In this paper, an interpretable classifier using an interval type-2 fuzzy neural network for detecting patients suffering from Parkinson's Disease (PD) based on analyzing the gait cycle is presented. The proposed method utilizes clinical features extracted from the vertical Ground Reaction Force (vGRF), measured by 16 wearable sensors placed in the soles of subjects' shoes and learns interpretable fuzzy rules. Therefore, experts can verify the decision made by the proposed method based on investigating the firing strength of interpretable fuzzy rules. Moreover, experts can utilize the extracted fuzzy rules for patient diagnosing or adjust them based on their knowledge. To improve the robustness of the proposed method against uncertainty and noisy sensor measurements, Interval Type-2 Fuzzy Logic is applied. To learn fuzzy rules, two paradigms are proposed: 1- A batch learning approach based on clustering available samples is applied to extract initial fuzzy rules, 2- A complementary online learning is proposed to improve the rule base encountering new labeled samples. The performance of the method is evaluated for classifying patients and healthy subjects in different conditions including the presence of noise or observing new instances. Moreover, the performance of the model is compared to some previous supervised and unsupervised machine learning approaches. The final Accuracy, Precision, Recall, and F1 Score of the proposed method are 88.74%, 89.41%, 95.10%, and 92.16%. Finally, the extracted fuzzy sets for each feature are reported.

Markov Chain Monte Carlo (MCMC) methods are promising solutions to sample from target distributions in high dimensions. While MCMC methods enjoy nice theoretical properties, like guaranteed convergence and mixing to the true target, in practice their sampling efficiency depends on the choice of the proposal distribution and the target at hand. This work considers using machine learning to adapt the proposal distribution to the target, in order to improve the sampling efficiency in the purely discrete domain. Specifically, (i) it proposes a new parametrization for a family of proposal distributions, called locally balanced proposals, (ii) it defines an objective function based on mutual information and (iii) it devises a learning procedure to adapt the parameters of the proposal to the target, thus achieving fast convergence and fast mixing. We call the resulting sampler as the Locally Self-Balancing Sampler (LSB). We show through experimental analysis on the Ising model and Bayesian networks that LSB is indeed able to improve the efficiency over a state-of-the-art sampler based on locally balanced proposals, thus reducing the number of iterations required to converge, while achieving comparable mixing performance.

A novel procedure to perform fuzzy clustering of multivariate time series generated from different dependence models is proposed. Different amounts of dissimilarity between the generating models or changes on the dynamic behaviours over time are some arguments justifying a fuzzy approach, where each series is associated to all the clusters with specific membership levels. Our procedure considers quantile-based cross-spectral features and consists of three stages: (i) each element is characterized by a vector of proper estimates of the quantile cross-spectral densities, (ii) principal component analysis is carried out to capture the main differences reducing the effects of the noise, and (iii) the squared Euclidean distance between the first retained principal components is used to perform clustering through the standard fuzzy C-means and fuzzy C-medoids algorithms. The performance of the proposed approach is evaluated in a broad simulation study where several types of generating processes are considered, including linear, nonlinear and dynamic conditional correlation models. Assessment is done in two different ways: by directly measuring the quality of the resulting fuzzy partition and by taking into account the ability of the technique to determine the overlapping nature of series located equidistant from well-defined clusters. The procedure is compared with the few alternatives suggested in the literature, substantially outperforming all of them whatever the underlying process and the evaluation scheme. Two specific applications involving air quality and financial databases illustrate the usefulness of our approach.

Bayesian methods have become a popular way to incorporate prior knowledge and a notion of uncertainty into machine learning models. At the same time, the complexity of modern machine learning makes it challenging to comprehend a model's reasoning process, let alone express specific prior assumptions in a rigorous manner. While primarily interested in the former issue, recent developments in transparent machine learning could also broaden the range of prior information that we can provide to complex Bayesian models. Inspired by the idea of selfexplaining models, we introduce a corresponding concept for variational Gaussian Processes. On the one hand, our contribution improves transparency for these types of models. More importantly though, our proposed self-explaining variational posterior distribution allows to incorporate both general prior knowledge about a target function as a whole and prior knowledge about the contribution of individual features.

We consider the use of Gaussian process (GP) priors for solving inverse problems in a Bayesian framework. As is well known, the computational complexity of GPs scales cubically in the number of datapoints. We here show that in the context of inverse problems involving integral operators, one faces additional difficulties that hinder inversion on large grids. Furthermore, in that context, covariance matrices can become too large to be stored. By leveraging results about sequential disintegrations of Gaussian measures, we are able to introduce an implicit representation of posterior covariance matrices that reduces the memory footprint by only storing low rank intermediate matrices, while allowing individual elements to be accessed on-the-fly without needing to build full posterior covariance matrices. Moreover, it allows for fast sequential inclusion of new observations. These features are crucial when considering sequential experimental design tasks. We demonstrate our approach by computing sequential data collection plans for excursion set recovery for a gravimetric inverse problem, where the goal is to provide fine resolution estimates of high density regions inside the Stromboli volcano, Italy. Sequential data collection plans are computed by extending the weighted integrated variance reduction (wIVR) criterion to inverse problems. Our results show that this criterion is able to significantly reduce the uncertainty on the excursion volume, reaching close to minimal levels of residual uncertainty. Overall, our techniques allow the advantages of probabilistic models to be brought to bear on large-scale inverse problems arising in the natural sciences.

We introduce semiparametric Bayesian networks that combine parametric and nonparametric conditional probability distributions. Their aim is to incorporate the advantages of both components: the bounded complexity of parametric models and the flexibility of nonparametric ones. We demonstrate that semiparametric Bayesian networks generalize two well-known types of Bayesian networks: Gaussian Bayesian networks and kernel density estimation Bayesian networks. For this purpose, we consider two different conditional probability distributions required in a semiparametric Bayesian network. In addition, we present modifications of two well-known algorithms (greedy hill-climbing and PC) to learn the structure of a semiparametric Bayesian network from data. To realize this, we employ a score function based on cross-validation. In addition, using a validation dataset, we apply an early-stopping criterion to avoid overfitting. To evaluate the applicability of the proposed algorithm, we conduct an exhaustive experiment on synthetic data sampled by mixing linear and nonlinear functions, multivariate normal data sampled from Gaussian Bayesian networks, real data from the UCI repository, and bearings degradation data. As a result of this experiment, we conclude that the proposed algorithm accurately learns the combination of parametric and nonparametric components, while achieving a performance comparable with those provided by state-of-the-art methods.

In this paper, we explore adaptive inference based on variational Bayes. Although a number of studies have been conducted to analyze contraction properties of variational posteriors, there is still a lack of a general and computationally tractable variational Bayes method that can achieve adaptive optimal contraction of the variational posterior. We propose a novel variational Bayes framework, called adaptive variational Bayes, which can operate on a collection of models with varying dimensions and structures. The proposed framework combines variational posteriors over individual models with certain weights to obtain a variational posterior over the entire model. It turns out that this combined variational posterior minimizes the Kullback-Leibler divergence to the original posterior distribution. We show that the proposed variational posterior achieves optimal contraction rates adaptively under very general conditions and attains model selection consistency when the true model structure exists. We apply the general results obtained for the adaptive variational Bayes to several examples including deep learning models and derive some new and adaptive inference results. Moreover, we consider the use of quasi-likelihood in our framework. We formulate conditions on the quasi-likelihood to ensure the adaptive optimality and discuss specific applications to stochastic block models and nonparametric regression with sub-Gaussian errors.

One major drawback of state-of-the-art artificial intelligence is its lack of explainability. One approach to solve the problem is taking causality into account. Causal mechanisms can be described by structural causal models. In this work, we propose a method for estimating bivariate structural causal models using a combination of normalising flows applied to density estimation and variational Gaussian process regression for post-nonlinear models. It facilitates causal discovery, i.e. distinguishing cause and effect, by either the independence of cause and residual or a likelihood ratio test. Our method which estimates post-nonlinear models can better explain a variety of real-world cause-effect pairs than a simple additive noise model. Though it remains difficult to exploit this benefit regarding all pairs from the T\"ubingen benchmark database, we demonstrate that combining the additive noise model approach with our method significantly enhances causal discovery.

Discovering causal structures from data is a challenging inference problem of fundamental importance in all areas of science. The appealing scaling properties of neural networks have recently led to a surge of interest in differentiable neural network-based methods for learning causal structures from data. So far differentiable causal discovery has focused on static datasets of observational or interventional origin. In this work, we introduce an active intervention-targeting mechanism which enables a quick identification of the underlying causal structure of the data-generating process. Our method significantly reduces the required number of interactions compared with random intervention targeting and is applicable for both discrete and continuous optimization formulations of learning the underlying directed acyclic graph (DAG) from data. We examine the proposed method across a wide range of settings and demonstrate superior performance on multiple benchmarks from simulated to real-world data. Learning causal structure from data is a challenging but important task that lies at the heart of scientific reasoning and accompanying progress in many disciplines (Sachs et al., 2005; Hill et al., 2016; Lauritzen & Spiegelhalter, 1988; Korb & Nicholson, 2010). While there exists a plethora of methods for the task, computationally and statistically more efficient algorithms are highly desired (Heinze-Deml et al., 2018). As a result, there has been a surge in interest in differentiable structure learning and the combination of deep learning and causal inference (Schölkopf et al., 2021). However, the improvement critically depends on the experiments and interventions available. Despite advances in high-throughput methods for interventional data in specific fields (Dixit et al., 2016), the acquisition of interventional samples in the general settings tends to be costly, technically impossible or even unethical for specific interventions.

Nowadays, the increase in data acquisition and availability and complexity around optimization make it imperative to jointly use artificial intelligence (AI) and optimization for devising data-driven and intelligent decision support systems (DSS). A DSS can be successful if large amounts of interactive data proceed fast and robustly and extract useful information and knowledge to help decision-making. In this context, the data-driven approach has gained prominence due to its provision of insights for decision-making and easy implementation. The data-driven approach can discover various database patterns without relying on prior knowledge while also handling flexible objectives and multiple scenarios. This chapter reviews recent advances in data-driven optimization, highlighting the promise of data-driven optimization that integrates mathematical programming and machine learning (ML) for decision-making under uncertainty and identifies potential research opportunities. This chapter provides guidelines and implications for researchers, managers, and practitioners in operations research who want to advance their decision-making capabilities under uncertainty concerning data-driven optimization. Then, a comprehensive review and classification of the relevant publications on the data-driven stochastic program, data-driven robust optimization, and data-driven chance-constrained are presented. This chapter also identifies fertile avenues for future research that focus on deep-data-driven optimization, deep data-driven models, as well as online learning-based data-driven optimization. Perspectives on reinforcement learning (RL)-based data-driven optimization and deep RL for solving NP-hard problems are discussed. We investigate the application of data-driven optimization in different case studies to demonstrate improvements in operational performance over conventional optimization methodology. Finally, some managerial implications and some future directions are provided.