Experts do not always feel very, comfortable when they have to give precise numerical estimations of certainty degrees. In this paper we present a qualitative approach which allows for attaching partially ordered symbolic grades to logical formulas. Uncertain information is expressed by means of parameterized modal operators. We propose a semantics for this multimodal logic and give a sound and complete axiomatization. We study the links with related approaches and suggest how this framework might be used to manage both uncertain and incomplere knowledge.

Influence diagram is a graphical representation of belief networks with uncertainty. This article studies the structural properties of a probabilistic model in an influence diagram. In particular, structural controllability theorems and structural observability theorems are developed and algorithms are formulated. Controllability and observability are fundamental concepts in dynamic systems (Luenberger 1979). Controllability corresponds to the ability to control a system while observability analyzes the inferability of its variables. Both properties can be determined by the ranks of the system matrices. Structural controllability and observability, on the other hand, analyze the property of a system with its structure only, without the specific knowledge of the values of its elements (tin 1974, Shields and Pearson 1976). The structural analysis explores the connection between the structure of a model and the functional dependence among its elements. It is useful in comprehending problem and formulating solution by challenging the underlying intuitions and detecting inconsistency in a model. This type of qualitative reasoning can sometimes provide insight even when there is insufficient numerical information in a model.

Possibilistic logic has been proposed as a numerical formalism for reasoning with uncertainty. There has been interest in developing qualitative accounts of possibility, as well as an explanation of the relationship between possibility and modal logics. We present two modal logics that can be used to represent and reason with qualitative statements of possibility and necessity. Within this modal framework, we are able to identify interesting relationships between possibilistic logic, beliefs and conditionals. In particular, the most natural conditional definable via possibilistic means for default reasoning is identical to Pearl's conditional for e-semantics.

In this paper we describe a novel method for evidential reasoning [1]. It involves modelling the process of evidential reasoning in three steps, namely, evidence structure construction, evidence accumulation, and decision making. The proposed method, called RES, is novel in that evidence strength is associated with an evidential support relationship (an argument) between a pair of statements and such strength is carried by comparison between arguments. This is in contrast to the onventional approaches, where evidence strength is represented numerically and is associated with a statement.

We study the problem of learning Markov decision processes with finite state and action spaces when the transition probability distributions and loss functions are chosen adversarially and are allowed to change with time. We introduce an algorithm whose regret with respect to any policy in a comparison class grows as the square root of the number of rounds of the game, provided the transition probabilities satisfy a uniform mixing condition. Our approach is efficient as long as the comparison class is polynomial and we can compute expectations over sample paths for each policy. Designing an efficient algorithm with small regret for the general case remains an open problem.

The classical approach to linear system identification is given by parametric Prediction Error Methods (PEM). In this context, model complexity is often unknown so that a model order selection step is needed to suitably trade-off bias and variance. Recently, a different approach to linear system identification has been introduced, where model order determination is avoided by using a regularized least squares framework. In particular, the penalty term on the impulse response is defined by so called stable spline kernels. They embed information on regularity and BIBO stability, and depend on a small number of parameters which can be estimated from data. In this paper, we provide new nonsmooth formulations of the stable spline estimator. In particular, we consider linear system identification problems in a very broad context, where regularization functionals and data misfits can come from a rich set of piecewise linear quadratic functions. Moreover, our anal- ysis includes polyhedral inequality constraints on the unknown impulse response. For any formulation in this class, we show that interior point methods can be used to solve the system identification problem, with complexity O(n3)+O(mn2) in each iteration, where n and m are the number of impulse response coefficients and measurements, respectively. The usefulness of the framework is illustrated via a numerical experiment where output measurements are contaminated by outliers.

We consider the problem of adaptive stratified sampling for Monte Carlo integration of a noisy function, given a finite budget n of noisy evaluations to the function. We tackle in this paper the problem of adapting to the function at the same time the number of samples into each stratum and the partition itself. More precisely, it is interesting to refine the partition of the domain in area where the noise to the function, or where the variations of the function, are very heterogeneous. On the other hand, having a (too) refined stratification is not optimal. Indeed, the more refined the stratification, the more difficult it is to adjust the allocation of the samples to the stratification, i.e. sample more points where the noise or variations of the function are larger. We provide in this paper an algorithm that selects online, among a large class of partitions, the partition that provides the optimal trade-off, and allocates the samples almost optimally on this partition.

Mean-field variational methods are widely used for approximate posterior inference in many probabilistic models. In a typical application, mean-field methods approximately compute the posterior with a coordinate-ascent optimization algorithm. When the model is conditionally conjugate, the coordinate updates are easily derived and in closed form. However, many models of interest---like the correlated topic model and Bayesian logistic regression---are nonconjuate. In these models, mean-field methods cannot be directly applied and practitioners have had to develop variational algorithms on a case-by-case basis. In this paper, we develop two generic methods for nonconjugate models, Laplace variational inference and delta method variational inference. Our methods have several advantages: they allow for easily derived variational algorithms with a wide class of nonconjugate models; they extend and unify some of the existing algorithms that have been derived for specific models; and they work well on real-world datasets. We studied our methods on the correlated topic model, Bayesian logistic regression, and hierarchical Bayesian logistic regression.

We consider the problem of creating fair course timetables in the setting of a university. Our motivation is to improve the overall satisfaction of individuals concerned (students, teachers, etc.) by providing a fair timetable to them. The central idea is that undesirable arrangements in the course timetable, i.e., violations of soft constraints, should be distributed in a fair way among the individuals. We propose two formulations for the fair course timetabling problem that are based on max-min fairness and Jain's fairness index, respectively. Furthermore, we present and experimentally evaluate an optimization algorithm based on simulated annealing for solving max-min fair course timetabling problems. The new contribution is concerned with measuring the energy difference between two timetables, i.e., how much worse a timetable is compared to another timetable with respect to max-min fairness. We introduce three different energy difference measures and evaluate their impact on the overall algorithm performance. The second proposed problem formulation focuses on the tradeoff between fairness and the total amount of soft constraint violations. Our experimental evaluation shows that the known best solutions to the ITC2007 curriculum-based course timetabling instances are quite fair with respect to Jain's fairness index. However, the experiments also show that the fairness can be improved further for only a rather small increase in the total amount of soft constraint violations.

The concept of refinement from probability elicitation is considered for proper scoring rules. Taking directions from the axioms of probability, refinement is further clarified using a Hilbert space interpretation and reformulated into the underlying data distribution setting where connections to maximal marginal diversity and conditional entropy are considered and used to derive measures that provide arbitrarily tight bounds on the Bayes error. Refinement is also reformulated into the classifier output setting and its connections to calibrated classifiers and proper margin losses are established.