We study the pricing problem faced by a firm that sells a large number of products, described via a wide range of features, to customers that arrive over time. Customers independently make purchasing decisions according to a general choice model that includes products features and customers' characteristics, encoded as $d$-dimensional numerical vectors, as well as the price offered. The parameters of the choice model are a priori unknown to the firm, but can be learned as the (binary-valued) sales data accrues over time. The firm's objective is to minimize the regret, i.e., the expected revenue loss against a clairvoyant policy that knows the parameters of the choice model in advance, and always offers the revenue-maximizing price. This setting is motivated in part by the prevalence of online marketplaces that allow for real-time pricing. We assume a structured choice model, parameters of which depend on $s_0$ out of the $d$ product features. We propose a dynamic policy, called Regularized Maximum Likelihood Pricing (RMLP) that leverages the (sparsity) structure of the high-dimensional model and obtains a logarithmic regret in $T$. More specifically, the regret of our algorithm is of $O(s_0 \log d \cdot \log T)$. Furthermore, we show that no policy can obtain regret better than $O(s_0 (\log d + \log T))$.

We establish the equivalence of two very general theories: the first is the decision-theoretic formalisation of incomplete preferences based on the mixture independence axiom; the second is the theory of coherent sets of desirable gambles (bounded variables) developed in the context of imprecise probability and extended here to vector-valued gambles. Such an equivalence allows us to analyse the theory of incomplete preferences from the point of view of desirability. Among other things, this leads us to uncover an unexpected and clarifying relation: that the notion of `state independence'---the traditional assumption that we can have separate models for beliefs (probabilities) and values (utilities)---coincides with that of `strong independence' in imprecise probability; this connection leads us also to propose much weaker, and arguably more realistic, notions of state independence. Then we simplify the treatment of complete beliefs and values by putting them on a more equal footing. We study the role of the Archimedean condition---which allows us to actually talk of expected utility---, identify some weaknesses and propose alternatives that solve these. More generally speaking, we show that desirability is a valuable alternative foundation to preferences for decision theory that streamlines and unifies a number of concepts while preserving great generality. In addition, the mentioned equivalence shows for the first time how to extend the theory of desirability to imprecise non-linear utility, thus enabling us to formulate one of the most powerful self-consistent theories of reasoning and decision-making available today.

We propose a dynamic network model where two mechanisms control the probability of a link between two nodes: (i) the existence or absence of this link in the past, and (ii) node-specific latent variables (dynamic fitnesses) describing the propensity of each node to create links. Assuming a Markov dynamics for both mechanisms, we propose an Expectation-Maximization algorithm for model estimation and inference of the latent variables. The estimated parameters and fitnesses can be used to forecast the presence of a link in the future. We apply our methodology to the e-MID interbank network for which the two linkage mechanisms are associated with two different trading behaviors in the process of network formation, namely preferential trading and trading driven by node-specific characteristics. The empirical results allow to recognise preferential lending in the interbank market and indicate how a method that does not account for time-varying network topologies tends to overestimate preferential linkage.

A novel data-driven stochastic robust optimization (DDSRO) framework is proposed for optimization under uncertainty leveraging labeled multi-class uncertainty data. Uncertainty data in large datasets are often collected from various conditions, which are encoded by class labels. Machine learning methods including Dirichlet process mixture model and maximum likelihood estimation are employed for uncertainty modeling. A DDSRO framework is further proposed based on the data-driven uncertainty model through a bi-level optimization structure. The outer optimization problem follows a two-stage stochastic programming approach to optimize the expected objective across different data classes; adaptive robust optimization is nested as the inner problem to ensure the robustness of the solution while maintaining computational tractability. A decomposition-based algorithm is further developed to solve the resulting multi-level optimization problem efficiently. Case studies on process network design and planning are presented to demonstrate the applicability of the proposed framework and algorithm.

Learning probability distributions on the weights of neural networks (NNs) has recently proven beneficial in many applications. Bayesian methods, such as Stein variational gradient descent (SVGD), offer an elegant framework to reason about NN model uncertainty. However, by assuming independent Gaussian priors for the individual NN weights (as often applied), SVGD does not impose prior knowledge that there is often structural information (dependence) among weights. We propose efficient posterior learning of structural weight uncertainty, within an SVGD framework, by employing matrix variate Gaussian priors on NN parameters. We further investigate the learned structural uncertainty in sequential decision-making problems, including contextual bandits and reinforcement learning. Experiments on several synthetic and real datasets indicate the superiority of our model, compared with state-of-the-art methods.

Reinforcement learning is a body of theory and techniques for optimal sequential decision making developed in the last thirty years primarily within the machine learning and operations research communities, and which has separately become important in psychology and neuroscience. This tutorial will develop an intuitive understanding of the underlying formal problem (Markov decision processes) and its core solution methods, including dynamic programming, Monte Carlo methods, and temporal-difference learning. It will focus on how these methods have been combined with parametric function approximation, including deep learning, to find good approximate solutions to problems that are otherwise too large to be addressed at all.

Remarkably general method for bounding the statistical risk of penalized likelihood estimators comes from work on two-part coding, one of the minimum description length (MDL) approaches to statistical inference. Two-part coding MDL prescribes assigning codelengths to a model (or model class) then selecting the distribution that provides the most efficient description of one's data [1]. The total description length has two parts: the part that specifies a distribution within the model (as well as a model within the model class if necessary) and the part that specifies the data with reference to the specified distribution. If the codelengths are exactly Kraft-valid, this approach is equivalent to Bayesian maximum a posteriori (MAP) estimation, in that the two parts correspond to log reciprocal of prior and log reciprocal of likelihood respectively. More generally, one can call the part of the codelength specifying the distribution a penalty term; it is called the complexity in MDL literature. Let (Θ, L) denote a discrete set indexing distributions along with a complexity function. With X P, the (pointwise) redundancy of any θ Θ is its two-part codelength minus log(1/p(X)), the codelength one gets by using P as the coding distribution.

Artificial intelligence (AI) generally and machine learning (ML) specifically demonstrate impressive practical success in many different application domains, e.g. in autonomous driving, speech recognition, or recommender systems. Deep learning approaches, trained on extremely large data sets or using reinforcement learning methods have even exceeded human performance in visual tasks, particularly on playing games such as Atari, or mastering the game of Go. Even in the medical domain there are remarkable results. The central problem of such models is that they are regarded as black-box models and even if we understand the underlying mathematical principles, they lack an explicit declarative knowledge representation, hence have difficulty in generating the underlying explanatory structures. This calls for systems enabling to make decisions transparent, understandable and explainable. A huge motivation for our approach are rising legal and privacy aspects. The new European General Data Protection Regulation entering into force on May 25th 2018, will make black-box approaches difficult to use in business. This does not imply a ban on automatic learning approaches or an obligation to explain everything all the time, however, there must be a possibility to make the results re-traceable on demand. In this paper we outline some of our research topics in the context of the relatively new area of explainable-AI with a focus on the application in medicine, which is a very special domain. This is due to the fact that medical professionals are working mostly with distributed heterogeneous and complex sources of data. In this paper we concentrate on three sources: images, *omics data and text. We argue that research in explainable-AI would generally help to facilitate the implementation of AI/ML in the medical domain, and specifically help to facilitate transparency and trust.

In X-ray binary star systems consisting of a compact object that accretes material from an orbiting secondary star, there is no straightforward means to decide if the compact object is a black hole or a neutron star. To assist this classification, we develop a Bayesian statistical model that makes use of the fact that X-ray binary systems appear to cluster based on their compact object type when viewed from a 3-dimensional coordinate system derived from X-ray spectral data. The first coordinate of this data is the ratio of counts in mid to low energy band (color 1), the second coordinate is the ratio of counts in high to low energy band (color 2), and the third coordinate is the sum of counts in all three bands. We use this model to estimate the probabilities that an X-ray binary system contains a black hole, non-pulsing neutron star, or pulsing neutron star. In particular, we utilize a latent variable model in which the latent variables follow a Gaussian process prior distribution, and hence we are able to induce the spatial correlation we believe exists between systems of the same type. The utility of this approach is evidenced by the accurate prediction of system types using Rossi X-ray Timing Explorer All Sky Monitor data, but it is not flawless. In particular, non-pulsing neutron systems containing "bursters" that are close to the boundary demarcating systems containing black holes tend to be classified as black hole systems. As a byproduct of our analyses, we provide the astronomer with public R code that can be used to predict the compact object type of X-ray binaries given training data.

In this work we introduce declarative statistics, a suite of declarative modelling tools for statistical analysis. Statistical constraints represent the key building block of declarative statistics. First, we introduce a range of relevant counting and matrix constraints and associated decompositions, some of which novel, that are instrumental in the design of statistical constraints. Second, we introduce a selection of novel statistical constraints and associated decompositions, which constitute a self-contained toolbox that can be used to tackle a wide range of problems typically encountered by statisticians. Finally, we deploy these statistical constraints to a wide range of application areas drawn from classical statistics and we contrast our framework against established practices.