Showing papers by "Rina Foygel Barber published in 2023"

De Finetti's Theorem and Related Results for Infinite Weighted Exchangeable Sequences

Abstract: De Finetti's theorem, also called the de Finetti-Hewitt-Savage theorem, is a foundational result in probability and statistics. Roughly, it says that an infinite sequence of exchangeable random variables can always be written as a mixture of independent and identically distributed (i.i.d.) sequences of random variables. In this paper, we consider a weighted generalization of exchangeability that allows for weight functions to modify the individual distributions of the random variables along the sequence, provided that -- modulo these weight functions -- there is still some common exchangeable base measure. We study conditions under which a de Finetti-type representation exists for weighted exchangeable sequences, as a mixture of distributions which satisfy a weighted form of the i.i.d. property. Our approach establishes a nested family of conditions that lead to weighted extensions of other well-known related results as well, in particular, extensions of the zero-one law and the law of large numbers.

Selective inference for clustering with unknown variance

Abstract: In many modern statistical problems, the limited available data must be used both to develop the hypotheses to test, and to test these hypotheses-that is, both for exploratory and confirmatory data analysis. Reusing the same dataset for both exploration and testing can lead to massive selection bias, leading to many false discoveries. Selective inference is a framework that allows for performing valid inference even when the same data is reused for exploration and testing. In this work, we are interested in the problem of selective inference for data clustering, where a clustering procedure is used to hypothesize a separation of the data points into a collection of subgroups, and we then wish to test whether these data-dependent clusters in fact represent meaningful differences within the data. Recent work by Gao et al. [2022] provides a framework for doing selective inference for this setting, where a hierarchical clustering algorithm is used for producing the cluster assignments, which was then extended to k-means clustering by Chen and Witten [2022]. Both these works rely on assuming a known covariance structure for the data, but in practice, the noise level needs to be estimated-and this is particularly challenging when the true cluster structure is unknown. In our work, we extend this work to the setting of noise with unknown variance, and provide a selective inference method for this more general setting. Empirical results show that our new method is better able to maintain high power while controlling Type I error when the true noise level is unknown.

Bagging Provides Assumption-free Stability

Abstract: Bagging is an important technique for stabilizing machine learning models. In this paper, we derive a finite-sample guarantee on the stability of bagging for any model. Our result places no assumptions on the distribution of the data, on the properties of the base algorithm, or on the dimensionality of the covariates. Our guarantee applies to many variants of bagging and is optimal up to a constant. Empirical results validate our findings, showing that bagging successfully stabilizes even highly unstable base algorithms.

Distribution-free inference with hierarchical data

Abstract: This paper studies distribution-free inference in settings where the data set has a hierarchical structure -- for example, groups of observations, or repeated measurements. In such settings, standard notions of exchangeability may not hold. To address this challenge, a hierarchical form of exchangeability is derived, facilitating extensions of distribution-free methods, including conformal prediction and jackknife+. While the standard theoretical guarantee obtained by the conformal prediction framework is a marginal predictive coverage guarantee, in the special case of independent repeated measurements, it is possible to achieve a stronger form of coverage -- the"second-moment coverage"property -- to provide better control of conditional miscoverage rates, and distribution-free prediction sets that achieve this property are constructed. Simulations illustrate that this guarantee indeed leads to uniformly small conditional miscoverage rates. Empirically, this stronger guarantee comes at the cost of a larger width of the prediction set in scenarios where the fitted model is poorly calibrated, but this cost is very mild in cases where the fitted model is accurate.

Pharmacokinetic Analysis of Enhancement-Constrained Acceleration (ECA) reconstruction-based high temporal resolution breast DCE-MRI

Abstract: The high spatial and temporal resolution of dynamic contrast-enhanced MRI (DCE-MRI) can improve the diagnostic accuracy of breast cancer screening in patients who have dense breasts or are at high risk of breast cancer. However, the spatiotemporal resolution of DCE-MRI is limited by technical issues in clinical practice. Our earlier work demonstrated the use of image reconstruction with enhancement-constrained acceleration (ECA) to increase temporal resolution. ECA exploits the correlation in k-space between successive image acquisitions. Because of this correlation, and due to the very sparse enhancement at early times after contrast media injection, we can reconstruct images from highly under-sampled k-space data. Our previous results showed that ECA reconstruction at 0.25 seconds per image (4 Hz) can estimate bolus arrival time (BAT) and initial enhancement slope (iSlope) more accurately than a standard inverse fast Fourier transform (IFFT) when k-space data is sampled following a Cartesian based sampling trajectory with adequate signal-to-noise ratio (SNR). In this follow-up study, we investigated the effect of different Cartesian based sampling trajectories, SNRs and acceleration rates on the performance of ECA reconstruction in estimating contrast media kinetics in lesions (BAT, iSlope and Ktrans) and in arteries (Peak signal intensity of first pass, time to peak, and BAT). We further validated ECA reconstruction with a flow phantom experiment. Our results show that ECA reconstruction of k-space data acquired with ‘Under-sampling with Repeated Advancing Phase’ (UnWRAP) trajectories with an acceleration factor of 14, and temporal resolution of 0.5 s/image and high SNR (SNR ≥ 30 dB, noise standard deviation (std) < 3%) ensures minor errors (5% or 1 s error) in lesion kinetics. Medium SNR (SNR ≥ 20 dB, noise std ≤ 10%) was needed to accurately measure arterial enhancement kinetics. Our results also suggest that accelerated temporal resolution with ECA with 0.5 s/image is practical.

Iterative Approximate Cross-Validation

Abstract: Cross-validation (CV) is one of the most popular tools for assessing and selecting predictive models. However, standard CV suffers from high computational cost when the number of folds is large. Recently, under the empirical risk minimization (ERM) framework, a line of works proposed efficient methods to approximate CV based on the solution of the ERM problem trained on the full dataset. However, in large-scale problems, it can be hard to obtain the exact solution of the ERM problem, either due to limited computational resources or due to early stopping as a way of preventing overfitting. In this paper, we propose a new paradigm to efficiently approximate CV when the ERM problem is solved via an iterative first-order algorithm, without running until convergence. Our new method extends existing guarantees for CV approximation to hold along the whole trajectory of the algorithm, including at convergence, thus generalizing existing CV approximation methods. Finally, we illustrate the accuracy and computational efficiency of our method through a range of empirical studies.

Simultaneous activity and attenuation estimation in TOF-PET with TV-constrained nonconvex optimization

Abstract: An alternating direction method of multipliers (ADMM) framework is developed for nonsmooth biconvex optimization for inverse problems in imaging. In particular, the simultaneous estimation of activity and attenuation (SAA) problem in time-of-flight positron emission tomography (TOF-PET) has such a structure when maximum likelihood estimation (MLE) is employed. The ADMM framework is applied to MLE for SAA in TOF-PET, resulting in the ADMM-SAA algorithm. This algorithm is extended by imposing total variation (TV) constraints on both the activity and attenuation map, resulting in the ADMM-TVSAA algorithm. The performance of these algorithms is illustrated using the standard maximum likelihood activity and attenuation estimation (MLAA) algorithm as a reference.

Integrating Uncertainty Awareness into Conformalized Quantile Regression

Abstract: Conformalized Quantile Regression (CQR) is a recently proposed method for constructing prediction intervals for a response $Y$ given covariates $X$, without making distributional assumptions. However, as we demonstrate empirically, existing constructions of CQR can be ineffective for problems where the quantile regressors perform better in certain parts of the feature space than others. The reason is that the prediction intervals of CQR do not distinguish between two forms of uncertainty: first, the variability of the conditional distribution of $Y$ given $X$ (i.e., aleatoric uncertainty), and second, our uncertainty in estimating this conditional distribution (i.e., epistemic uncertainty). This can lead to uneven coverage, with intervals that are overly wide (or overly narrow) in regions where epistemic uncertainty is low (or high). To address this, we propose a new variant of the CQR methodology, Uncertainty-Aware CQR (UACQR), that explicitly separates these two sources of uncertainty to adjust quantile regressors differentially across the feature space. Compared to CQR, our methods enjoy the same distribution-free theoretical guarantees for coverage properties, while demonstrating in our experiments stronger conditional coverage in simulated settings and tighter intervals on a range of real-world data sets.

Conformalized matrix completion

Abstract: Matrix completion aims to estimate missing entries in a data matrix, using the assumption of a low-complexity structure (e.g., low rank) so that imputation is possible. While many effective estimation algorithms exist in the literature, uncertainty quantification for this problem has proved to be challenging, and existing methods are extremely sensitive to model misspecification. In this work, we propose a distribution-free method for predictive inference in the matrix completion problem. Our method adapts the framework of conformal prediction, which provides confidence intervals with guaranteed distribution-free validity in the setting of regression, to the problem of matrix completion. Our resulting method, conformalized matrix completion (cmc), offers provable predictive coverage regardless of the accuracy of the low-rank model. Empirical results on simulated and real data demonstrate that cmc is robust to model misspecification while matching the performance of existing model-based methods when the model is correct.