Measures of rank correlation are commonly used in statistics to capture the degree of concordance between two orderings of the same set of items. Standard measures like Kendall's tau and Spearman's rho coefficient put equal emphasis on each position of a ranking. Yet, motivated by applications in which some of the positions (typically those on the top) are more important than others, a few weighted variants of these measures have been proposed. Most of these generalizations fail to meet desirable formal properties, however. Besides, they are often quite inflexible in the sense of committing to a fixed weighing scheme. In this paper, we propose a weighted rank correlation measure on the basis of fuzzy order relations. Our measure, called scaled gamma, is related to Goodman and Kruskal's gamma rank correlation. It is parametrized by a fuzzy equivalence relation on the rank positions, which in turn is specified conveniently by a so-called scaling function. This approach combines soundness with flexibility: it has a sound formal foundation and allows for weighing rank positions in a flexible way.

Language change is a cultural evolutionary process in which variants of linguistic variables change in frequency through processes analogous to mutation, selection and genetic drift. In this work, we apply a recently-introduced method to corpus data to quantify the strength of selection in specific instances of historical language change. We first demonstrate, in the context of English irregular verbs, that this method is more reliable and interpretable than similar methods that have previously been applied. We further extend this study to demonstrate that a bias towards phonological simplicity overrides that favouring grammatical simplicity when these are in conflict. Finally, with reference to Spanish spelling reforms, we show that the method can also detect points in time at which selection strengths change, a feature that is generically expected for socially-motivated language change. Together, these results indicate how hypotheses for mechanisms of language change can be tested quantitatively using historical corpus data.

The field of Artificial Intelligence (AI) is focusing on creating automated decision-making (ADM) systems that operate as close as possible to human-like intelligence. This effort has pushed AI researchers into exploring cognitive fields like psychology. The work of Daniel Kahneman and the late Amos Tversky on biased human decision-making, including the study of the conjunction fallacy, has experienced a second revival because of this. Under the conjunction fallacy a human decision-maker will go against basic probability laws and rank as more likely a conjunction over one of its parts. It has been proven overtime through a set of experiments with the Linda Problem being the most famous one. Although this interdisciplinary effort is welcomed, we fear that AI researchers ignore the driving force behind the conjunction fallacy as captured by the Linda Problem: the fact that Linda must be stereotypically described as a woman. In this paper we revisit the Linda Problem and formulate it as a fairness problem. In doing so we introduce perception as a parameter of interest through the structural causal perception framework. Using an illustrative decision-making example, we showcase the proposed conceptual framework and its potential impact for developing fair ADM systems.

We consider Linear Stochastic Approximation (LSA) with a constant stepsize and Markovian data. Viewing the joint process of the data and LSA iterate as a time-homogeneous Markov chain, we prove its convergence to a unique limiting and stationary distribution in Wasserstein distance and establish non-asymptotic, geometric convergence rates. Furthermore, we show that the bias vector of this limit admits an infinite series expansion with respect to the stepsize. Consequently, the bias is proportional to the stepsize up to higher order terms. This result stands in contrast with LSA under i.i.d. data, for which the bias vanishes. In the reversible chain setting, we provide a general characterization of the relationship between the bias and the mixing time of the Markovian data, establishing that they are roughly proportional to each other. While Polyak-Ruppert tail-averaging reduces the variance of the LSA iterates, it does not affect the bias. The above characterization allows us to show that the bias can be reduced using Richardson-Romberg extrapolation with $m\ge 2$ stepsizes, which eliminates the $m-1$ leading terms in the bias expansion. This extrapolation scheme leads to an exponentially smaller bias and an improved mean squared error, both in theory and empirically. Our results immediately apply to the Temporal Difference learning algorithm with linear function approximation, Markovian data, and constant stepsizes.

Many problems in science and engineering involve optimizing an expensive black-box function over a high-dimensional space. For such black-box optimization (BBO) problems, we typically assume a small budget for online function evaluations, but also often have access to a fixed, offline dataset for pretraining. Prior approaches seek to utilize the offline data to approximate the function or its inverse but are not sufficiently accurate far from the data distribution. We propose BONET, a generative framework for pretraining a novel black-box optimizer using offline datasets. In BONET, we train an autoregressive model on fixed-length trajectories derived from an offline dataset. We design a sampling strategy to synthesize trajectories from offline data using a simple heuristic of rolling out monotonic transitions from low-fidelity to high-fidelity samples. Empirically, we instantiate BONET using a causally masked Transformer and evaluate it on Design-Bench, where we rank the best on average, outperforming state-of-the-art baselines.

Counting and sampling directed acyclic graphs from a Markov equivalence class are fundamental tasks in graphical causal analysis. In this paper we show that these tasks can be performed in polynomial time, solving a long-standing open problem in this area. Our algorithms are effective and easily implementable. As we show in experiments, these breakthroughs make thought-to-be-infeasible strategies in active learning of causal structures and causal effect identification with regard to a Markov equivalence class practically applicable.

Quantum State Tomography (QST) is a fundamental technique in Quantum Information Processing (QIP) for reconstructing unknown quantum states. However, the conventional QST methods are limited by the number of measurements required, which makes them impractical for large-scale quantum systems. To overcome this challenge, we propose the integration of Quantum Machine Learning (QML) techniques to enhance the efficiency of QST. In this paper, we conduct a comprehensive investigation into various approaches for QST, encompassing both classical and quantum methodologies; We also implement different QML approaches for QST and demonstrate their effectiveness on various simulated and experimental quantum systems, including multi-qubit networks. Our results show that our QML-based QST approach can achieve high fidelity (98%) with significantly fewer measurements than conventional methods, making it a promising tool for practical QIP applications.

Discriminative learning effectively predicts true object class for image classification. However, it often results in false positives for outliers, posing critical concerns in applications like autonomous driving and video surveillance systems. Previous attempts to address this challenge involved training image classifiers through contrastive learning using actual outlier data or synthesizing outliers for self-supervised learning. Furthermore, unsupervised generative modeling of inliers in pixel space has shown limited success for outlier detection. In this work, we introduce a quantile-based maximum likelihood objective for learning the inlier distribution to improve the outlier separation during inference. Our approach fits a normalizing flow to pre-trained discriminative features and detects the outliers according to the evaluated log-likelihood. The experimental evaluation demonstrates the effectiveness of our method as it surpasses the performance of the state-of-the-art unsupervised methods for outlier detection. The results are also competitive compared with a recent self-supervised approach for outlier detection. Our work allows to reduce dependency on well-sampled negative training data, which is especially important for domains like medical diagnostics or remote sensing.

Complex decision-making systems rarely have direct access to the current state of the world and they instead rely on opinions to form an understanding of what the ground truth could be. Even in problems where experts provide opinions without any intention to manipulate the decision maker, it is challenging to decide which expert's opinion is more reliable -- a challenge that is further amplified when decision-maker has limited, delayed, or no access to the ground truth after the fact. This paper explores a statistical approach to infer the competence of each expert based on their opinions without any need for the ground truth. Echoing the logic behind what is commonly referred to as \textit{the wisdom of crowds}, we propose measuring the competence of each expert by their likeliness to agree with their peers. We further show that the more reliable an expert is the more likely it is that they agree with their peers. We leverage this fact to propose a completely unsupervised version of the na\"{i}ve Bayes classifier and show that the proposed technique is asymptotically optimal for a large class of problems. In addition to aggregating a large block of opinions, we further apply our technique for online opinion aggregation and for decision-making based on a limited the number of opinions.

It is well know that basic conditionals satisfy some desirable basic logical and probabilistic properties, such as the compound probability theorem, but checking the validity of these becomes trickier when we switch to compound and iterated conditionals. We consider de Finetti's notion of conditional as a three-valued object and as a conditional random quantity in the betting framework. We recall the notions of conjunction and disjunction among conditionals in selected trivalent logics. First, in the framework of specific three-valued logics we analyze the notions of iterated conditioning introduced by Cooper-Calabrese, de Finetti and Farrell, respectively. We show that the compound probability theorem and other basic properties are not preserved by these objects, by also computing some probability propagation rules. Then, for each trivalent logic we introduce an iterated conditional as a suitable random quantity which satisfies the compound prevision theorem and some of the desirable properties. We also check the validity of two generalized versions of Bayes' Rule for iterated conditionals. We study the p-validity of generalized versions of Modus Ponens and two-premise centering for iterated conditionals. Finally, we observe that all the basic properties are satisfied only by the iterated conditional mainly developed in recent papers by Gilio and Sanfilippo in the setting of conditional random quantities.