arXiv Statistics @arxiv_stats@qoto.org

In a criminal investigation, an inferential error occurs when the probability that a suspect is the source of some evidence -- such as a fingerprint -- is taken as the probability of guilt. This is known as the ultimate issue error, and the same error occurs in statistical inference when the probability that a parameter equals some value is incorrectly taken to be the probability of a hypothesis. Almost all statistical inference in the social and biological sciences is subject to this error, and replacing every instance of "hypothesis testing" with "parameter testing" in these fields would more accurately describe the target of inference. The relationship between parameter values and quantities derived from them, such as p-values or Bayes factors, have no direct quantitative relationship with scientific hypotheses. Here, we describe the problem, its consequences, and suggest options for improving scientific inference.

This work derives extremal tail bounds for the Gaussian trace estimator applied to a real symmetric matrix. We define a partial ordering on the eigenvalues, so that when a matrix has greater spectrum under this ordering, its estimator will have worse tail bounds. This is done for two families of matrices: positive semidefinite matrices with bounded effective rank, and indefinite matrices with bounded 2-norm and fixed Frobenius norm. In each case, the tail region is defined rigorously and is constant for a given family.

We present a procedure for handling asymmetric errors. Many results in particle physics are presented as values with different positive and negative errors, and there is no consistent procedure for handling them. We consider the difference between errors quoted using pdfs and using likelihoods, and the difference between the rms spread of a measurement and the 68\% central confidence region. We provide a comprehensive analysis of the possibilities, and software tools to enable their use.

Hourly maxima of 3-second wind gust speeds are prominent indicators of the severity of wind storms, and accurately forecasting them is thus essential for populations, civil authorities and insurance companies. Space-time max-stable models appear as natural candidates for this, but those explored so far are not suited for forecasting and, more generally, the forecasting literature for max-stable fields is limited. To fill this gap, we consider a specific space-time max-stable model, more precisely a max-autoregressive model with advection, that is well-adapted to model and forecast atmospheric variables. We apply it, as well as our related forecasting strategy, to reanalysis 3-second wind gust data for France in 1999, and show good performance compared to a competitor model. On top of demonstrating the practical relevance of our model, we meticulously study its theoretical properties and show the consistency and asymptotic normality of the space-time pairwise likelihood estimator which is used to calibrate the model.

Biomanufacturing plays an important role in supporting public health and the growth of the bioeconomy. Modeling and studying the interaction effects among various input variables is very critical for obtaining a scientific understanding and process specification in biomanufacturing. In this paper, we use the ShapleyOwen indices to measure the interaction effects for the policy-augmented Bayesian network (PABN) model, which characterizes the risk- and science-based understanding of production bioprocess mechanisms. In order to facilitate efficient interaction effect quantification, we propose a sampling-based simulation estimation framework. In addition, to further improve the computational efficiency, we develop a non-nested simulation algorithm with sequential sampling, which can dynamically allocate the simulation budget to the interactions with high uncertainty and therefore estimate the interaction effects more accurately under a total fixed budget setting.

Multi-regional clinical trial (MRCT) has been common practice for drug development and global registration. The FDA guidance "Demonstrating Substantial Evidence of Effectiveness for Human Drug and Biological Products Guidance for Industry" (FDA, 2019) requires that substantial evidence of effectiveness of a drug/biologic product to be demonstrated for market approval. In the situations where two pivotal MRCTs are needed to establish effectiveness of a specific indication for a drug or biological product, a systematic approach of consistency evaluation for regional effect is crucial. In this paper, we first present some existing regional consistency evaluations in a unified way that facilitates regional sample size calculation under the simple fixed effect model. Second, we extend the two commonly used consistency assessment criteria of MHLW (2007) in the context of two MRCTs and provide their evaluation and regional sample size calculation. Numerical studies demonstrate the proposed regional sample size attains the desired probability of showing regional consistency. A hypothetical example is provided for illustration of application. We provide an R package for implementation.

This paper is in the field of stochastic Multi-Armed Bandits (MABs), i.e. those sequential selection techniques able to learn online using only the feedback given by the chosen option (a.k.a. $arm$). We study a particular case of the rested bandits in which the arms' expected reward is monotonically non-decreasing and concave. We study the inherent sample complexity of the regret minimization problem by deriving suitable regret lower bounds. Then, we design an algorithm for the rested case $\textit{R-ed-UCB}$, providing a regret bound depending on the properties of the instance and, under certain circumstances, of $\widetilde{\mathcal{O}}(T^{\frac{2}{3}})$. We empirically compare our algorithms with state-of-the-art methods for non-stationary MABs over several synthetically generated tasks and an online model selection problem for a real-world dataset

Methods to handle missing data have been extensively explored in the context of estimation and descriptive studies, with multiple imputation being the most widely used method in clinical research. However, in the context of clinical risk prediction models, where the goal is often to achieve high prediction accuracy and to make predictions for future patients, there are different considerations regarding the handling of missing data. As a result, deterministic imputation is better suited to the setting of clinical risk prediction models, since the outcome is not included in the imputation model and the imputation method can be easily applied to future patients. In this paper, we provide a tutorial demonstrating how to conduct bootstrapping followed by deterministic imputation of missing data to construct and internally validate the performance of a clinical risk prediction model in the presence of missing data. Extensive simulation study results are provided to help guide decision-making in real-world applications.

We propose a random-effects approach to missing values for linear mixed model (LMM) analysis. The method converts a LMM with missing covariates to another LMM without missing covariates. The standard LMM analysis tools for longitudinal data then apply. Performance of the method is evaluated empirically, and compared with alternative approaches, including the popular MICE procedure of multiple imputation. Theoretical explanations are given for the patterns observed in the simulation studies. A real-data example is discussed.

Variational empirical Bayes (VEB) methods provide a practically attractive approach to fitting large, sparse, multiple regression models. These methods usually use coordinate ascent to optimize the variational objective function, an approach known as coordinate ascent variational inference (CAVI). Here we propose alternative optimization approaches based on gradient-based (quasi-Newton) methods, which we call gradient-based variational inference (GradVI). GradVI exploits a recent result from Kim et. al. [arXiv:2208.10910] which writes the VEB regression objective function as a penalized regression. Unfortunately the penalty function is not available in closed form, and we present and compare two approaches to dealing with this problem. In simple situations where CAVI performs well, we show that GradVI produces similar predictive performance, and GradVI converges in fewer iterations when the predictors are highly correlated. Furthermore, unlike CAVI, the key computations in GradVI are simple matrix-vector products, and so GradVI is much faster than CAVI in settings where the design matrix admits fast matrix-vector products (e.g., as we show here, trendfiltering applications) and lends itself to parallelized implementations in ways that CAVI does not. GradVI is also very flexible, and could exploit automatic differentiation to easily implement different prior families. Our methods are implemented in an open-source Python software, GradVI (available from https://github.com/stephenslab/gradvi ).

In this paper, we developed a nonparametric relative entropy (RlEn) for modelling loss of complexity in intermittent time series. This technique consists of two steps. First, we carry out a nonlinear autoregressive model where the lag order is determined by a Bayesian Information Criterion (BIC), and complexity of each intermittent time series is obtained by our novel relative entropy. Second, change-points in complexity were detected by using the cumulative sum (CUSUM) based method. Using simulations and compared to the popular method appropriate entropy (ApEN), the performance of RlEn was assessed for its (1) ability to localise complexity change-points in intermittent time series; (2) ability to faithfully estimate underlying nonlinear models. The performance of the proposal was then examined in a real analysis of fatigue-induced changes in the complexity of human motor outputs. The results demonstrated that the proposed method outperformed the ApEn in accurately detecting complexity changes in intermittent time series segments.

We give a dimension-independent sparsification result for suprema of centered Gaussian processes: Let $T$ be any (possibly infinite) bounded set of vectors in $\mathbb{R}^n$, and let $\{\boldsymbol{X}_t\}_{t\in T}$ be the canonical Gaussian process on $T$. We show that there is an $O_\varepsilon(1)$-size subset $S \subseteq T$ and a set of real values $\{c_s\}_{s \in S}$ such that $\sup_{s \in S} \{\boldsymbol{X}_s + c_s\}$ is an $\varepsilon$-approximator of $\sup_{t \in T} {\boldsymbol{X}}_t$. Notably, the size of $S$ is completely independent of both the size of $T$ and of the ambient dimension $n$. We use this to show that every norm is essentially a junta when viewed as a function over Gaussian space: Given any norm $ν(x)$ on $\mathbb{R}^n$, there is another norm $ψ(x)$ which depends only on the projection of $x$ along $O_\varepsilon(1)$ directions, for which $ψ({\boldsymbol{g}})$ is a multiplicative $(1 \pm \varepsilon)$-approximation of $ν({\boldsymbol{g}})$ with probability $1-\varepsilon$ for ${\boldsymbol{g}} \sim N(0,I_n)$. We also use our sparsification result for suprema of centered Gaussian processes to give a sparsification lemma for convex sets of bounded geometric width: Any intersection of (possibly infinitely many) halfspaces in $\mathbb{R}^n$ that are at distance $O(1)$ from the origin is $\varepsilon$-close, under $N(0,I_n)$, to an intersection of only $O_\varepsilon(1)$ many halfspaces. We describe applications to agnostic learning and tolerant property testing.

Adaptive causal representation learning from observational data is presented, integrated with an efficient sample splitting technique within the semiparametric estimating equation framework. The support points sample splitting (SPSS), a subsampling method based on energy distance, is employed for efficient double machine learning (DML) in causal inference. The support points are selected and split as optimal representative points of the full raw data in a random sample, in contrast to the traditional random splitting, and providing an optimal sub-representation of the underlying data generating distribution. They offer the best representation of a full big dataset, whereas the unit structural information of the underlying distribution via the traditional random data splitting is most likely not preserved. Three machine learning estimators were adopted for causal inference, support vector machine (SVM), deep learning (DL), and a hybrid super learner (SL) with deep learning (SDL), using SPSS. A comparative study is conducted between the proposed SVM, DL, and SDL representations using SPSS, and the benchmark results from Chernozhukov et al. (2018), which employed random forest, neural network, and regression trees with a random k-fold cross-fitting technique on the 401(k)-pension plan real data. The simulations show that DL with SPSS and the hybrid methods of DL and SL with SPSS outperform SVM with SPSS in terms of computational efficiency and the estimation quality, respectively.

Existing methods to summarize posterior inference for mixture models focus on identifying a point estimate of the implied random partition for clustering, with density estimation as a secondary goal (Wade and Ghahramani, 2018; Dahl et al., 2022). We propose a novel approach for summarizing posterior inference in nonparametric Bayesian mixture models, prioritizing density estimation of the mixing measure (or mixture) as an inference target. One of the key features is the model-agnostic nature of the approach, which remains valid under arbitrarily complex dependence structures in the underlying sampling model. Using a decision-theoretic framework, our method identifies a point estimate by minimizing posterior expected loss. A loss function is defined as a discrepancy between mixing measures. Estimating the mixing measure implies inference on the mixture density and the random partition. Exploiting the discrete nature of the mixing measure, we use a version of sliced Wasserstein distance. We introduce two specific variants for Gaussian mixtures. The first, mixed sliced Wasserstein, applies generalized geodesic projections on the product of the Euclidean space and the manifold of symmetric positive definite matrices. The second, sliced mixture Wasserstein, leverages the linearity of Gaussian mixture measures for efficient projection.

Computer models are used as a way to explore complex physical systems. Stationary Gaussian process emulators, with their accompanying uncertainty quantification, are popular surrogates for computer models. However, many computer models are not well represented by stationary Gaussian processes models. Deep Gaussian processes have been shown to be capable of capturing non-stationary behaviors and abrupt regime changes in the computer model response. In this paper, we explore the properties of two deep Gaussian process formulations within the context of computer model emulation. For one of these formulations, we introduce a new parameter that controls the amount of smoothness in the deep Gaussian process layers. We adapt a stochastic variational approach to inference for this model, allowing for prior specification and posterior exploration of the smoothness of the response surface. Our approach can be applied to a large class of computer models, and scales to arbitrarily large simulation designs. The proposed methodology was motivated by the need to emulate an astrophysical model of the formation of binary black hole mergers.

In this paper, we propose an new the CUSUM sequential test (control chart, stopping time) with the observation-adjusted control limits (CUSUM-OAL) for monitoring quickly and adaptively the change in distribution of a sequential observations. We give the estimation of the in-control and the out-of-control average run lengths (ARLs) of the CUSUM-OAL test. The theoretical results are illustrated by numerical simulations in detecting $α$ shifts of the extreme heavy-tailed distribution observations sequence.

Among all metrics based on p-norms, the Manhattan (p=1), euclidean (p=2) and Chebyshev distances (p=infinity) are the most widely used for their interpretability, simplicity and technical convenience. But these are not the only arguments for the ubiquity of these three p-norms. This article proves that there is a volume-surface correspondence property that is unique to them. More precisely, it is shown that sampling uniformly from the volume of an n-dimensional p-ball and projecting to its surface is equivalent to directly sampling uniformly from its surface if and only if p is 1, 2 or infinity. Sampling algorithms and their implementations in Python are also provided.

In the recent two decades, the Mexico City Metropolitan Area (MCMA) has been plagued by high concentrations of air pollutants, risking the health integrity of its inhabitants. Although some policies have been undertaken, they have been insufficient to deplete high air pollutants. Environmental contingencies are commonly imposed when the ozone concentration overpasses a certain threshold, which is well above the recommended maximum by the WHO. This work used a causal version of a generalized Morlet wavelet to characterize the dynamics of daily ozone concentration in the MCMA. The results indicated that the formation of dangerous ozone concentration levels is a consequence of accumulation and incomplete dissipation effects acting over a wide range of time scales. Ozone contingencies occurred when the wavelet coefficient power is increasing, which was linked to an inti-persistence behavior. It was proposed that the wavelet methodology could be used as a further tool for signaling the potential formation of adverse ozone pollution scenarios.

Bayesian inference is used to estimate continuous parameter values given measured data in many fields of science. The method relies on conditional probability densities to describe information about both data and parameters, yet the notion of conditional densities is inadmissible: probabilities of the same physical event, computed from conditional densities under different parameterizations, may be inconsistent. We show that this inconsistency, together with acausality in hierarchical methods, invalidate a variety of commonly applied Bayesian methods when applied to problems in the physical world, including trans-dimensional inference, general Bayesian dimensionality reduction methods, and hierarchical and empirical Bayes. Models in parameter spaces of different dimensionalities cannot be compared, invalidating the concept of natural parsimony, the probabilistic counterpart to Occams Razor. Bayes theorem itself is inadmissible, and Bayesian inference applied to parameters that characterize physical properties requires reformulation.

The quality of numerical computations can be measured through their forward error, for which finding good error bounds is challenging in general. For several algorithms and using stochastic rounding (SR), probabilistic analysis has been shown to be an effective alternative for obtaining tight error bounds. This analysis considers the distribution of errors and evaluates the algorithm's performance on average. Using martingales and the Azuma-Hoeffding inequality, it provides error bounds that are valid with a certain probability and in $\mathcal{O}(\sqrt{n}u)$ instead of deterministic worst-case bounds in $\mathcal{O}(nu)$, where $n$ is the number of operations and $u$ is the unit roundoff.In this paper, we present a general method that automatically constructs a martingale for any computation scheme with multi-linear errors based on additions, subtractions, and multiplications. We apply this generalization to algorithms previously studied with SR, such as pairwise summation and the Horner algorithm, and prove equivalent results. We also analyze a previously unstudied algorithm, Karatsuba polynomial multiplication, which illustrates that the method can handle reused intermediate computations.