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.

Foundational image-language models have generated considerable interest due to their efficient adaptation to downstream tasks by prompt learning. Prompt learning treats part of the language model input as trainable while freezing the rest, and optimizes an Empirical Risk Minimization objective. However, Empirical Risk Minimization is known to suffer from distributional shifts which hurt generalizability to prompts unseen during training. By leveraging the regularization ability of Bayesian methods, we frame prompt learning from the Bayesian perspective and formulate it as a variational inference problem. Our approach regularizes the prompt space, reduces overfitting to the seen prompts and improves the prompt generalization on unseen prompts. Our framework is implemented by modeling the input prompt space in a probabilistic manner, as an a priori distribution which makes our proposal compatible with prompt learning approaches that are unconditional or conditional on the image. We demonstrate empirically on 15 benchmarks that Bayesian prompt learning provides an appropriate coverage of the prompt space, prevents learning spurious features, and exploits transferable invariant features. This results in better generalization of unseen prompts, even across different datasets and domains. Code available at: https://github.com/saic-fi/Bayesian-Prompt-Learning

Deep learning (DL) has achieved remarkable successes in many disciplines such as computer vision and natural language processing due to the availability of ``big data''. However, such success cannot be easily replicated in many nuclear engineering problems because of the limited amount of training data, especially when the data comes from high-cost experiments. To overcome such a data scarcity issue, this paper explores the applications of deep generative models (DGMs) that have been widely used for image data generation to scientific data augmentation. DGMs, such as generative adversarial networks (GANs), normalizing flows (NFs), variational autoencoders (VAEs), and conditional VAEs (CVAEs), can be trained to learn the underlying probabilistic distribution of the training dataset. Once trained, they can be used to generate synthetic data that are similar to the training data and significantly expand the dataset size. By employing DGMs to augment TRACE simulated data of the steady-state void fractions based on the NUPEC Boiling Water Reactor Full-size Fine-mesh Bundle Test (BFBT) benchmark, this study demonstrates that VAEs, CVAEs, and GANs have comparable generative performance with similar errors in the synthetic data, with CVAEs achieving the smallest errors. The findings shows that DGMs have a great potential to augment scientific data in nuclear engineering, which proves effective for expanding the training dataset and enabling other DL models to be trained more accurately.

Reciprocity, or the tendency of individuals to mirror behavior, is a key measure that describes information exchange in a social network. Users in social networks tend to engage in different levels of reciprocal behavior. Differences in such behavior may indicate the existence of communities that reciprocate links at varying rates. In this paper, we develop methodology to model the diverse reciprocal behavior in growing social networks. In particular, we present a preferential attachment model with heterogeneous reciprocity that imitates the attraction users have for popular users, plus the heterogeneous nature by which they reciprocate links. We compare Bayesian and frequentist model fitting techniques for large networks, as well as computationally efficient variational alternatives. Cases where the number of communities are known and unknown are both considered. We apply the presented methods to the analysis of a Facebook wallpost network where users have non-uniform reciprocal behavior patterns. The fitted model captures the heavy-tailed nature of the empirical degree distributions in the Facebook data and identifies multiple groups of users that differ in their tendency to reply to and receive responses to wallposts.

This paper presents a new accelerated proximal Markov chain Monte Carlo methodology to perform Bayesian inference in imaging inverse problems with an underlying convex geometry. The proposed strategy takes the form of a stochastic relaxed proximal-point iteration that admits two complementary interpretations. For models that are smooth or regularised by Moreau-Yosida smoothing, the algorithm is equivalent to an implicit midpoint discretisation of an overdamped Langevin diffusion targeting the posterior distribution of interest. This discretisation is asymptotically unbiased for Gaussian targets and shown to converge in an accelerated manner for any target that is $\kappa$-strongly log-concave (i.e., requiring in the order of $\sqrt{\kappa}$ iterations to converge, similarly to accelerated optimisation schemes), comparing favorably to [M. Pereyra, L. Vargas Mieles, K.C. Zygalakis, SIAM J. Imaging Sciences, 13, 2 (2020), pp. 905-935] which is only provably accelerated for Gaussian targets and has bias. For models that are not smooth, the algorithm is equivalent to a Leimkuhler-Matthews discretisation of a Langevin diffusion targeting a Moreau-Yosida approximation of the posterior distribution of interest, and hence achieves a significantly lower bias than conventional unadjusted Langevin strategies based on the Euler-Maruyama discretisation. For targets that are $\kappa$-strongly log-concave, the provided non-asymptotic convergence analysis also identifies the optimal time step which maximizes the convergence speed. The proposed methodology is demonstrated through a range of experiments related to image deconvolution with Gaussian and Poisson noise, with assumption-driven and data-driven convex priors.

We investigate a framework for binary image denoising via restricted Boltzmann machines (RBMs) that introduces a denoising objective in quadratic unconstrained binary optimization (QUBO) form and is well-suited for quantum annealing. The denoising objective is attained by balancing the distribution learned by a trained RBM with a penalty term for derivations from the noisy image. We derive the statistically optimal choice of the penalty parameter assuming the target distribution has been well-approximated, and further suggest an empirically supported modification to make the method robust to that idealistic assumption. We also show under additional assumptions that the denoised images attained by our method are, in expectation, strictly closer to the noise-free images than the noisy images are. While we frame the model as an image denoising model, it can be applied to any binary data. As the QUBO formulation is well-suited for implementation on quantum annealers, we test the model on a D-Wave Advantage machine, and also test on data too large for current quantum annealers by approximating QUBO solutions through classical heuristics.