A salient approach to interpretable machine learning is to restrict modeling to simple and hence understandable models. In the Bayesian framework, this can be pursued by restricting the model structure and prior to favor interpretable models. Fundamentally, however, interpretability is about users' preferences, not the data generation mechanism: it is more natural to formulate interpretability as a utility function. In this work, we propose an interpretability utility, which explicates the trade-off between explanation fidelity and interpretability in the Bayesian framework. The method consists of two steps. First, a reference model, possibly a black-box Bayesian predictive model compromising no accuracy, is constructed and fitted to the training data. Second, a proxy model from an interpretable model family that best mimics the predictive behaviour of the reference model is found by optimizing the interpretability utility function. The approach is model agnostic - neither the interpretable model nor the reference model are restricted to be from a certain class of models - and the optimization problem can be solved using standard tools in the chosen model family. Through experiments on real-word data sets using decision trees as interpretable models and Bayesian additive regression models as reference models, we show that for the same level of interpretability, our approach generates more accurate models than the earlier alternative of restricting the prior. We also propose a systematic way to measure stabilities of interpretabile models constructed by different interpretability approaches and show that our proposed approach generates more stable models.

This chapter presents probability logic as a rationality framework for human reasoning under uncertainty. Selected formal-normative aspects of probability logic are discussed in the light of experimental evidence. Specifically, probability logic is characterized as a generalization of bivalent truth-functional propositional logic (short "logic"), as being connexive, and as being nonmonotonic. The chapter discusses selected argument forms and associated uncertainty propagation rules. Throughout the chapter, the descriptive validity of probability logic is compared to logic, which was used as the gold standard of reference for assessing the rationality of human reasoning in the 20th century.

For example, we might expect to encounter sensor malfunctions in a wireless sensor network at a rate proportional to the size of the network. Therefore, there is a growing need to develop machine learning techniques that enable satisfactory training and inference from incomplete data. Imputation, where missing data values are filled with suitable values inferred from observations, represents a promising technique for extending machine learning methods to handle missing data. Given their explicit representation of underlying data distributions, Restricted Boltzmann Machines (RBMs) are an appealing choice for imputing missing values. With a well trained RBM, the conditional probabilities of the missing values given the observed values remain accessible via either direct calculation (in a theoretical sense) or indirect Gibbs sampling. A variety of training and imputing procedures have been proposed to allow the application of RBMs to handle missing data, with various computational costs.

We consider the task of learning a causal graph in the presence of latent confounders given i.i.d.~samples from the model. While current algorithms for causal structure discovery in the presence of latent confounders are constraint-based, we here propose a score-based approach. We prove that under assumptions weaker than faithfulness, any sparsest independence map (IMAP) of the distribution belongs to the Markov equivalence class of the true model. This motivates the \emph{Sparsest Poset} formulation - that posets can be mapped to minimal IMAPs of the true model such that the sparsest of these IMAPs is Markov equivalent to the true model. Motivated by this result, we propose a greedy algorithm over the space of posets for causal structure discovery in the presence of latent confounders and compare its performance to the current state-of-the-art algorithms FCI and FCI+ on synthetic data.

The concept of causality has a controversial history. The question of whether it is possible to represent and address causal problems with probability theory, or if fundamentally new mathematics such as the do-calculus is required has been hotly debated, In this paper we demonstrate that, while it is critical to explicitly model our assumptions on the impact of intervening in a system, provided we do so, estimating causal effects can be done entirely within the standard Bayesian paradigm. The invariance assumptions underlying causal graphical models can be encoded in ordinary Probabilistic graphical models, allowing causal estimation with Bayesian statistics, equivalent to the do-calculus.

While often stated as an instance of the likelihood ratio trick [Rubinstein, 1989], the original policy gradient theorem [Sutton, 1999] involves an integral over the action space. When this integral can be computed, the resulting "all-action" estimator [Sutton, 2001] provides a conditioning effect [Bratley, 1987] reducing the variance significantly compared to the REINFORCE estimator [Williams, 1992]. In this paper, we adopt a numerical integration perspective to broaden the applicability of the all-action estimator to general spaces and to any function class for the policy or critic components, beyond the Gaussian case considered by [Ciosek, 2018]. In addition, we provide a new theoretical result on the effect of using a biased critic which offers more guidance than the previous "compatible features" condition of [Sutton, 1999]. We demonstrate the benefit of our approach in continuous control tasks with nonlinear function approximation. Our results show improved performance and sample efficiency.

Existing approaches to amortized inference in probabilistic programs with unbounded loops can produce estimators with infinite variance. An instance of this is importance sampling inference in programs that explicitly include rejection sampling as part of the user-programmed generative procedure. In this paper we develop a new and efficient amortized importance sampling estimator. We prove finite variance of our estimator and empirically demonstrate our method's correctness and efficiency compared to existing alternatives on generative programs containing rejection sampling loops and discuss how to implement our method in a generic probabilistic programming framework.

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Algorithms are often grouped by similarity in terms of their function (how they work). For example, tree-based methods, and neural network inspired methods. I think this is the most useful way to group algorithms and it is the approach we will use here. This is a useful grouping method, but it is not perfect. There are still algorithms that could just as easily fit into multiple categories like Learning Vector Quantization that is both a neural network inspired method and an instance-based method.

V ARIBAD: A V ERY G OOD M ETHOD FOR B AYES-A DAPTIVE D EEP RL VIA M ETA-L EARNING Luisa Zintgraf University of Oxford Kyriacos Shiarlis Latent Logic Maximilian Igl University of Oxford Sebastian Schulze University of Oxford Y arin Gal OA TML Group, University of Oxford Katja Hofmann Microsoft Research Shimon Whiteson University of Oxford Latent Logic A BSTRACT Trading off exploration and exploitation in an unknown environment is key to maximising expected return during learning. A Bayes-optimal policy, which does so optimally, conditions its actions not only on the environment state but on the agent's uncertainty about the environment. Computing a Bayes-optimal policy is however intractable for all but the smallest tasks. In this paper, we introduce variational Bayes-Adaptive Deep RL (variBAD), a way to meta-learn to perform approximate inference in an unknown environment, and incorporate task uncertainty directly during action selection. In a grid-world domain, we illustrate how variBAD performs structured online exploration as a function of task uncertainty. We also evaluate variBAD on MuJoCo domains widely used in meta-RL and show that it achieves higher return during training than existing methods. 1 I NTRODUCTION Reinforcement learning (RL) is typically concerned with finding an optimal policy that maximises expected return for a given Markov decision process (MDP) with an unknown reward and transition function. If these were known, the optimal policy could in theory be computed without interacting with the environment. By contrast, learning in an unknown environment typically requires trading off exploration (learning about the environment) and exploitation (taking promising actions). Balancing this tradeoff is key to maximising expected return during learning . A Bayes-optimal policy, which does so optimally, conditions actions not only on the environment state but on the agent's own uncertainty about the current MDP . In principle, a Bayes-optimal policy can be computed using the framework of Bayes-adaptive Markov decision processes (BAMDPs) (Martin, 1967; Duff & Barto, 2002). The agent maintains a belief, i.e., a posterior distribution, over possible environments. Augmenting the state space of the underlying MDP with this posterior distribution yields a BAMDP, a special case of a belief MDP (Kaelbling et al., 1998).