Motivated by a real failure dataset in a two-dimensional context, this paper presents an extension of the Markov modulated Poisson process (MMPP) to two dimensions. The one-dimensional MMPP has been proposed for the modeling of dependent and non-exponential inter-failure times (in contexts as queuing, risk or reliability, among others). The novel two-dimensional MMPP allows for dependence between the two sequences of inter-failure times, while at the same time preserves the MMPP properties, marginally. The generalization is based on the Marshall-Olkin exponential distribution. Inference is undertaken for the new model through a method combining a matching moments approach with an Approximate Bayesian Computation (ABC) algorithm. The performance of the method is shown on simulated and real datasets representing times and distances covered between consecutive failures in a public transport company. For the real dataset, some quantities of importance associated with the reliability of the system are estimated as the probabilities and expected number of failures at different times and distances covered by trains until the occurrence of a failure.
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Maximum entropy (Maxent) models are a class of statistical models that use the maximum entropy principle to estimate probability distributions from data. Due to the size of modern data sets, Maxent models need efficient optimization algorithms to scale well for big data applications. State-of-the-art algorithms for Maxent models, however, were not originally designed to handle big data sets; these algorithms either rely on technical devices that may yield unreliable numerical results, scale poorly, or require smoothness assumptions that many practical Maxent models lack. In this paper, we present novel optimization algorithms that overcome the shortcomings of state-of-the-art algorithms for training large-scale, non-smooth Maxent models. Our proposed first-order algorithms leverage the Kullback-Leibler divergence to train large-scale and non-smooth Maxent models efficiently. For Maxent models with discrete probability distribution of $n$ elements built from samples, each containing $m$ features, the stepsize parameters estimation and iterations in our algorithms scale on the order of $O(mn)$ operations and can be trivially parallelized. Moreover, the strong $\ell_{1}$ convexity of the Kullback--Leibler divergence allows for larger stepsize parameters, thereby speeding up the convergence rate of our algorithms. To illustrate the efficiency of our novel algorithms, we consider the problem of estimating probabilities of fire occurrences as a function of ecological features in the Western US MTBS-Interagency wildfire data set. Our numerical results show that our algorithms outperform the state of the arts by one order of magnitude and yield results that agree with physical models of wildfire occurrence and previous statistical analyses of wildfire drivers.
In decision-making, maxitive functions are used for worst-case and best-case evaluations. Maxitivity gives rise to a rich structure that is well-studied in the context of the pointwise order. In this article, we investigate maxitivity with respect to general preorders and provide a representation theorem for such functionals. The results are illustrated for different stochastic orders in the literature, including the usual stochastic order, the increasing convex/concave order, and the dispersive order.
The purpose of anonymizing structured data is to protect the privacy of individuals in the data while retaining the statistical properties of the data. There is a large body of work that examines anonymization vulnerabilities. Focusing on strong anonymization mechanisms, this paper examines a number of prominent attack papers and finds several problems, all of which lead to overstating risk. First, some papers fail to establish a correct statistical inference baseline (or any at all), leading to incorrect measures. Notably, the reconstruction attack from the US Census Bureau that led to a redesign of its disclosure method made this mistake. We propose the non-member framework, an improved method for how to compute a more accurate inference baseline, and give examples of its operation. Second, some papers don't use a realistic membership base rate, leading to incorrect precision measures if precision is reported. Third, some papers unnecessarily report measures in such a way that it is difficult or impossible to assess risk. Virtually the entire literature on membership inference attacks, dozens of papers, make one or both of these errors. We propose that membership inference papers report precision/recall values using a representative range of base rates.
This article is concerned with the multilevel Monte Carlo (MLMC) methods for approximating expectations of some functions of the solution to the Heston 3/2-model from mathematical finance, which takes values in $(0, \infty)$ and possesses superlinearly growing drift and diffusion coefficients. To discretize the SDE model, a new Milstein-type scheme is proposed to produce independent sample paths. The proposed scheme can be explicitly solved and is positivity-preserving unconditionally, i.e., for any time step-size $h>0$. This positivity-preserving property for large discretization time steps is particularly desirable in the MLMC setting. Furthermore, a mean-square convergence rate of order one is proved in the non-globally Lipschitz regime, which is not trivial, as the diffusion coefficient grows super-linearly. The obtained order-one convergence in turn promises the desired relevant variance of the multilevel estimator and justifies the optimal complexity $\mathcal{O}(\epsilon^{-2})$ for the MLMC approach, where $\epsilon > 0$ is the required target accuracy. Numerical experiments are finally reported to confirm the theoretical findings.
Mendelian randomization is an instrumental variable method that utilizes genetic information to investigate the causal effect of a modifiable exposure on an outcome. In most cases, the exposure changes over time. Understanding the time-varying causal effect of the exposure can yield detailed insights into mechanistic effects and the potential impact of public health interventions. Recently, a growing number of Mendelian randomization studies have attempted to explore time-varying causal effects. However, the proposed approaches oversimplify temporal information and rely on overly restrictive structural assumptions, limiting their reliability in addressing time-varying causal problems. This paper considers a novel approach to estimate time-varying effects through continuous-time modelling by combining functional principal component analysis and weak-instrument-robust techniques. Our method effectively utilizes available data without making strong structural assumptions and can be applied in general settings where the exposure measurements occur at different timepoints for different individuals. We demonstrate through simulations that our proposed method performs well in estimating time-varying effects and provides reliable inference results when the time-varying effect form is correctly specified. The method could theoretically be used to estimate arbitrarily complex time-varying effects. However, there is a trade-off between model complexity and instrument strength. Estimating complex time-varying effects requires instruments that are unrealistically strong. We illustrate the application of this method in a case study examining the time-varying effects of systolic blood pressure on urea levels.
Multi-contrast (MC) Magnetic Resonance Imaging (MRI) reconstruction aims to incorporate a reference image of auxiliary modality to guide the reconstruction process of the target modality. Known MC reconstruction methods perform well with a fully sampled reference image, but usually exhibit inferior performance, compared to single-contrast (SC) methods, when the reference image is missing or of low quality. To address this issue, we propose DuDoUniNeXt, a unified dual-domain MRI reconstruction network that can accommodate to scenarios involving absent, low-quality, and high-quality reference images. DuDoUniNeXt adopts a hybrid backbone that combines CNN and ViT, enabling specific adjustment of image domain and k-space reconstruction. Specifically, an adaptive coarse-to-fine feature fusion module (AdaC2F) is devised to dynamically process the information from reference images of varying qualities. Besides, a partially shared shallow feature extractor (PaSS) is proposed, which uses shared and distinct parameters to handle consistent and discrepancy information among contrasts. Experimental results demonstrate that the proposed model surpasses state-of-the-art SC and MC models significantly. Ablation studies show the effectiveness of the proposed hybrid backbone, AdaC2F, PaSS, and the dual-domain unified learning scheme.
This paper presents a novel stochastic optimisation methodology to perform empirical Bayesian inference in semi-blind image deconvolution problems. Given a blurred image and a parametric class of possible operators, the proposed optimisation approach automatically calibrates the parameters of the blur model by maximum marginal likelihood estimation, followed by (non-blind) image deconvolution by maximum-a-posteriori estimation conditionally to the estimated model parameters. In addition to the blur model, the proposed approach also automatically calibrates the noise variance as well as any regularisation parameters. The marginal likelihood of the blur, noise variance, and regularisation parameters is generally computationally intractable, as it requires calculating several integrals over the entire solution space. Our approach addresses this difficulty by using a stochastic approximation proximal gradient optimisation scheme, which iteratively solves such integrals by using a Moreau-Yosida regularised unadjusted Langevin Markov chain Monte Carlo algorithm. This optimisation strategy can be easily and efficiently applied to any model that is log-concave, and by using the same gradient and proximal operators that are required to compute the maximum-a-posteriori solution by convex optimisation. We provide convergence guarantees for the proposed optimisation scheme under realistic and easily verifiable conditions and subsequently demonstrate the effectiveness of the approach with a series of deconvolution experiments and comparisons with alternative strategies from the state of the art.
In this article, we introduce a notion of depth functions for data types that are not given in statistical standard data formats. Data depth functions have been intensively studied for normed vector spaces. However, a discussion on depth functions on data where one specific data structure cannot be presupposed is lacking. We call such data non-standard data. To define depth functions for non-standard data, we represent the data via formal concept analysis which leads to a unified data representation. Besides introducing these depth functions, we give a systematic basis of depth functions for non-standard using formal concept analysis by introducing structural properties. Furthermore, we embed the generalised Tukey depth into our concept of data depth and analyse it using the introduced structural properties. Thus, this article provides the mathematical formalisation of centrality and outlyingness for non-standard data. Thereby, we increase the number of spaces in which centrality can be discussed. In particular, it gives a basis to define further depth functions and statistical inference methods for non-standard data.
Charts, figures, and text derived from data play an important role in decision making, from data-driven policy development to day-to-day choices informed by online articles. Making sense of, or fact-checking, outputs means understanding how they relate to the underlying data. Even for domain experts with access to the source code and data sets, this poses a significant challenge. In this paper we introduce a new program analysis framework which supports interactive exploration of fine-grained I/O relationships directly through computed outputs, making use of dynamic dependence graphs. Our main contribution is a novel notion in data provenance which we call related inputs, a relation of mutual relevance or "cognacy" which arises between inputs when they contribute to common features of the output. Queries of this form allow readers to ask questions like "What outputs use this data element, and what other data elements are used along with it?". We show how Jonsson and Tarski's concept of conjugate operators on Boolean algebras appropriately characterises the notion of cognacy in a dependence graph, and give a procedure for computing related inputs over such a graph.
Robust Markov Decision Processes (RMDPs) are a widely used framework for sequential decision-making under parameter uncertainty. RMDPs have been extensively studied when the objective is to maximize the discounted return, but little is known for average optimality (optimizing the long-run average of the rewards obtained over time) and Blackwell optimality (remaining discount optimal for all discount factors sufficiently close to 1). In this paper, we prove several foundational results for RMDPs beyond the discounted return. We show that average optimal policies can be chosen stationary and deterministic for sa-rectangular RMDPs but, perhaps surprisingly, that history-dependent (Markovian) policies strictly outperform stationary policies for average optimality in s-rectangular RMDPs. We also study Blackwell optimality for sa-rectangular RMDPs, where we show that {\em approximate} Blackwell optimal policies always exist, although Blackwell optimal policies may not exist. We also provide a sufficient condition for their existence, which encompasses virtually any examples from the literature. We then discuss the connection between average and Blackwell optimality, and we describe several algorithms to compute the optimal average return. Interestingly, our approach leverages the connections between RMDPs and stochastic games.
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