SIPTA Seminars

Optimal Transport (OT) seeks the most efficient way to morph one probability distribution into another one, and Distributionally Robust Optimization (DRO) studies worst-case risk minimization problems under distributional ambiguity. It is well known that OT gives rise to a rich class of data-driven DRO models, where the decision-maker plays a zero-sum game against nature who can adversely reshape the empirical distribution of the uncertain problem parameters within a prescribed transportation budget. Even though generic OT problems are computationally hard, the Nash strategies of the decision-maker and nature in OT-based DRO problems can often be computed efficiently. In this talk we will uncover deep connections between robustification and regularization, and we will disclose striking properties of nature’s Nash strategy, which implicitly constructs an adversarial training dataset. We will also show that OT-based DRO offers a principled approach to deal with distribution shifts and heterogeneous data sources, and we will highlight new applications of OT-based DRO in machine learning, statistics, risk management and control. Finally, we will argue that, while OT is useful for DRO, ideas from DRO can also help us to solve challenging OT problems.

Statisticians develop methods to assist in building probability statements that will be used to make inference on relevant unknowns. Popper argued that probability statements themselves can’t be falsified, but what about the statistical methods that use data to generate them? Science today is largely empirical, so if statistical methods’ conversion of data into scientific judgments can’t be scrutinized, then it’s not fair to expect society to “trust the science.”

Fisher’s underworld of probability concerns layers below the textbook surface level, where knowledge is vague and imprecise. Roughly, suppose that an agent quantifies his uncertainty about a relevant unknown via (imprecise) probability statements, which defines his betting odds. Now suppose that a second agent, who may not have her own probability statements about the relevant unknown, believes that the first agent’s assessments are wrong and can formulate odds at which she’d bet against the first agent’s wagers. If the second agent wins in these side-bets, then she reveals a shortcoming in the first agent’s assessments. I claim that the statistical method and “society” above are like the first and second agents here, respectively, and that scrutiny of a statistical method proceeds by giving “society” an opportunity to bet against its claims.

In this talk, I’ll carry out this scrutiny formally/mathematically and present some key take-aways. No surprise, a statistical method that’s falsification-proof in this sense is the behaviorally most reliable and conservative generalized Bayes rule. More surprising, however, is that a necessary condition for being falsification-proof is a statistical reliability property – called validity – that I’ve been advocating for recently. It follows, then, from the false confidence theorem that statistical methods quantifying uncertainty via precise probabilities can typically be falsified in this sense. More generally, since validity also implies certain behavioral reliability properties and needn’t be overly conservative, my new possibilistic inference framework (which I’ll describe and illustrate) is a promising way to balance the behavioral and statistical reliability properties.

There’s no paper yet on the exact contents of this talk, but some relevant material can be found at https://arxiv.org/abs/2203.06703 and https://arxiv.org/abs/2211.14567.

Engineers design components, structures and systems and plan activities to extend their service lives despite a limited understanding of the underlying physics and/or the availability of sufficient informative data. A big challenge is to deal with unknown and uncontrollable variables such as changes on the environmental conditions, deliberated threats, change of intended use, etc. As a result of this, large safety factors are usually adopted in order to mitigate the use of approximate methods and deal with uncertainty. Often the methods for dealing with uncertainty assume a complete knowledge of the underlying stochastic process. This wide availability of information is however rarely the case in practice.

Although imprecise probability offers the tools to cope with lack of knowledge and data, it is not largely adopted in practice. One of the main reasons is the lack of accessible and efficient tools, both analytical and numerical, for uncertainty quantification. On top, there exists still a lack of awareness of the potential capabilities of imprecise probability theory and its applications.

In this seminar, we are presenting the challenges in the application of imprecise probability to practical engineering problems These challenges have been the driver for several novel algorithms and approaches that are going to be presented.

Lower probabilities, defined as normalised and monotone set functions, constitute one of the basic models within Imprecise Probability theory. One of their interpretations allows building a bridge with coalitional game theory: the possibility space is regarded as a set of players who must share a reward, events represent coalitions of players who collaborate in order to obtain a greater reward, and the lower probability of a coalition represents the minimum reward that this collaboration can guarantee.

This correspondence makes lower probabilities and coalitional games formally equivalent, being the notation, terminology and interpretation the only difference. As an example, coherent lower probabilities are the same as exact games, the credal set of the lower probability is referred to as the core of the game,…

In this presentation I dig into this connection, paying special attention to game solutions and their interpretation as centroids of the credal set. In addition, I show that if we move to the more general setting of lower previsions, it is possible to represent information about the coalitions and their rewards that cannot be captured by the standard coalitional game theory. This shows that lower previsions constitute a more general framework than the classical theory of coalitional games.