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This paper presents a novel approach to Bayesian nonparametric spectral analysis of stationary multivariate time series. Starting with a parametric vector-autoregressive model, the parametric likelihood is nonparametrically adjusted in the frequency domain to account for potential deviations from parametric assumptions. We show mutual contiguity of the nonparametrically corrected likelihood, the multivariate Whittle likelihood approximation and the exact likelihood for Gaussian time series. A multivariate extension of the nonparametric Bernstein-Dirichlet process prior for univariate spectral densities to the space of Hermitian positive definite spectral density matrices is specified directly on the correction matrices. An infinite series representation of this prior is then used to develop a Markov chain Monte Carlo algorithm to sample from the posterior distribution. The code is made publicly available for ease of use and reproducibility. With this novel approach we provide a generalization of the multivariate Whittle-likelihood-based method of Meier et al. (2020) as well as an extension of the nonparametrically corrected likelihood for univariate stationary time series of Kirch et al. (2019) to the multivariate case. We demonstrate that the nonparametrically corrected likelihood combines the efficiencies of a parametric with the robustness of a nonparametric model. Its numerical accuracy is illustrated in a comprehensive simulation study. We illustrate its practical advantages by a spectral analysis of two environmental time series data sets: a bivariate time series of the Southern Oscillation Index and fish recruitment and time series of windspeed data at six locations in California.

We propose to use L\'evy {\alpha}-stable distributions for constructing priors for Bayesian inverse problems. The construction is based on Markov fields with stable-distributed increments. Special cases include the Cauchy and Gaussian distributions, with stability indices {\alpha} = 1, and {\alpha} = 2, respectively. Our target is to show that these priors provide a rich class of priors for modelling rough features. The main technical issue is that the {\alpha}-stable probability density functions do not have closed-form expressions in general, and this limits their applicability. For practical purposes, we need to approximate probability density functions through numerical integration or series expansions. Current available approximation methods are either too time-consuming or do not function within the range of stability and radius arguments needed in Bayesian inversion. To address the issue, we propose a new hybrid approximation method for symmetric univariate and bivariate {\alpha}-stable distributions, which is both fast to evaluate, and accurate enough from a practical viewpoint. Then we use approximation method in the numerical implementation of {\alpha}-stable random field priors. We demonstrate the applicability of the constructed priors on selected Bayesian inverse problems which include the deconvolution problem, and the inversion of a function governed by an elliptic partial differential equation. We also demonstrate hierarchical {\alpha}-stable priors in the one-dimensional deconvolution problem. We employ maximum-a-posterior-based estimation at all the numerical examples. To that end, we exploit the limited-memory BFGS and its bounded variant for the estimator.

After just a few hundred training updates, a standard probabilistic model for language generation has likely not yet learnt many semantic or syntactic rules of natural language, making it difficult to estimate the probability distribution over next tokens. Yet around this point, these models have identified a simple, loss-minimising behaviour: to output the unigram distribution of the target training corpus. The use of such a heuristic raises the question: Can we initialise our models with this behaviour and save precious compute resources and model capacity? Here we show that we can effectively endow standard neural language generation models with a separate module that reflects unigram frequency statistics as prior knowledge, simply by initialising the bias term in a model's final linear layer with the log-unigram distribution. We use neural machine translation as a test bed for this simple technique and observe that it: (i) improves learning efficiency; (ii) achieves better overall performance; and perhaps most importantly (iii) appears to disentangle strong frequency effects by encouraging the model to specialise in non-frequency-related aspects of language.

Causal effect estimation has been studied by many researchers when only observational data is available. Sound and complete algorithms have been developed for pointwise estimation of identifiable causal queries. For non-identifiable causal queries, researchers developed polynomial programs to estimate tight bounds on causal effect. However, these are computationally difficult to optimize for variables with large support sizes. In this paper, we analyze the effect of "weak confounding" on causal estimands. More specifically, under the assumption that the unobserved confounders that render a query non-identifiable have small entropy, we propose an efficient linear program to derive the upper and lower bounds of the causal effect. We show that our bounds are consistent in the sense that as the entropy of unobserved confounders goes to zero, the gap between the upper and lower bound vanishes. Finally, we conduct synthetic and real data simulations to compare our bounds with the bounds obtained by the existing work that cannot incorporate such entropy constraints and show that our bounds are tighter for the setting with weak confounders.

Many multivariate data sets exhibit a form of positive dependence, which can either appear globally between all variables or only locally within particular subgroups. A popular notion of positive dependence that allows for localized positivity is positive association. In this work we introduce the notion of extremal positive association for multivariate extremes from threshold exceedances. Via a sufficient condition for extremal association, we show that extremal association generalizes extremal tree models. For H\"usler--Reiss distributions the sufficient condition permits a parametric description that we call the metric property. As the parameter of a H\"usler--Reiss distribution is a Euclidean distance matrix, the metric property relates to research in electrical network theory and Euclidean geometry. We show that the metric property can be localized with respect to a graph and study surrogate likelihood inference. This gives rise to a two-step estimation procedure for locally metrical H\"usler--Reiss graphical models. The second step allows for a simple dual problem, which is implemented via a gradient descent algorithm. Finally, we demonstrate our results on simulated and real data.

This article presents a new three-degree-of-freedom (3-DOF) parallel mechanism (PM) with two translations and one rotation (2T1R), designed based on the topological design theory of the parallel mechanism using position and orientation characteristics (POC). The PM is primarily intended for use in package sorting and delivery. The mobile platform of the PM moves along a translation axis, picks up objects from a conveyor belt, and tilts them to either side of the axis. We first calculate the PM's topological characteristics, such as the degree of freedom (DOF) and the degree of coupling, and provide its topological analytical formula to represent the topological information of the PM. Next, we solve the direct and inverse kinematic models based on the kinematic modelling principle using the topological features. The models are purely analytic and are broken down into a series of quadratic equations, making them suitable for use in an industrial robot. We also study the singular configurations to identify the serial and parallel singularities. Using the decoupling properties, we size the mechanism to address the package sorting and depositing problem using an algebraic approach. To determine the smallest segment lengths, we use a cylindrical algebraic decomposition to solve a system with inequalities.

State-space models (SSMs) are a powerful statistical tool for modelling time-varying systems via a latent state. In these models, the latent state is never directly observed. Instead, a sequence of data points related to the state are obtained. The linear-Gaussian state-space model is widely used, since it allows for exact inference when all model parameters are known, however this is rarely the case. The estimation of these parameters is a very challenging but essential task to perform inference and prediction. In the linear-Gaussian model, the state dynamics are described via a state transition matrix. This model parameter is known to behard to estimate, since it encodes the relationships between the state elements, which are never observed. In many applications, this transition matrix is sparse since not all state components directly affect all other state components. However, most parameter estimation methods do not exploit this feature. In this work we propose SpaRJ, a fully probabilistic Bayesian approach that obtains sparse samples from the posterior distribution of the transition matrix. Our method explores sparsity by traversing a set of models that exhibit differing sparsity patterns in the transition matrix. Moreover, we also design new effective rules to explore transition matrices within the same level of sparsity. This novel methodology has strong theoretical guarantees, and unveils the latent structure of the data generating process, thereby enhancing interpretability. The performance of SpaRJ is showcased in example with dimension 144 in the parameter space, and in a numerical example with real data.

Forecasts of multivariate probability distributions are required for a variety of applications. Scoring rules enable the evaluation of forecast accuracy, and comparison between forecasting methods. We propose a theoretical framework for scoring rules for multivariate distributions, which encompasses the existing quadratic score and multivariate continuous ranked probability score. We demonstrate how this framework can be used to generate new scoring rules. In some multivariate contexts, it is a forecast of a level set that is needed, such as a density level set for anomaly detection or the level set of the cumulative distribution as a measure of risk. This motivates consideration of scoring functions for such level sets. For univariate distributions, it is well-established that the continuous ranked probability score can be expressed as the integral over a quantile score. We show that, in a similar way, scoring rules for multivariate distributions can be decomposed to obtain scoring functions for level sets. Using this, we present scoring functions for different types of level set, including density level sets and level sets for cumulative distributions. To compute the scores, we propose a simple numerical algorithm. We perform a simulation study to support our proposals, and we use real data to illustrate usefulness for forecast combining and CoVaR estimation.

Mental distress like depression and anxiety contribute to the largest proportion of the global burden of diseases. Automated diagnosis systems of such disorders, empowered by recent innovations in Artificial Intelligence, can pave the way to reduce the sufferings of the affected individuals. Development of such systems requires information-rich and balanced corpora. In this work, we introduce a novel mental distress analysis audio dataset DEPAC, labeled based on established thresholds on depression and anxiety standard screening tools. This large dataset comprises multiple speech tasks per individual, as well as relevant demographic information. Alongside, we present a feature set consisting of hand-curated acoustic and linguistic features, which were found effective in identifying signs of mental illnesses in human speech. Finally, we justify the quality and effectiveness of our proposed audio corpus and feature set in predicting depression severity by comparing the performance of baseline machine learning models built on this dataset with baseline models trained on other well-known depression corpora.