This chapter offers an accessible introduction to the channel-based approach to Bayesian probability theory. This framework rests on algebraic and logical foundations, inspired by the methodologies of programming language semantics. It offers a uniform, structured and expressive language for describing Bayesian phenomena in terms of familiar programming concepts, like channel, predicate transformation and state transformation. The introduction also covers inference in Bayesian networks, which will be modelled by a suitable calculus of string diagrams.

In scenarios such as natural disasters, seasonal agriculture, and other short-duration operations, a rapidly deployable, autonomous mobile sensing system that decides where to take sensor measurements can be more versatile and costeffective than installing stationary sensors. In this work, we are interested in formulating a solution for persistent sensing of real-world stochastic phenomena using a team of mobile robots, even when the underlying covariance structure changes sharply across time, such as sunlight variation in a forest understory (Figure 1). Assuming no prior knowledge on the underlying model of the phenomenon dynamics, this presents two challenges: 1) adapting a belief on the underlying model based on recently observed phenomenon dynamics and 2) correspondingly optimizing the next sensing locations. While exactly modeling stochastic real-world phenomena remains a significant challenge, this work deals mainly with modeling the underlying covariance structure. The underlying covariance structure directly corresponds to information metrics such as entropy, required for evaluating the informativeness or representativeness of sensor readings across a set of locations [9, 13, 30]. Gaussian processes (GP) have emerged as a favored choice for this specific modeling goal primarily because of their nonparametric nature [14, 20, 33, 42].

Every year, thousands of people receive consumer product related injuries. Research indicates that online customer reviews can be processed to autonomously identify product safety issues. Early identification of safety issues can lead to earlier recalls, and thus fewer injuries and deaths. A dataset of product reviews from Amazon.com was compiled, along with \emph{SaferProducts.gov} complaints and recall descriptions from the Consumer Product Safety Commission (CPSC) and European Commission Rapid Alert system. A system was built to clean the collected text and to extract relevant features. Dimensionality reduction was performed by computing feature relevance through a Random Forest and discarding features with low information gain. Various classifiers were analyzed, including Logistic Regression, SVMs, Na{\"i}ve-Bayes, Random Forests, and an Ensemble classifier. Experimentation with various features and classifier combinations resulted in a logistic regression model with 70.2\% precision in the top 50 reviews surfaced. This classifier outperforms all benchmarks set by related literature and consumer product safety professionals.

Deep neural network (DNN) has achieved remarkable success in classification tasks such as image classification [1]. The network output can mimic the posterior probabilities of target classes for the input observation when the nonlinear activation function in the output layer is defined as a soft-max function [2]. The learning objective is to minimize the difference between the predicted distribution and the true datagenerating distribution. In information theory, the cross entropy between two probability distributions over a common event set of events measures the average number of bits needed to identify an event if coding follows a learned probability distribution rather than the true but unknow distribution [3]. Therefore, cross entropy is a reasonable loss function for the DNN-based classification. However, in practice the true data-generating probability distribution is unknown and replaced by the empirical probability distribution over a training set where each sample is drawn independently and identically distributed (i.i.d.) from the data space [4]. Under assumptions of uniform distributions of feature and label spaces, minimizing cross-entropy is equivalent to maximum likelihood, i.e., the learning problem aims to maximize likelihood of correct class for each of training samples [2]. Maximum likelihood is a generative training criterion by which the model learns the likelihood of correct class for the observation. The model makes predictions by using Bayes rules to calculate posterior probabilities of target classes for the observation and then select the most likely class.

Wasserstein distances have a clear intuitive meaning: What is the minimum cost if we want to obtain ν by transporting the probability mass in µ? Here the cost is defined as the product of probability mass moved and the distance moved raised to the pth power. Therefore, the Wasserstein distance is also called "optimal transport distance" or "earth mover's distance". The problem of optimal transport can be traced back to [18] and [13]. 1 Since the introduction in [25], the Wasserstein distances have become an important tool in computer vision and statistical machine learning. In addition to the connection with optimal transport, Wasserstein distances have some desirable features. For example, they can be meaningfully defined for any two distributions without any requirement on the existence of density or absolute continuity. Other measurements, such as the Kullback-Leibler divergence, have more stringent requirements on µ and ν. See [21] for a more thorough historical review of Wasserstein distances and their applications in statistics, and [27] for further details about Wasserstein distances and optimal transport in a broader context. In statistics and machine learning, we often do not have access to µ but only its empirical version ˆµ, which puts 1/n mass at each one of n independent samples from µ.

Many web systems rank and present a list of items to users, from recommender systems to search and advertising. An important problem in practice is to evaluate new ranking policies offline and optimize them before they are deployed. We address this problem by proposing new evaluation algorithms for estimating the expected number of clicks on ranked lists from stored logs of past results. The existing algorithms are not guaranteed to be statistically efficient in our problem because the number of recommended lists can grow exponentially with their length. To overcome this challenge, we use models of user interaction with the list of items, the so-called click models, to construct estimators that learn statistically efficiently. We analyze our estimators and prove that they are more efficient than the estimators that do not use the structure of the click model, under the assumption that the click model holds. We evaluate our estimators in a series of experiments on a real-world dataset and show that they consistently outperform prior estimators.

We tackle the problem of penalty selection of regularization on the basis of the minimum description length (MDL) principle. In particular, we consider that the design space of the penalty function is high-dimensional. In this situation, the luckiness-normalized-maximum-likelihood(LNML)-minimization approach is favorable, because LNML quantifies the goodness of regularized models with any forms of penalty functions in view of the minimum description length principle, and guides us to a good penalty function through the high-dimensional space. However, the minimization of LNML entails two major challenges: 1) the computation of the normalizing factor of LNML and 2) its minimization in high-dimensional spaces. In this paper, we present a novel regularization selection method (MDL-RS), in which a tight upper bound of LNML (uLNML) is minimized with local convergence guarantee. Our main contribution is the derivation of uLNML, which is a uniform-gap upper bound of LNML in an analytic expression. This solves the above challenges in an approximate manner because it allows us to accurately approximate LNML and then efficiently minimize it. The experimental results show that MDL-RS improves the generalization performance of regularized estimates specifically when the model has redundant parameters.

Crowdsourcing has become very popular among the machine learning community as a way to obtain labels that allow a ground truth to be estimated for a given dataset. In most of the approaches that use crowdsourced labels, annotators are asked to provide, for each presented instance, a single class label. Such a request could be inefficient, that is, considering that the labelers may not be experts, that way to proceed could fail to take real advantage of the knowledge of the labelers. In this paper, the use of weak labeling for crowd learning is proposed, where the annotators may provide more than a single label per instance to try not to miss the real label. The main hypothesis is that, by allowing weak labeling, knowledge can be extracted from the labelers more efficiently by than in the standard crowd learning scenario. Empirical evidence which supports that hypothesis is presented.

Approximate linear programming (ALP) represents one of the major algorithmic families to solve large-scale Markov decision processes (MDP). In this work, we study a primal-dual formulation of the ALP, and develop a scalable, model-free algorithm called bilinear $\pi$ learning for reinforcement learning when a sampling oracle is provided. This algorithm enjoys a number of advantages. First, it adopts (bi)linear models to represent the high-dimensional value function and state-action distributions, using given state and action features. Its run-time complexity depends on the number of features, not the size of the underlying MDPs. Second, it operates in a fully online fashion without having to store any sample, thus having minimal memory footprint. Third, we prove that it is sample-efficient, solving for the optimal policy to high precision with a sample complexity linear in the dimension of the parameter space.

We examine a network of learners which address the same classification task but must learn from different data sets. The learners can share a limited portion of their data sets so as to preserve the network load. We introduce DELCO (standing for Decentralized Ensemble Learning with COpulas), a new approach in which the shared data and the trained models are sent to a central machine that allows to build an ensemble of classifiers. The proposed method aggregates the base classifiers using a probabilistic model relying on Gaussian copulas. Experiments on logistic regressor ensembles demonstrate competing accuracy and increased robustness as compared to gold standard approaches. A companion python implementation can be downloaded at https://github.com/john-klein/DELCO