We use an algebraic viewpoint, namely a matrix framework to deal with the problem of resource allocation under uncertainty in the context of a qualitative approach. Our basic qualitative data are a plausibility relation over the resources, a hierarchical relation over the agents and of course the preference that the agents have over the resources. With this data we propose a qualitative binary relation $\unrhd$ between allocations such that $\mathcal{F}\unrhd \mathcal{G}$ has the following intended meaning: the allocation $\mathcal{F}$ produces more or equal social welfare than the allocation $\mathcal{G}$. We prove that there is a family of allocations which are maximal with respect to $\unrhd$. We prove also that there is a notion of simple deal such that optimal allocations can be reached by sequences of simple deals. Finally, we introduce some mechanism for discriminating {optimal} allocations.

About this course: This course describes Bayesian statistics, in which one's inferences about parameters or hypotheses are updated as evidence accumulates. You will learn to use Bayes' rule to transform prior probabilities into posterior probabilities, and be introduced to the underlying theory and perspective of the Bayesian paradigm. The course will apply Bayesian methods to several practical problems, to show end-to-end Bayesian analyses that move from framing the question to building models to eliciting prior probabilities to implementing in R (free statistical software) the final posterior distribution. Additionally, the course will introduce credible regions, Bayesian comparisons of means and proportions, Bayesian regression and inference using multiple models, and discussion of Bayesian prediction. We assume learners in this course have background knowledge equivalent to what is covered in the earlier three courses in this specialization: "Introduction to Probability and Data," "Inferential Statistics," and "Linear Regression and Modeling."

One of the largest growing demographics in the EU is individuals aged 65 and over, and two thirds of this group are in situation of multimorbidity, i.e., perons who suffer from two or more chronic diseases. The ineffective treatment of multimorbidity has been pointed out as an urgent problem to address by the Academy of Medical Sciences in a recently released report. As part of an EU H2020 funded project called ProACT, our team at IBM Research – Ireland is working with partners in academia and industry to find new ways to use IoT, AI and cloud technologies to advance self-management capabilities and home-based integrated care for Persons with Multimorbidity (PwM). The ProACT project is investigating ways wearable, home sensors and tablet applications can be used to help persons with multimorbidity, as well as their support actors, which include informal caregivers (e.g. The project includes proof-of-concept trials in Ireland and Belgium, involving national health services, with a number of patients equipped with wearable and home sensors, and their support actors.

Pricing is a common problem faced by businesses, and one that can be addressed effectively by Bayesian statistical methods. We'll step through a simple example and build the background necessary to extend get involved with this approach. Let's start with some hypothetical data. A small company has tried a few different price points (say, one week each) and recorded the demand at each price. We'll abstract away some economic issues in order to focus on the statistical approach.

Artificial intelligence owes a lot of its smarts to Judea Pearl. In the 1980s he led efforts that allowed machines to reason probabilistically. In his latest book, "The Book of Why: The New Science of Cause and Effect," he argues that artificial intelligence has been handicapped by an incomplete understanding of what intelligence really is. Three decades ago, a prime challenge in artificial intelligence research was to program machines to associate a potential cause to a set of observable conditions. Pearl figured out how to do that using a scheme called Bayesian networks.

We present a simple model of a non-equilibrium self-organizing market where asset prices are partially driven by investment decisions of a bounded-rational agent. The agent acts in a stochastic market environment driven by various exogenous "alpha" signals, agent's own actions (via market impact), and noise. Unlike traditional agent-based models, our agent aggregates all traders in the market, rather than being a representative agent. Therefore, it can be identified with a bounded-rational component of the market itself, providing a particular implementation of an Invisible Hand market mechanism. In such setting, market dynamics are modeled as a fictitious self-play of such bounded-rational market-agent in its adversarial stochastic environment. As rewards obtained by such self-playing market agent are not observed from market data, we formulate and solve a simple model of such market dynamics based on a neuroscience-inspired Bounded Rational Information Theoretic Inverse Reinforcement Learning (BRIT-IRL). This results in effective asset price dynamics with a non-linear mean reversion - which in our model is generated dynamically, rather than being postulated. We argue that our model can be used in a similar way to the Black-Litterman model. In particular, it represents, in a simple modeling framework, market views of common predictive signals, market impacts and implied optimal dynamic portfolio allocations, and can be used to assess values of private signals. Moreover, it allows one to quantify a "market-implied" optimal investment strategy, along with a measure of market rationality. Our approach is numerically light, and can be implemented using standard off-the-shelf software such as TensorFlow.

Aiming at financial applications, we study the problem of learning the volatility under market microstructure noise. Specifically, we consider noisy discrete time observations from a stochastic differential equation and develop a novel computational method to learn the diffusion coefficient of the equation. We take a nonparametric Bayesian approach, where we model the volatility function a priori as piecewise constant. Its prior is specified via the inverse Gamma Markov chain. Sampling from the posterior is accomplished by incorporating the Forward Filtering Backward Simulation algorithm in the Gibbs sampler. Good performance of the method is demonstrated on two representative synthetic data examples. Finally, we apply the method on the EUR/USD exchange rate dataset.

In this paper, we introduce a novel algorithm to solve projected model counting (PMC). PMC asks to count solutions of a Boolean formula with respect to a given set of projected variables, where multiple solutions that are identical when restricted to the projected variables count as only one solution. Our algorithm exploits small treewidth of the primal graph of the input instance. It runs in time $O({2^{2^{k+4}} n^2})$ where k is the treewidth and n is the input size of the instance. In other words, we obtain that the problem PMC is fixed-parameter tractable when parameterized by treewidth. Further, we take the exponential time hypothesis (ETH) into consideration and establish lower bounds of bounded treewidth algorithms for PMC, yielding asymptotically tight runtime bounds of our algorithm.

Approximate Bayesian Computation (ABC) is typically used when the likelihood is either unavailable or intractable but where data can be simulated under different parameter settings using a forward model. Despite the recent interest in ABC, high-dimensional data and costly simulations still remain a bottleneck. There is also no consensus as to how to best assess the performance of such methods without knowing the true posterior. We show how a nonparametric conditional density estimation (CDE) framework, which we refer to as ABC-CDE, help address three key challenges in ABC: (i) how to efficiently estimate the posterior distribution with limited simulations and different types of data, (ii) how to tune and compare the performance of ABC and related methods in estimating the posterior itself, rather than just certain properties of the density, and (iii) how to efficiently choose among a large set of summary statistics based on a CDE surrogate loss. We provide theoretical and empirical evidence that justify ABC-CDE procedures that directly estimate and assess the posterior based on an initial ABC sample, and we describe settings where standard ABC and regression-based approaches are inadequate.

Many data science issues have to be addressed through unbalanced datasets. Indeed, it may be quite affordable to gather data on the representatives of a given pathology in medicine, or positive operating scenarios of machines in the industry [1]. The related complementary occurrences are, by contrast, scarce and/or expensive to raise. The practice of One-Class Classification (OCC) has been developed within this consideration [1, 2]. One-class classifiers are trained on a single class sample, in the possible presence of a few counterexamples. The related issue consists of understanding and isolating a given class from the rest of the universe. The resulting model allows to predict target (or positive) patterns and to reject outlier (or negative) ones. One-Class Support Vector Machine (OCSVM) is a popular OCC method [3, 4]. Statistics-based techniques such as Gaussian models and Kernel Density Estimation (KDE) [5] are also commonly considered as respectively parametric and nonparametric approaches to estimate a sample distribution.