With the increase in the number of active satellites and space debris in orbit, the problem of initial orbit determination (IOD) becomes increasingly important, demanding a high accuracy. Over the years, different approaches have been presented such as filtering methods (for example, Extended Kalman Filter), differential algebra or solving Lambert's problem. In this work, we consider a setting of three monostatic radars, where all available measurements are taken approximately at the same instant. This follows a similar setting as trilateration, a state-of-the-art approach, where each radar is able to obtain a single measurement of range and range-rate. Differently, and due to advances in Multiple-Input Multiple-Output (MIMO) radars, we assume that each location is able to obtain a larger set of range, angle and Doppler shift measurements. Thus, our method can be understood as an extension of trilateration leveraging more recent technology and incorporating additional data. We formulate the problem as a Maximum Likelihood Estimator (MLE), which for some number of observations is asymptotically unbiased and asymptotically efficient. Through numerical experiments, we demonstrate that our method attains the same accuracy as the trilateration method for the same number of measurements and offers an alternative and generalization, returning a more accurate estimation of the satellite's state vector, as the number of available measurements increases.

An optimal randomized strategy for design of balanced, normalized mass transport plans is developed. It replaces -- but specializes to -- the deterministic, regularized optimal transport (OT) strategy, which yields only a certainty-equivalent plan. The incompletely specified -- and therefore uncertain -- transport plan is acknowledged to be a random process. Therefore, hierarchical fully probabilistic design (HFPD) is adopted, yielding an optimal hyperprior supported on the set of possible transport plans, and consistent with prior mean constraints on the marginals of the uncertain plan. This Bayesian resetting of the design problem for transport plans -- which we call HFPD-OT -- confers new opportunities. These include (i) a strategy for the generation of a random sample of joint transport plans; (ii) randomized marginal contracts for individual source-target pairs; and (iii) consistent measures of uncertainty in the plan and its contracts. An application in algorithmic fairness is outlined, where HFPD-OT enables the recruitment of a more diverse subset of contracts -- than is possible in classical OT -- into the delivery of an expected plan. Also, it permits fairness proxies to be endowed with uncertainty quantifiers.

The vast majority of the literature on learning dynamical systems or stochastic processes from time series has focused on stable or ergodic systems, for both Bayesian and frequentist inference procedures. However, most real-world systems are only metastable, that is, the dynamics appear to be stable on some time scale, but are in fact unstable over longer time scales. Consistency of inference for metastable systems may not be possible, but one can ask about metaconsistency: Do inference procedures converge when observations are taken over a large but finite time interval, but diverge on longer time scales? In this paper we introduce, discuss, and quantify metaconsistency in a Bayesian framework. We discuss how metaconsistency can be exploited to efficiently infer a model for a sub-system of a larger system, where inference on the global behavior may require much more data. We also discuss the relation between meta-consistency and the spectral properties of the model dynamical system in the case of uniformly ergodic diffusions.

In modern and personalised education, there is a growing interest in developing learners' competencies and accurately assessing them. In a previous work, we proposed a procedure for deriving a learner model for automatic skill assessment from a task-specific competence rubric, thus simplifying the implementation of automated assessment tools. The previous approach, however, suffered two main limitations: (i) the ordering between competencies defined by the assessment rubric was only indirectly modelled; (ii) supplementary skills, not under assessment but necessary for accomplishing the task, were not included in the model. In this work, we address issue (i) by introducing dummy observed nodes, strictly enforcing the skills ordering without changing the network's structure. In contrast, for point (ii), we design a network with two layers of gates, one performing disjunctive operations by noisy-OR gates and the other conjunctive operations through logical ANDs. Such changes improve the model outcomes' coherence and the modelling tool's flexibility without compromising the model's compact parametrisation, interpretability and simple experts' elicitation. We used this approach to develop a learner model for Computational Thinking (CT) skills assessment. The CT-cube skills assessment framework and the Cross Array Task (CAT) are used to exemplify it and demonstrate its feasibility.

Estimating treatment effects from observational data is of central interest across numerous application domains. Individual treatment effect offers the most granular measure of treatment effect on an individual level, and is the most useful to facilitate personalized care. However, its estimation and inference remain underdeveloped due to several challenges. In this article, we propose a novel conformal diffusion model-based approach that addresses those intricate challenges. We integrate the highly flexible diffusion modeling, the model-free statistical inference paradigm of conformal inference, along with propensity score and covariate local approximation that tackle distributional shifts. We unbiasedly estimate the distributions of potential outcomes for individual treatment effect, construct an informative confidence interval, and establish rigorous theoretical guarantees. We demonstrate the competitive performance of the proposed method over existing solutions through extensive numerical studies.

We present two new approaches for point prediction with streaming data. One is based on the Count-Min sketch (CMS) and the other is based on Gaussian process priors with a random bias. These methods are intended for the most general predictive problems where no true model can be usefully formulated for the data stream. In statistical contexts, this is often called the $\mathcal{M}$-open problem class. Under the assumption that the data consists of i.i.d samples from a fixed distribution function $F$, we show that the CMS-based estimates of the distribution function are consistent. We compare our new methods with two established predictors in terms of cumulative $L^1$ error. One is based on the Shtarkov solution (often called the normalized maximum likelihood) in the normal experts setting and the other is based on Dirichlet process priors. These comparisons are for two cases. The first is one-pass meaning that the updating of the predictors is done using the fact that the CMS is a sketch. For predictors that are not one-pass, we use streaming $K$-means to give a representative subset of fixed size that can be updated as data accumulate. Preliminary computational work suggests that the one-pass median version of the CMS method is rarely outperformed by the other methods for sufficiently complex data. We also find that predictors based on Gaussian process priors with random biases perform well. The Shtarkov predictors we use here did not perform as well probably because we were only using the simplest example. The other predictors seemed to perform well mainly when the data did not look like they came from an M-open data generator.

The upcoming high-luminosity runs of the LHC will push the quantitative frontier of data taking to over 25-times its current rates. To ensure precision gains from such high statistics, this increase in experimental data needs to be met by an equal amount of simulation. The required computational power is predicted to outgrow the increase in budget in the coming years [1, 2]. One solution to this predicament is the augmentation of the expensive, Monte Carlo-based, simulation chain with generative machine learning. A special focus is often put on the costly detector simulation [3, 4]. This approach is only viable under the assumption that the generated data is not statistically limited to the size of the simulated training data. Previous studies have shown, for both toy data [5] and calorimeter images [6], that samples generated with generative neural networks can surpass the training statistics due to powerful interpolation abilities of the network in data space. These studies rely on comparing a distance measure between histograms of generated data and true hold-out data to the distance between smaller, statistically limited sets of Monte Carlo data and the hold-out set. The phenomenon of a generative model surpassing the precision of its training set is also known as amplification.

Bayesian estimation is a vital tool in robotics as it allows systems to update the belief of the robot state using incomplete information from noisy sensors. To render the state estimation problem tractable, many systems assume that the motion and measurement noise, as well as the state distribution, are all unimodal and Gaussian. However, there are numerous scenarios and systems that do not comply with these assumptions. Existing non-parametric filters that are used to model multimodal distributions have drawbacks that limit their ability to represent a diverse set of distributions. In this paper, we introduce a novel approach to nonparametric Bayesian filtering to cope with multimodal distributions using harmonic exponential distributions. This approach leverages two key insights of harmonic exponential distributions: a) the product of two distributions can be expressed as the element-wise addition of their log-likelihood Fourier coefficients, and b) the convolution of two distributions can be efficiently computed as the tensor product of their Fourier coefficients. These observations enable the development of an efficient and exact solution to the Bayes filter up to the band limit of a Fourier transform. We demonstrate our filter's superior performance compared with established nonparametric filtering methods across a range of simulated and real-world localization tasks.

Recent advancements in machine learning have accelerated its widespread adoption across various real-world applications. However, in safety-critical domains, the deployment of machine learning models is riddled with challenges due to their complexity, lack of interpretability, and absence of formal guarantees regarding their behavior. In this paper, we introduce a verification framework tailored for Bayesian networks, designed to address these drawbacks. Our framework comprises two key components: (1) a two-step compilation and encoding scheme that translates Bayesian networks into Boolean logic literals, and (2) formal verification queries that leverage these literals to verify various properties encoded as constraints. Specifically, we introduce two verification queries: if-then rules (ITR) and feature monotonicity (FMO). We benchmark the efficiency of our verification scheme and demonstrate its practical utility in real-world scenarios.

AI has been dealing with uncertainty to have highly accurate results. This becomes even worse with reasonably small data sets or a variation in the data sets. This has far-reaching effects on decision-making, forecasting and learning mechanisms. This study seeks to unpack the nature of uncertainty that exists within AI by drawing ideas from established works, the latest developments and practical applications and provide a novel total uncertainty definition in AI. From inception theories up to current methodologies, this paper provides an integrated view of dealing with better total uncertainty as well as complexities of uncertainty in AI that help us understand its meaning and value across different domains.