We study the problem of learning Bayesian networks where an $\epsilon$-fraction of the samples are adversarially corrupted. We focus on the fully-observable case where the underlying graph structure is known. In this work, we present the first nearly-linear time algorithm for this problem with a dimension-independent error guarantee. Previous robust algorithms with comparable error guarantees are slower by at least a factor of $(d/\epsilon)$, where $d$ is the number of variables in the Bayesian network and $\epsilon$ is the fraction of corrupted samples. Our algorithm and analysis are considerably simpler than those in previous work. We achieve this by establishing a direct connection between robust learning of Bayesian networks and robust mean estimation. As a subroutine in our algorithm, we develop a robust mean estimation algorithm whose runtime is nearly-linear in the number of nonzeros in the input samples, which may be of independent interest.

We propose a Multiscale Invertible Generative Network (MsIGN) and associated training algorithm that leverages multiscale structure to solve high-dimensional Bayesian inference. To address the curse of dimensionality, MsIGN exploits the low-dimensional nature of the posterior, and generates samples from coarse to fine scale (low to high dimension) by iteratively upsampling and refining samples. MsIGN is trained in a multi-stage manner to minimize the Jeffreys divergence, which avoids mode dropping in high-dimensional cases. On two high-dimensional Bayesian inverse problems, we show superior performance of MsIGN over previous approaches in posterior approximation and multiple mode capture. On the natural image synthesis task, MsIGN achieves superior performance in bits-per-dimension over baseline models and yields great interpret-ability of its neurons in intermediate layers.

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.

In classification, the goal is to learn a model that uses observed feature measurements to make a class prediction on the categorical outcome. However, for safety-critical areas such as medicine and finance, a single class prediction might be insufficient and reliable measures of confidence or certainty may be desired. Such uncertainty quantification is often provided by predictors that produce not just a class label, but a probability distribution over the labels. If the predicted probability distribution is consistent with observed empirical frequencies of labels, the predictor is said to be calibrated [Dawid, 1982]. In this paper we study the problem of calibration for binary classification; let X and Y " t0, 1u denote the feature and label spaces. We focus on the recalibration or post-hoc calibration setting, a standard statistical setting where the goal is to recalibrate existing ('pre-learnt') classifiers that are powerful and (statistically) efficient for classification accuracy, but do not satisfy calibration properties out-of-the-box. This setup is popular for recalibrating pre-trained deep nets. For example, Guo et al. [2017, Figure 4] demonstrated that a pre-learnt ResNet is initially miscalibrated, but can be effectively post-hoc calibrated. In the case of binary classification, the pre-learnt model can be any arbitrary function that provides a classification'score' g: X Ñ r0, 1s.

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].

Uncertainty awareness is crucial to develop reliable machine learning models. In this work, we propose the Natural Posterior Network (NatPN) for fast and high-quality uncertainty estimation for any task where the target distribution belongs to the exponential family. Thus, NatPN finds application for both classification and general regression settings. Unlike many previous approaches, NatPN does not require out-of-distribution (OOD) data at training time. Instead, it leverages Normalizing Flows to fit a single density on a learned low-dimensional and task-dependent latent space. For any input sample, NatPN uses the predicted likelihood to perform a Bayesian update over the target distribution. Theoretically, NatPN assigns high uncertainty far away from training data. Empirically, our extensive experiments on calibration and OOD detection show that NatPN delivers highly competitive performance for classification, regression and count prediction tasks.