arXiv subjects

Michael Stanley

Publications and source records attributed to Michael Stanley.

8 recordsLinked to original sources

arXiv2026

Sample correlation adjustments for robust Multi-fidelity Monte Carlo under limited pilot sampling

Multi-fidelity Monte Carlo (MFMC) is a variance reduction method that leverages a multi-fidelity ensemble of models of varying cost and accuracy levels. Constructing an MFMC estimator with optimal variance requires knowledge of the correlation coefficients between the different fidelity models which are not usually known in practice. The correlations are typically estimated using offline pilot samples and the sample correlation formula, after which the MFMC method proceeds as if the estimated correlations are the true correlations. Computational cost often restricts the number of pilot samples used leading to poor correlation estimates and suboptimal estimators. Leveraging the MFMC problem setting and probabilistic information about the sample covariance matrix, we present a method to improve standard sample-based correlation estimates in the presence of limited pilot samples. We define a novel discrepancy function quantifying the estimator suboptimality which in turn facilitates selecting a correlation estimator minimizing the worst-case expected discrepancy, where the expectation is taken with respect to the pilot sampling variability. Through a simple bivariate Gaussian example and a multi-fidelity modeling application from a NASA Entry, Descent, and Landing (EDL) problem, we show that this method produces better MFMC estimators than the standard sample covariance under small pilot sample sizes and limited total budgets.

stat.ME↗

arXiv2025

Simultaneous Frequentist Calibration of Confidence Regions for Multiple Functionals in Constrained Inverse Problems

Many scientific analyses require simultaneous comparison of multiple functionals of an unknown signal at once, calling for multidimensional confidence regions with guaranteed simultaneous frequentist under structural constraints (e.g., non-negativity, shape, or physics-based). This paper unifies and extends many previous optimization-based approaches to constrained confidence region construction in linear inverse problems through the lens of statistical test inversion. We begin by reviewing the historical development of optimization-based confidence intervals for the single-functional setting, from "strict bounds" to the Burrus conjecture and its recent refutation via the aforementioned test inversion framework. We then extend this framework to the multiple-functional setting. This framework can be used to: (i) improve the calibration constants of previous methods, yielding smaller confidence regions that still preserve frequentist coverage, (ii) obtain tractable multidimensional confidence regions that need not be hyper-rectangles to better capture functional dependence structure, and (iii) generalize beyond Gaussian error distributions to generic log-concave error distributions. We provide theory establishing nominal simultaneous coverage of our methods and show quantitative volume improvements relative to prior approaches using numerical experiments.

math.ST↗

arXiv2025

RealDriveSim: A Realistic Multi-Modal Multi-Task Synthetic Dataset for Autonomous Driving

As perception models continue to develop, the need for large-scale datasets increases. However, data annotation remains far too expensive to effectively scale and meet the demand. Synthetic datasets provide a solution to boost model performance with substantially reduced costs. However, current synthetic datasets remain limited in their scope, realism, and are designed for specific tasks and applications. In this work, we present RealDriveSim, a realistic multi-modal synthetic dataset for autonomous driving that not only supports popular 2D computer vision applications but also their LiDAR counterparts, providing fine-grained annotations for up to 64 classes. We extensively evaluate our dataset for a wide range of applications and domains, demonstrating state-of-the-art results compared to existing synthetic benchmarks. The dataset is publicly available at https://realdrivesim.github.io/.

cs.CV↗

arXiv2025

Confidence intervals for functionals in constrained inverse problems via data-adaptive sampling-based calibration

We address functional uncertainty quantification for ill-posed inverse problems where it is possible to evaluate a possibly rank-deficient forward model, the observation noise distribution is known, and there are known parameter constraints. We present four constraint-aware confidence intervals extending the work of Batlle et al. (2023) by making the intervals both computationally feasible and less conservative. Our approach first shrinks the potentially unbounded constraint set compact in a data-adaptive way, obtains samples of the relevant test statistic inside this set to estimate a quantile function, and then uses these computed quantities to produce the intervals. Our data-adaptive bounding approach is based on the approach by Berger and Boos (1994), and involves defining a subset of the constraint set where the true parameter exists with high probability. This probabilistic guarantee is then incorporated into the final coverage guarantee in the form of an uncertainty budget. We then propose custom sampling algorithms to efficiently sample from this subset, even when the parameter space is high-dimensional. Optimization-based interval methods formulate confidence interval computation as two endpoint optimizations, where the optimization constraints can be set to achieve different types of interval calibration while seamlessly incorporating parameter constraints. However, choosing valid optimization constraints has been elusive. We show that all four proposed intervals achieve nominal coverage for a particular functional both theoretically and in practice, with numerical examples demonstrating superior performance of our intervals over the OSB interval in terms of both coverage and expected length. In particular, we show the superior performance in a realistic unfolding simulation from high-energy physics that is severely ill-posed and involves a rank-deficient forward model.

stat.ME↗

arXiv2023

Optimization-based frequentist confidence intervals for functionals in constrained inverse problems: Resolving the Burrus conjecture

We present an optimization-based framework to construct confidence intervals for functionals in constrained inverse problems, ensuring valid one-at-a-time frequentist coverage guarantees. Our approach builds upon the now-called strict bounds intervals, originally pioneered by Burrus (1965) and Rust and Burrus (1972), which offer ways to directly incorporate any side information about the parameters during inference without introducing external biases. This family of methods allows for uncertainty quantification in ill-posed inverse problems without needing to select a regularizing prior. By tying optimization-based intervals to an inversion of a constrained likelihood ratio test, we translate interval coverage guarantees into type I error control and characterize the resulting interval via solutions to optimization problems. Along the way, we refute the Burrus conjecture, which posited that, for possibly rank-deficient linear Gaussian models with positivity constraints, a correction based on the quantile of the chi-squared distribution with one degree of freedom suffices to shorten intervals while maintaining frequentist coverage guarantees. Our framework provides a novel approach to analyzing the conjecture, and we construct a counterexample employing a stochastic dominance argument, which we also use to disprove a general form of the conjecture. We illustrate our framework with several numerical examples and provide directions for extensions beyond the Rust-Burrus method for nonlinear, non-Gaussian settings with general constraints.

math.ST↗

arXiv2023

Posterior Uncertainty Estimation via a Monte Carlo Procedure Specialized for Data Assimilation

Through the Bayesian lens of data assimilation, uncertainty on model parameters is traditionally quantified through the posterior covariance matrix. However, in modern settings involving high-dimensional and computationally expensive forward models, posterior covariance knowledge must be relaxed to deterministic or stochastic approximations. In the carbon flux inversion literature, Chevallier et al. proposed a stochastic method capable of approximating posterior variances of linear functionals of the model parameters that is particularly well-suited for large-scale Earth-system data assimilation tasks. This note formalizes this algorithm and clarifies its properties. We provide a formal statement of the algorithm, demonstrate why it converges to the desired posterior variance quantity of interest, and provide additional uncertainty quantification allowing incorporation of the Monte Carlo sampling uncertainty into the method's Bayesian credible intervals. The methodology is demonstrated using toy simulations and a realistic carbon flux inversion observing system simulation experiment.

stat.CO↗

arXiv2022

Uncertainty Quantification of the 4th kind; optimal posterior accuracy-uncertainty tradeoff with the minimum enclosing ball

There are essentially three kinds of approaches to Uncertainty Quantification (UQ): (A) robust optimization, (B) Bayesian, (C) decision theory. Although (A) is robust, it is unfavorable with respect to accuracy and data assimilation. (B) requires a prior, it is generally brittle and posterior estimations can be slow. Although (C) leads to the identification of an optimal prior, its approximation suffers from the curse of dimensionality and the notion of risk is one that is averaged with respect to the distribution of the data. We introduce a 4th kind which is a hybrid between (A), (B), (C), and hypothesis testing. It can be summarized as, after observing a sample $x$, (1) defining a likelihood region through the relative likelihood and (2) playing a minmax game in that region to define optimal estimators and their risk. The resulting method has several desirable properties (a) an optimal prior is identified after measuring the data, and the notion of risk is a posterior one, (b) the determination of the optimal estimate and its risk can be reduced to computing the minimum enclosing ball of the image of the likelihood region under the quantity of interest map (which is fast and not subject to the curse of dimensionality). The method is characterized by a parameter in $ [0,1]$ acting as an assumed lower bound on the rarity of the observed data (the relative likelihood). When that parameter is near $1$, the method produces a posterior distribution concentrated around a maximum likelihood estimate with tight but low confidence UQ estimates. When that parameter is near $0$, the method produces a maximal risk posterior distribution with high confidence UQ estimates. In addition to navigating the accuracy-uncertainty tradeoff, the proposed method addresses the brittleness of Bayesian inference by navigating the robustness-accuracy tradeoff associated with data assimilation.

stat.ME↗

arXiv2022

Uncertainty quantification for wide-bin unfolding: one-at-a-time strict bounds and prior-optimized confidence intervals

Unfolding is an ill-posed inverse problem in particle physics aiming to infer a true particle-level spectrum from smeared detector-level data. For computational and practical reasons, these spaces are typically discretized using histograms, and the smearing is modeled through a response matrix corresponding to a discretized smearing kernel of the particle detector. This response matrix depends on the unknown shape of the true spectrum, leading to a fundamental systematic uncertainty in the unfolding problem. To handle the ill-posed nature of the problem, common approaches regularize the problem either directly via methods such as Tikhonov regularization, or implicitly by using wide-bins in the true space that match the resolution of the detector. Unfortunately, both of these methods lead to a non-trivial bias in the unfolded estimator, thereby hampering frequentist coverage guarantees for confidence intervals constructed from these methods. We propose two new approaches to addressing the bias in the wide-bin setting through methods called One-at-a-time Strict Bounds (OSB) and Prior-Optimized (PO) intervals. The OSB intervals are a bin-wise modification of an existing guaranteed-coverage procedure, while the PO intervals are based on a decision-theoretic view of the problem. Importantly, both approaches provide well-calibrated frequentist confidence intervals even in constrained and rank-deficient settings. These methods are built upon a more general answer to the wide-bin bias problem, involving unfolding with fine bins first, followed by constructing confidence intervals for linear functionals of the fine-bin counts. We test and compare these methods to other available methodologies in a wide-bin deconvolution example and a realistic particle physics simulation of unfolding a steeply falling particle spectrum.

stat.AP↗