Inhomogeneities in the aortic wall can lead to localized stress accumulations, possibly initiating dissection. In many cases, a dissection results from pathological changes such as fragmentation or loss of elastic fibers. But it has been shown that even the healthy aortic wall has an inherent heterogeneous microstructure. Some parts of the aorta are particularly susceptible to the development of inhomogeneities due to pathological changes, however, the distribution in the aortic wall and the spatial extent, such as size, shape, and type, are difficult to predict. Motivated by this observation, we describe the heterogeneous distribution of elastic fiber degradation in the dissected aortic wall using a stochastic constitutive model. For this purpose, random field realizations, which model the stochastic distribution of degraded elastic fibers, are generated over a non-equidistant grid. The random field then serves as input for a uni-axial extension test of the pathological aortic wall, solved with the finite-element (FE) method. To include the microstructure of the dissected aortic wall, a constitutive model developed in a previous study is applied, which also includes an approach to model the degradation of inter-lamellar elastic fibers. Then to assess the uncertainty in the output stress distribution due to this stochastic constitutive model, a convolutional neural network, specifically a Bayesian encoder-decoder, was used as a surrogate model that maps the random input fields to the output stress distribution obtained from the FE analysis. The results show that the neural network is able to predict the stress distribution of the FE analysis while significantly reducing the computational time. In addition, it provides the probability for exceeding critical stresses within the aortic wall, which could allow for the prediction of delamination or fatal rupture.

Bayesian additive regression trees have seen increased interest in recent years due to their ability to combine machine learning techniques with principled uncertainty quantification. The Bayesian backfitting algorithm used to fit BART models, however, limits their application to a small class of models for which conditional conjugacy exists. In this article, we greatly expand the domain of applicability of BART to arbitrary \emph{generalized BART} models by introducing a very simple, tuning-parameter-free, reversible jump Markov chain Monte Carlo algorithm. Our algorithm requires only that the user be able to compute the likelihood and (optionally) its gradient and Fisher information. The potential applications are very broad; we consider examples in survival analysis, structured heteroskedastic regression, and gamma shape regression.

We propose a network architecture capable of reliably estimating uncertainty of regression based predictions without sacrificing accuracy. The current state-of-the-art uncertainty algorithms either fall short of achieving prediction accuracy comparable to the mean square error optimization or underestimate the variance of network predictions. We propose a decoupled network architecture that is capable of accomplishing both at the same time. We achieve this by breaking down the learning of prediction and prediction interval (PI) estimations into a two-stage training process. We use a custom loss function for learning a PI range around optimized mean estimation with a desired coverage of a proportion of the target labels within the PI range. We compare the proposed method with current state-of-the-art uncertainty quantification algorithms on synthetic datasets and UCI benchmarks, reducing the error in the predictions by 23 to 34% while maintaining 95% Prediction Interval Coverage Probability (PICP) for 7 out of 9 UCI benchmark datasets. We also examine the quality of our predictive uncertainty by evaluating on Active Learning and demonstrating 17 to 36% error reduction on UCI benchmarks.

The Latent Dirichlet Allocation (LDA) model is a popular method for creating mixed-membership clusters. Despite having been originally developed for text analysis, LDA has been used for a wide range of other applications. We propose a new formulation for the LDA model which incorporates covariates. In this model, a negative binomial regression is embedded within LDA, enabling straight-forward interpretation of the regression coefficients and the analysis of the quantity of cluster-specific elements in each sampling units (instead of the analysis being focused on modeling the proportion of each cluster, as in Structural Topic Models). We use slice sampling within a Gibbs sampling algorithm to estimate model parameters. We rely on simulations to show how our algorithm is able to successfully retrieve the true parameter values and the ability to make predictions for the abundance matrix using the information given by the covariates. The model is illustrated using real data sets from three different areas: text-mining of Coronavirus articles, analysis of grocery shopping baskets, and ecology of tree species on Barro Colorado Island (Panama). This model allows the identification of mixed-membership clusters in discrete data and provides inference on the relationship between covariates and the abundance of these clusters.

We revisit convergence rates for maximum likelihood estimation (MLE) under finite mixture models. The Wasserstein distance has become a standard loss function for the analysis of parameter estimation in these models, due in part to its ability to circumvent label switching and to accurately characterize the behaviour of fitted mixture components with vanishing weights. However, the Wasserstein metric is only able to capture the worst-case convergence rate among the remaining fitted mixture components. We demonstrate that when the log-likelihood function is penalized to discourage vanishing mixing weights, stronger loss functions can be derived to resolve this shortcoming of the Wasserstein distance. These new loss functions accurately capture the heterogeneity in convergence rates of fitted mixture components, and we use them to sharpen existing pointwise and uniform convergence rates in various classes of mixture models. In particular, these results imply that a subset of the components of the penalized MLE typically converge significantly faster than could have been anticipated from past work. We further show that some of these conclusions extend to the traditional MLE. Our theoretical findings are supported by a simulation study to illustrate these improved convergence rates.

Bayesian inference is based on the fact that we often don't know the underlying distribution of data, so we need to build a model and then iteratively adjust it as we get more data. In parametric Bayesian inference you start with picking a general form of the probability distribution f(x;θ) defined by parameters θ. A good example of the distribution could be a Normal distribution with two parameters μ and σ 2. The probability of the data under a hypothetical distribution, assuming independent data examples, is: This function is called likelihood function. The parameter θ is itself a random variable, and its probability distribution can be found using Bayes' theorem: Here p(θ) is called posterior distribution, π(θ) is prior distribution and expresses our beliefs about parameter θ before we see any data. The term in the denominator is called evidence and represents probability of data.

Air pollution is one of the leading causes of mortality globally, resulting in millions of deaths each year. Efficient monitoring is important to measure exposure and enforce legal limits. New low-cost sensors can be deployed in greater numbers and in more varied locations, motivating the problem of efficient automated placement. Previous work suggests Bayesian optimisation is an appropriate method, but only considered a satellite data set, with data aggregated over all altitudes. It is ground-level pollution, that humans breathe, which matters most. We improve on those results using hierarchical models and evaluate our models on urban pollution data in London to show that Bayesian optimisation can be successfully applied to the problem.

Anomaly detection is a widely studied task for a broad variety of data types; among them, multiple time series appear frequently in applications, including for example, power grids and traffic networks. Detecting anomalies for multiple time series, however, is a challenging subject, owing to the intricate interdependencies among the constituent series. We hypothesize that anomalies occur in low density regions of a distribution and explore the use of normalizing flows for unsupervised anomaly detection, because of their superior quality in density estimation. Moreover, we propose a novel flow model by imposing a Bayesian network among constituent series. A Bayesian network is a directed acyclic graph (DAG) that models causal relationships; it factorizes the joint probability of the series into the product of easy-to-evaluate conditional probabilities. We call such a graph-augmented normalizing flow approach GANF and propose joint estimation of the DAG with flow parameters. We conduct extensive experiments on real-world datasets and demonstrate the effectiveness of GANF for density estimation, anomaly detection, and identification of time series distribution drift.

Gene expression datasets consist of thousand of genes with relatively small samplesizes (i.e. are large-$p$-small-$n$). Moreover, dependencies of various orders co-exist in the datasets. In the Undirected probabilistic Graphical Model (UGM) framework the Glasso algorithm has been proposed to deal with high dimensional micro-array datasets forcing sparsity. Also, modifications of the default Glasso algorithm are developed to overcome the problem of complex interaction structure. In this work we advocate the use of a simple score-based Hill Climbing algorithm (HC) that learns Gaussian Bayesian Networks (BNs) leaning on Directed Acyclic Graphs (DAGs). We compare HC with Glasso and its modifications in the UGM framework on their capability to reconstruct GRNs from micro-array data belonging to the Escherichia Coli genome. We benefit from the analytical properties of the Joint Probability Density (JPD) function on which both directed and undirected PGMs build to convert DAGs to UGMs. We conclude that dependencies in complex data are learned best by the HC algorithm, presenting them most accurately and efficiently, simultaneously modelling strong local and weaker but significant global connections coexisting in the gene expression dataset. The HC algorithm adapts intrinsically to the complex dependency structure of the dataset, without forcing a specific structure in advance. On the contrary, Glasso and modifications model unnecessary dependencies at the expense of the probabilistic information in the network and of a structural bias in the JPD function that can only be relieved including many parameters.

In former articles, we have been looking at many different concepts related to Bayesian Networks: how to represent them in a compact manner, their reasoning patterns, and I-Map. In this article, we will take a look at D-separation and try to understand it both intuitively and mathematically.