When used as a surrogate objective for maximum likelihood estimation in latent variable models, the evidence lower bound (ELBO) produces state-of-the-art results. Inspired by this, we consider the extension of the ELBO to a family of lower bounds defined by a particle filter's estimator of the marginal likelihood, the filtering variational objectives (FIVOs). FIVOs take the same arguments as the ELBO, but can exploit a model's sequential structure to form tighter bounds. We present results that relate the tightness of FIVO's bound to the variance of the particle filter's estimator by considering the generic case of bounds defined as log-transformed likelihood estimators. Experimentally, we show that training with FIVO results in substantial improvements over training the same model architecture with the ELBO on sequential data.

Epileptic seizure activity shows complicated dynamics in both space and time. To understand the evolution and propagation of seizures spatially extended sets of data need to be analysed. We have previously described an efficient filtering scheme using variational Laplace that can be used in the Dynamic Causal Modelling (DCM) framework [Friston, 2003] to estimate the temporal dynamics of seizures recorded using either invasive or non-invasive electrical recordings (EEG/ECoG). Spatiotemporal dynamics are modelled using a partial differential equation -- in contrast to the ordinary differential equation used in our previous work on temporal estimation of seizure dynamics [Cooray, 2016]. We provide the requisite theoretical background for the method and test the ensuing scheme on simulated seizure activity data and empirical invasive ECoG data. The method provides a framework to assimilate the spatial and temporal dynamics of seizure activity, an aspect of great physiological and clinical importance.

Sparse pseudo-point approximations for Gaussian process (GP) models provide a suite of methods that support deployment of GPs in the large data regime and enable analytic intractabilities to be sidestepped. However, the field lacks a principled method to handle streaming data in which both the posterior distribution over function values and the hyperparameter estimates are updated in an online fashion. The small number of existing approaches either use suboptimal hand-crafted heuristics for hyperparameter learning, or suffer from catastrophic forgetting or slow updating when new data arrive. This paper develops a new principled framework for deploying Gaussian process probabilistic models in the streaming setting, providing methods for learning hyperparameters and optimising pseudo-input locations. The proposed framework is assessed using synthetic and real-world datasets.

Variational inference (VI) is widely used as an efficient alternative to Markov chain Monte Carlo. It posits a family of approximating distributions $q$ and finds the closest member to the exact posterior $p$. Closeness is usually measured via a divergence $D(q || p)$ from $q$ to $p$. While successful, this approach also has problems. Notably, it typically leads to underestimation of the posterior variance. In this paper we propose CHIVI, a black-box variational inference algorithm that minimizes $D_{\chi}(p || q)$, the $\chi$-divergence from $p$ to $q$. CHIVI minimizes an upper bound of the model evidence, which we term the $\chi$ upper bound (CUBO). Minimizing the CUBO leads to improved posterior uncertainty, and it can also be used with the classical VI lower bound (ELBO) to provide a sandwich estimate of the model evidence. We study CHIVI on three models: probit regression, Gaussian process classification, and a Cox process model of basketball plays. When compared to expectation propagation and classical VI, CHIVI produces better error rates and more accurate estimates of posterior variance.

In this paper we deal with the problem of extending Zadeh's operators on fuzzy sets (FSs) to interval-valued (IVFSs), set-valued (SVFSs) and type-2 (T2FSs) fuzzy sets. Namely, it is known that seeing FSs as SVFSs, or T2FSs, whose membership degrees are singletons is not order-preserving. We then describe a family of lattice embeddings from FSs to SVFSs. Alternatively, if the former singleton viewpoint is required, we reformulate the intersection on hesitant fuzzy sets and introduce what we have called closed-valued fuzzy sets. This new type of fuzzy sets extends standard union and intersection on FSs. In addition, it allows handling together membership degrees of different nature as, for instance, closed intervals and finite sets. Finally, all these constructions are viewed as T2FSs forming a chain of lattices.

We develop SHOPPER, a sequential probabilistic model of market baskets. SHOPPER uses interpretable components to model the forces that drive how a customer chooses products; in particular, we designed SHOPPER to capture how items interact with other items. We develop an efficient posterior inference algorithm to estimate these forces from large-scale data, and we analyze a large dataset from a major chain grocery store. We are interested in answering counterfactual queries about changes in prices. We found that SHOPPER provides accurate predictions even under price interventions, and that it helps identify complementary and substitutable pairs of products.

The Bayes error rate (BER) is a central concept in the statistical theory of classification. It represents the error rate of the Bayes classifier, which assigns a label to an object corresponding to the class with the highest posterior probability. By definition, the Bayes error represents the smallest possible average error rate that can be achieved by any decision rule (Wald, 1947). Because of these properties, the BER is of great interest both for benchmarking classification algorithms as well as for the practical design of classification algorithms. For example, an accurate approximation of the BER can be used for classifier parameter selection, data dimensionality reduction, or variable selection. However, accurate BER approximation is difficult, especially in high dimension, and thus much attention has focused on tight and tractable BER bounds. This paper proposes a model-free approach to designing multiclass classifiers using a bias-corrected BER bound estimated directly from the multiclass data. There exists several useful bounds on the BER that are functions of the class-dependent feature distributions. These include information theoretic divergence measures such as the Chernoffα -divergence (Chernoff, 1952), the Bhattacharyya divergence (Kailath, 1967), or the Jensen-Shannon divergence (Lin, 1991).

Many countries are suffering from severe air pollution. Understanding how different air pollutants accumulate and propagate is critical to making relevant public policies. In this paper, we use urban big data (air quality data and meteorological data) to identify the \emph{spatiotemporal (ST) causal pathways} for air pollutants. This problem is challenging because: (1) there are numerous noisy and low-pollution periods in the raw air quality data, which may lead to unreliable causality analysis, (2) for large-scale data in the ST space, the computational complexity of constructing a causal structure is very high, and (3) the \emph{ST causal pathways} are complex due to the interactions of multiple pollutants and the influence of environmental factors. Therefore, we present \emph{p-Causality}, a novel pattern-aided causality analysis approach that combines the strengths of \emph{pattern mining} and \emph{Bayesian learning} to efficiently and faithfully identify the \emph{ST causal pathways}. First, \emph{Pattern mining} helps suppress the noise by capturing frequent evolving patterns (FEPs) of each monitoring sensor, and greatly reduce the complexity by selecting the pattern-matched sensors as "causers". Then, \emph{Bayesian learning} carefully encodes the local and ST causal relations with a Gaussian Bayesian network (GBN)-based graphical model, which also integrates environmental influences to minimize biases in the final results. We evaluate our approach with three real-world data sets containing 982 air quality sensors, in three regions of China from 01-Jun-2013 to 19-Dec-2015. Results show that our approach outperforms the traditional causal structure learning methods in time efficiency, inference accuracy and interpretability.

In this paper we analyse the benefits of incorporating interval-valued fuzzy sets into the Bousi-Prolog system. A syntax, declarative semantics and im- plementation for this extension is presented and formalised. We show, by using potential applications, that fuzzy logic programming frameworks enhanced with them can correctly work together with lexical resources and ontologies in order to improve their capabilities for knowledge representation and reasoning.

Both in statistics and machine learning there has been substantial interest in the design and study of distributed statistical or learning methods in recent years. One driving reason is the fact that in certain applications datasets have become so large that it is often unfeasible, or computationally undesirable, to carry out the analysis on a single machine. In a distributed method the data are divided over a cluster consisting of several machines and/or cores. The machines in the cluster then process their data locally, after which the local results are somehow aggregated on a central machine to finally produce the overall outcome of the statistical analysis. Distributed methods are not only used for computational reasons, but are for instance also of interest in situations where privacy is important and it is undesirable that all data are handled at a single location. Moreover, there are applications in which data are by construction gathered at multiple locations and first processed locally, before being combined at a central location. Over the last years a variety of distributed methods have been proposed. Recent examples include Consensus Monte Carlo (Scott et al. (2016)), WASP The research leading to these results has received funding from the Netherlands Science foundation NWO and from the European Research Council under ERC Grant Agreement 320637.