In this paper we derive an explicit formula for calculating the marginal likelihood of a given factorization of a categorical dataset. Since the marginal likelihood is proportional to the posterior probability of the factorization, these likelihoods can be used to order all possible factorizations and select the "best" way to factor the overall distribution from which the dataset is drawn. The best factorization can then be used to construct a Bayes classifier which benefits from factoring out mutually independent sets of variables.

In this paper, we propose a Bayesian framework to create two dimensional ionospheric images of high spatio-temporal resolution to monitor ionospheric irregularities as measured by the S4 index. Here, we recast the standard Bayesian recursive filtering for a linear Gaussian state-space model, also referred to as the Kalman filter, first by augmenting the (pierce point) observation model with connectivity information stemming from the insight and assumptions/standard modeling about the spatial distribution of the scintillation activity on the ionospheric shell at 350 km altitude. Thus, we achieve to handle the limited spatio-temporal observations. Then, by introducing a set of Kalman filters running in parallel, we mitigate the uncertainty related to a tuning parameter of the proposed augmented model. The output images are a weighted average of the state estimates of the individual filters. We demonstrate our approach by rendering two dimensional real-time ionospheric images of S4 amplitude scintillation at 350 km over South America with temporal resolution of one minute. Furthermore, we employ extra S4 data that was not used in producing these ionospheric images, to check and verify the ability of our images to predict this extra data in particular ionospheric pierce points. Our results show that in areas with a network of ground receivers with a relatively good coverage (e.g. within a couple of kilometers distance) the produced images can provide reliable real-time results. Our proposed algorithmic framework can be readily used to visualize real-time ionospheric images taking as inputs the available scintillation data provided from freely available web-servers.

Marginal-likelihood based model-selection, even though promising, is rarely used in deep learning due to estimation difficulties. Instead, most approaches rely on validation data, which may not be readily available. In this work, we present a scalable marginal-likelihood estimation method to select both the hyperparameters and network architecture based on the training data alone. Some hyperparameters can be estimated online during training, simplifying the procedure. Our marginal-likelihood estimate is based on Laplace's method and Gauss-Newton approximations to the Hessian, and it outperforms cross-validation and manual-tuning on standard regression and image classification datasets, especially in terms of calibration and out-of-distribution detection. Our work shows that marginal likelihoods can improve generalization and be useful when validation data is unavailable (e.g., in nonstationary settings).

Probabilistic machine learning models are often insufficient to help with decisions on interventions because those models find correlations - not causal relationships. If observational data is only available and experimentation are infeasible, the correct approach to study the impact of an intervention is to invoke Pearl's causality framework. Even that framework assumes that the underlying causal graph is known, which is seldom the case in practice. When the causal structure is not known, one may use out-of-the-box algorithms to find causal dependencies from observational data. However, there exists no method that also accounts for the decision-maker's prior knowledge when developing the causal structure either. The objective of this paper is to develop rational approaches for making decisions from observational data in the presence of causal graph uncertainty and prior knowledge from the decision-maker. We use ensemble methods like Bayesian Model Averaging (BMA) to infer set of causal graphs that can represent the data generation process. We provide decisions by computing the expected value and risk of potential interventions explicitly. We demonstrate our approach by applying them in different example contexts.

Designing efficient exploration is central to Reinforcement Learning due to the fundamental problem posed by the exploration-exploitation dilemma. Bayesian exploration strategies like Thompson Sampling resolve this trade-off in a principled way by modeling and updating the distribution of the parameters of the the action-value function, the outcome model of the environment. However, this technique becomes infeasible for complex environments due to the difficulty of representing and updating probability distributions over parameters of outcome models of corresponding complexity. Moreover, the approximation techniques introduced to mitigate this issue typically result in poor exploration-exploitation trade-offs, as observed in the case of deep neural network models with approximate posterior methods that have been shown to underperform in the deep bandit scenario. In this paper we introduce Sample Average Uncertainty (SAU), a simple and efficient uncertainty measure for contextual bandits. While Bayesian approaches like Thompson Sampling estimate outcomes uncertainty indirectly by first quantifying the variability over the parameters of the outcome model, SAU is a frequentist approach that directly estimates the uncertainty of the outcomes based on the value predictions. Importantly, we show theoretically that the uncertainty measure estimated by SAU asymptotically matches the uncertainty provided by Thompson Sampling, as well as its regret bounds. Because of its simplicity SAU can be seamlessly applied to deep contextual bandits as a very scalable drop-in replacement for epsilon-greedy exploration. Finally, we empirically confirm our theory by showing that SAU-based exploration outperforms current state-of-the-art deep Bayesian bandit methods on several real-world datasets at modest computation cost.

Bayesian inference has the potential to improve deep neural networks (DNNs) by providing 1) uncertainty estimates for robust prediction and downstream decision-making, and 2) an objective function (the marginal likelihood) for hyperparameter selection [MacKay, 1992a; 1992b; 2003]. The recent success of deep learning [Krizhevsky et al., 2012; Vaswani et al., 2017; Schrittwieser et al., 2020] has renewed interest in large-scale Bayesian Neural Networks (BNNs) as well, with effort mainly focused on obtaining useful uncertainty estimates [Blundell et al., 2015; Kingma et al., 2015; Gal and Ghahramani, 2016]. Despite already providing usable uncertainty estimates, there is significant evidence that current approximations to the uncertainty on neural network weights can still be significantly improved [Hron et al., 2018; Foong et al., 2020]. The accuracy of the uncertainty approximation is also linked to the quality of the marginal likelihood estimate [Blei et al., 2017]. Since hyperparameter learning using the marginal likelihood fails for most common approximations [e.g., Blundell et al., 2015], the accuracy of the uncertainty estimates is also questionable. Damianou and Lawrence [2013] used Gaussian processes [Rasmussen and Williams, 2006] as layers to create a different Bayesian analogue to a DNN: the Deep Gaussian process (DGP). Gaussian processes (GPs) are a different representation of a single layer neural network, which is promising because it allows high-quality approximations to uncertainty [Titsias, 2009; Burt et al., 2019].

This paper considers the problem of differentially private semi-supervised transfer learning. The notion of membership-mapping is developed using measure theory basis to learn data representation via a fuzzy membership function. An alternative conception of deep autoencoder, referred to as Conditionally Deep Membership-Mapping Autoencoder (CDMMA) (that consists of a nested compositions of membership-mappings), is considered. Under practice-oriented settings, an analytical solution for the learning of CDMFA can be derived by means of variational optimization. The paper proposes a transfer learning approach that combines CDMMA with a tailored noise adding mechanism to achieve a given level of privacy-loss bound with the minimum perturbation of the data. Numerous experiments were carried out using MNIST, USPS, Office, and Caltech256 datasets to verify the competitive robust performance of the proposed methodology.

Credal networks are a popular class of imprecise probabilistic graphical models obtained as a Bayesian network generalization based on, so-called credal, sets of probability mass functions. A Java library called CREMA has been recently released to model, process and query credal networks. Despite the NP-hardness of the (exact) task, a number of algorithms is available to approximate credal network inferences. In this paper we present CREPO, an open repository of synthetic credal networks, provided together with the exact results of inference tasks on these models. A Python tool is also delivered to load these data and interact with CREMA, thus making extremely easy to evaluate and compare existing and novel inference algorithms. To demonstrate such benchmarking scheme, we propose an approximate heuristic to be used inside variable elimination schemes to keep a bound on the maximum number of vertices generated during the combination step. A CREPO-based validation against approximate procedures based on linearization and exact techniques performed in CREMA is finally discussed.

Model selection, via penalized likelihood type criteria, is a standard task in many statistical inference and machine learning problems. Progress has led to deriving criteria with asymptotic consistency results and an increasing emphasis on introducing non-asymptotic criteria. We focus on the problem of modeling non-linear relationships in regression data with potential hidden graph-structured interactions between the high-dimensional predictors, within the mixture of experts modeling framework. In order to deal with such a complex situation, we investigate a block-diagonal localized mixture of polynomial experts (BLoMPE) regression model, which is constructed upon an inverse regression and block-diagonal structures of the Gaussian expert covariance matrices. We introduce a penalized maximum likelihood selection criterion to estimate the unknown conditional density of the regression model. This model selection criterion allows us to handle the challenging problem of inferring the number of mixture components, the degree of polynomial mean functions, and the hidden block-diagonal structures of the covariance matrices, which reduces the number of parameters to be estimated and leads to a trade-off between complexity and sparsity in the model. In particular, we provide a strong theoretical guarantee: a finite-sample oracle inequality satisfied by the penalized maximum likelihood estimator with a Jensen-Kullback-Leibler type loss, to support the introduced non-asymptotic model selection criterion. The penalty shape of this criterion depends on the complexity of the considered random subcollection of BLoMPE models, including the relevant graph structures, the degree of polynomial mean functions, and the number of mixture components.

A fundamental task in AI is to assess (in)dependence between mixed-type variables (text, image, sound). We propose a Bayesian kernelised correlation test of (in)dependence using a Dirichlet process model. The new measure of (in)dependence allows us to answer some fundamental questions: Based on data, are (mixed-type) variables independent? How likely is dependence/independence to hold? How high is the probability that two mixed-type variables are more than just weakly dependent? We theoretically show the properties of the approach, as well as algorithms for fast computation with it. We empirically demonstrate the effectiveness of the proposed method by analysing its performance and by comparing it with other frequentist and Bayesian approaches on a range of datasets and tasks with mixed-type variables.