamical systems. Generally lacking closed-form solutions, ODEs are traditionally approximated using deterministic solvers. However, there is a growing body of evidence to suggest that probabilistic ODE solvers produce more reliable parameter estimates by better accounting for numerical uncertainty. Here we present rodeo, a Python library providing a fast, lightweight, and extensible interface to a broad class of probabilistic ODE solvers, along with several associated methods for parameter inference. At its core, rodeo provides a probabilistic solver that scales linearly in both the number of evaluation points and system variables. Furthermore, by leveraging state-of-the-art automatic differentiation (AD) and just-in-time (JIT) compiling techniques, rodeo is shown across several examples to provide fast, accurate, and scalable parameter inference for a variety of ODE systems.