Rational approximations are introduced and studied in granular graded rough sets and generalizations thereof by the first author in recent research papers. The concept of rationality is determined by related ontologies and coherence between granularity, mereology and approximations in the context. In addition, a framework for rational approximations is introduced by her in the mentioned paper(s). Granular approximations constructed as per the procedures of variable precision rough sets (VPRS) are likely to be more rational than those constructed from a classical perspective under certain conditions. This may continue to hold for some generalizations of the former. However, a formal characterization of such conditions is not available in the previously published literature. In this research, theoretical aspects of the problem are critically examined, uniform generalizations of granular VPRS are introduced, new connections with granular graded rough sets are proved, appropriate concepts of substantial parthood are introduced, their extent of compatibility with the framework is accessed, and the framework is extended. Basic assumptions are explained in detail, and additional examples are constructed for readability. Furthermore, meta applications to cluster validation, image segmentation and dynamic sorting are invented. Extensions to direct generalizations of VPRS such as probabilistic rough sets are a natural consequence of the work.

"Prediction" is a broad and generic term that describes any process for obtaining guesses of unseen variables based on any available information, as well as each of these guesses. On the other hand, "forecasting" is a more specific term that describes any process for issuing predictions for future variables based on information (which most commonly takes the form of time series) about the present and the past, with these particular predictions being broadly called "forecasts". Forecasting is a key theme and topic for this study. Therefore, in what follows, the general focus will be on it and not on prediction in general, although many of the statements and methods that will be referring to it are equally relevant and applicable to other prediction types. The origins of forecasting trace back to the early humans and their pronounced need for certainty in the practical endeavour of supporting their various everyday life decisions (Petropoulos et al. 2022). Thus, forecasting has met until today and still meets numerous implementations, formal and informal. Independently of their exact categorization and features, the formal implementations of forecasting rely, in principal, on concepts, theory and practice that originate from or can be attributed to the predictive branch of statistical modelling, although forecasting is also considered as an entire field on its own because of the major role that the temporal dependence plays in the formulation of its methods. The predictive branch of statistical modelling exhibits profound and fundamental differences with respect to the descriptive and explanatory ones, as it is thoroughly explained in Shmueli (2010).

Shape constraints yield flexible middle grounds between fully nonparametric and fully parametric approaches to modeling distributions of data. The specific assumption of log-concavity is motivated by applications across economics, survival modeling, and reliability theory. However, there do not currently exist valid tests for whether the underlying density of given data is log-concave. The recent universal inference methodology provides a valid test. The universal test relies on maximum likelihood estimation (MLE), and efficient methods already exist for finding the log-concave MLE. This yields the first test of log-concavity that is provably valid in finite samples in any dimension, for which we also establish asymptotic consistency results. Empirically, we find that the highest power is obtained by using random projections to convert the d-dimensional testing problem into many one-dimensional problems, leading to a simple procedure that is statistically and computationally efficient.

However, before ice core data can have scientific value, the chronology must be inferred by estimating the age as a function of depth. Under certain conditions, chemicals locked in the ice display quasi-periodic cycles that delineate annual layers. Manually counting these noisy seasonal patterns to infer the chronology can be an imperfect and time-consuming process, and does not capture uncertainty in a principled fashion. In addition, several ice cores may be collected from a region, introducing an aspect of spatial correlation between them. We present an exploration of the use of probabilistic models for automatic dating of ice cores, using probabilistic programming to showcase its use for prototyping, automatic inference and maintainability, and demonstrate common failure modes of these tools.

Current methods for regularization in machine learning require quite specific model assumptions (e.g. a kernel shape) that are not derived from prior knowledge about the application, but must be imposed merely to make the method work. We show in this paper that regularization can indeed be achieved by assuming nothing but invariance principles (w.r.t. scaling, translation, and rotation of input and output space) and the degree of differentiability of the true function. Concretely, we derive a novel (non-Gaussian) stochastic process from the above minimal assumptions, and we present a corresponding Bayesian inference method for regression. The mean posterior turns out to be a polyharmonic spline, and the posterior process is a mixture of t-processes. Compared with Gaussian process regression, the proposed method shows equal performance and has the advantages of being (i) less arbitrary (no choice of kernel) (ii) potentially faster (no kernel parameter optimization), and (iii) having better extrapolation behavior. We believe that the proposed theory has central importance for the conceptual foundations of regularization and machine learning and also has great potential to enable practical advances in ML areas beyond regression.

We study the problem of inferring user intent from noninvasive electroencephalography (EEG) to restore communication for people with severe speech and physical impairments (SSPI). The focus of this work is improving the estimation of posterior symbol probabilities in a typing task. At each iteration of the typing procedure, a subset of symbols is chosen for the next query based on the current probability estimate. Evidence about the user's response is collected from event-related potentials (ERP) in order to update symbol probabilities, until one symbol exceeds a predefined confidence threshold. We provide a graphical model describing this task, and derive a recursive Bayesian update rule based on a discriminative probability over label vectors for each query, which we approximate using a neural network classifier. We evaluate the proposed method in a simulated typing task and show that it outperforms previous approaches based on generative modeling.

Uncertainty on human behaviors poses a significant challenge to autonomous driving in crowded urban environments. The partially observable Markov decision processes (POMDPs) offer a principled framework for planning under uncertainty, often leveraging Monte Carlo sampling to achieve online performance for complex tasks. However, sampling also raises safety concerns by potentially missing critical events. To address this, we propose a new algorithm, LEarning Attention over Driving bEhavioRs (LEADER), that learns to attend to critical human behaviors during planning. LEADER learns a neural network generator to provide attention over human behaviors in real-time situations. It integrates the attention into a belief-space planner, using importance sampling to bias reasoning towards critical events. To train the algorithm, we let the attention generator and the planner form a min-max game. By solving the min-max game, LEADER learns to perform risk-aware planning without human labeling.

When the fuzzy relation equations have a solution, we first propose an algorithm to find all minimal solutions of the fuzzy relation equations and also supply an algorithm to find all maximal solutions of the fuzzy relation equations, which will be illustrated, respectively, by numeral examples. Then we prove that every solution of the fuzzy relation equations is between a minimal solution and a maximal one, so that we describe the solution set of the fuzzy relation equations completely.

Bayesian inference applied to microseismic activity monitoring allows the accurate location of microseismic events from recorded seismograms and the estimation of the associated uncertainties. However, the forward modelling of these microseismic events, which is necessary to perform Bayesian source inversion, can be prohibitively expensive in terms of computational resources. A viable solution is to train a surrogate model based on machine learning techniques, to emulate the forward model and thus accelerate Bayesian inference. In this paper, we substantially enhance previous work, which considered only sources with isotropic moment tensors. We train a machine learning algorithm on the power spectrum of the recorded pressure wave and show that the trained emulator allows complete and fast event locations for $\textit{any}$ source mechanism. Moreover, we show that our approach is computationally inexpensive, as it can be run in less than 1 hour on a commercial laptop, while yielding accurate results using less than $10^4$ training seismograms. We additionally demonstrate how the trained emulators can be used to identify the source mechanism through the estimation of the Bayesian evidence. Finally, we demonstrate that our approach is robust to real noise as measured in field data. This work lays the foundations for efficient, accurate future joint determinations of event location and moment tensor, and associated uncertainties, which are ultimately key for accurately characterising human-induced and natural earthquakes, and for enhanced quantitative seismic hazard assessments.

Bayesian additive regression tree (BART) models have seen increased attention in recent years as a general-purpose nonparametric modeling technique. BART combines the flexibility of modern machine learning techniques with the principled uncertainty quantification of Bayesian inference, and it has been shown to be uniquely appropriate for addressing the high-noise problems that occur commonly in many areas of science, including medicine and the social sciences. This paper introduces the SoftBart package for fitting the Soft BART algorithm of Linero and Yang (2018). In addition to improving upon the predictive performance of other BART packages, a major goal of this package has been to facilitate the inclusion of BART in larger models, making it ideal for researchers in Bayesian statistics. I show both how to use this package for standard prediction tasks and how to embed BART models in larger models; I illustrate by using SoftBart to implement a nonparametric probit regression model, a semiparametric varying coefficient model, and a partial linear model.