Bayesian network structure learning is the notoriously difficult problem of discovering a Bayesian network (BN) that optimally represents a given set of training data [4]. Since the exact inference on a BN is exponential in the BN's treewidth [14], one is particularly interested in learning BNs of bounded treewidth. However, learning a BN of bounded treewidth that optimally fits the data (i.e., with the largest possible score) is, in turn, an NPhard task [13]. This predicament caused the research on treewidth-bounded BN structure learning to split into two branches: 1. Heuristic Learning (see, e.g., [5, 17, 23, 24]), which is scalable to large BNs with thousands of nodes (but with a score that can be far from optimal), and 2. Exact Learning (see, e.g., [2, 13, 19]), which learns optimal BNs (but is scalable only to a few dozen nodes). In this paper, we combine heuristic and exact learning and take the best of both worlds.

Consider learning a policy purely on the basis of demonstrated behavior---that is, with no access to reinforcement signals, no knowledge of transition dynamics, and no further interaction with the environment. This *strictly batch imitation learning* problem arises wherever live experimentation is costly, such as in healthcare. One solution is simply to retrofit existing algorithms for apprenticeship learning to work in the offline setting. But such an approach bargains heavily on model estimation or off-policy evaluation, and can be indirect and inefficient. We argue that a good solution should be able to explicitly parameterize a policy (i.e. respecting action conditionals), implicitly account for rollout dynamics (i.e. respecting state marginals), and---crucially---operate in an entirely offline fashion. To meet this challenge, we propose a novel technique by *energy-based distribution matching* (EDM): By identifying parameterizations of the (discriminative) model of a policy with the (generative) energy function for state distributions, EDM provides a simple and effective solution that equivalently minimizes a divergence between the occupancy measures of the demonstrator and the imitator. Through experiments with application to control tasks and healthcare settings, we illustrate consistent performance gains over existing algorithms for strictly batch imitation learning.

Principal Component Analysis (PCA) is a powerful tool in statistics and machine learning. While existing study of PCA focuses on the recovery of principal components and their associated eigenvalues, there are few precise characterizations of individual principal component scores that yield low-dimensional embedding of samples. That hinders the analysis of various spectral methods. In this paper, we first develop an $\ell_p$ perturbation theory for a hollowed version of PCA in Hilbert spaces which provably improves upon the vanilla PCA in the presence of heteroscedastic noises. Through a novel $\ell_p$ analysis of eigenvectors, we investigate entrywise behaviors of principal component score vectors and show that they can be approximated by linear functionals of the Gram matrix in $\ell_p$ norm, which includes $\ell_2$ and $\ell_\infty$ as special examples. For sub-Gaussian mixture models, the choice of $p$ giving optimal bounds depends on the signal-to-noise ratio, which further yields optimality guarantees for spectral clustering. For contextual community detection, the $\ell_p$ theory leads to a simple spectral algorithm that achieves the information threshold for exact recovery. These also provide optimal recovery results for Gaussian mixture and stochastic block models as special cases.

In this article we introduce theory and algorithms for learning discrete representations that take on a lattice that is embedded in an Euclidean space. Lattice representations possess an interesting combination of properties: a) they can be computed explicitly using lattice quantization, yet they can be learned efficiently using the ideas we introduce in this paper, b) they are highly related to Gaussian Variational Autoencoders, allowing designers familiar with the latter to easily produce discrete representations from their models and c) since lattices satisfy the axioms of a group, their adoption can lead into a way of learning simple algebras for modeling binary operations between objects through symbolic formalisms, yet learn these structures also formally using differentiation techniques. This article will focus on laying the groundwork for exploring and exploiting the first two properties, including a new mathematical result linking expressions used during training and inference time and experimental validation on two popular datasets.

We derive a family of loss functions to train models in the presence of sampling bias. Examples are when the prevalence of a pathology differs from its sampling rate in the training dataset, or when a machine learning practioner rebalances their training dataset. Sampling bias causes large discrepancies between model performance in the lab and in more realistic settings. It is omnipresent in medical imaging applications, yet is often overlooked at training time or addressed on an ad-hoc basis. Our approach is based on Bayesian risk minimization. For arbitrary likelihood models we derive the associated bias corrected loss for training, exhibiting a direct connection to information gain. The approach integrates seamlessly in the current paradigm of (deep) learning using stochastic backpropagation and naturally with Bayesian models. We illustrate the methodology on case studies of lung nodule malignancy grading.

Compartmental models are used for predicting the scale and duration of epidemics, estimating epidemiological parameters such as reproduction numbers, and guiding outbreak control measures [Brauer, 2008, O'Neill, 2010, Kucharski et al., 2020]. They are increasingly important because they allow joint modelling of disease dynamics and multimodal data, such as medical test results, cell phone and transport flow data [Rubrichi et al., 2018, Wu et al., 2020], census and demographic information [Prem et al., 2020]. However, statistical inference in stochastic variants of compartmental models is a major computational challenge [Bretó, 2018]. The likelihood function for model parameters is usually intractable because it involves summation over a prohibitively large number of configurations of latent variables representing counts of subpopulations in disease states which cannot be observed directly. This has lead to the recent development of sophisticated computational methods for approximate inference involving various forms of stochastic simulation [Funk and King, 2020].

We are often interested in identifying the feasible subset of a decision space under multiple constraints to permit effective design exploration. If determining feasibility required computationally expensive simulations, the cost of exploration would be prohibitive. Bayesian search is data-efficient for such problems: starting from a small dataset, the central concept is to use Bayesian models of constraints with an acquisition function to locate promising solutions that may improve predictions of feasibility when the dataset is augmented. At the end of this sequential active learning approach with a limited number of expensive evaluations, the models can accurately predict the feasibility of any solution obviating the need for full simulations. In this paper, we propose a novel acquisition function that combines the probability that a solution lies at the boundary between feasible and infeasible spaces (representing exploitation) and the entropy in predictions (representing exploration). Experiments confirmed the efficacy of the proposed function.

Behavior prediction plays an essential role in both autonomous driving systems and Advanced Driver Assistance Systems (ADAS), since it enhances vehicle's awareness of the imminent hazards in the surrounding environment. Many existing lane change prediction models take as input lateral or angle information and make short-term (< 5 seconds) maneuver predictions. In this study, we propose a longer-term (5~10 seconds) prediction model without any lateral or angle information. Three prediction models are introduced, including a logistic regression model, a multilayer perceptron (MLP) model, and a recurrent neural network (RNN) model, and their performances are compared by using the real-world NGSIM dataset. To properly label the trajectory data, this study proposes a new time-window labeling scheme by adding a time gap between positive and negative samples. Two approaches are also proposed to address the unstable prediction issue, where the aggressive approach propagates each positive prediction for certain seconds, while the conservative approach adopts a roll-window average to smooth the prediction. Evaluation results show that the developed prediction model is able to capture 75% of real lane change maneuvers with an average advanced prediction time of 8.05 seconds.

For testing conditional independence (CI) of a response $Y$ and a predictor $X$ given covariates $Z$, the recently introduced model-X (MX) framework has been the subject of active methodological research, especially in the context of MX knockoffs and their successful application to genome-wide association studies. In this paper, we build a theoretical foundation for the MX CI problem, yielding quantitative explanations for empirically observed phenomena and novel insights to guide the design of MX methodology. We focus our analysis on the conditional randomization test (CRT), whose validity conditional on $Y,Z$ allows us to view it as a test of a point null hypothesis involving the conditional distribution of $X$. We use the Neyman-Pearson lemma to derive the most powerful CRT statistic against a point alternative as well as an analogous result for MX knockoffs. We define CRT-style analogs of $t$- and $F$-tests with explicit critical values, and show that they have uniform asymptotic Type-I error control under the assumption that only the first two moments of $X$ given $Z$ are known, a significant relaxation of MX. We derive expressions for the power of these tests against local semiparametric alternatives using Le Cam's local asymptotic normality theory, explicitly capturing the prediction error of the underlying learning algorithm. Finally, we pave the way for estimation in the MX setting by drawing connections to semiparametric statistics and causal inference. Thus, this work forms explicit bridges from MX to both classical statistics (testing) and modern causal inference (estimation).

Graphs are ubiquitous in modelling relational structures. Recent endeavours in machine learning for graph-structured data have led to many architectures and learning algorithms. However, the graph used by these algorithms is often constructed based on inaccurate modelling assumptions and/or noisy data. As a result, it fails to represent the true relationships between nodes. A Bayesian framework which targets posterior inference of the graph by considering it as a random quantity can be beneficial. In this paper, we propose a novel non-parametric graph model for constructing the posterior distribution of graph adjacency matrices. The proposed model is flexible in the sense that it can effectively take into account the output of graph-based learning algorithms that target specific tasks. In addition, model inference scales well to large graphs. We demonstrate the advantages of this model in three different problem settings: node classification, link prediction and recommendation.