Presbyterian reverend Thomas Bayes had no reason to suspect he'd make any lasting contribution to humankind. Born in England at the beginning of the 18th century, Bayes was a quiet and questioning man. He published only two works in his lifetime. In 1731, he wrote a defense of God's--and the British monarchy's--"divine benevolence," and in 1736, an anonymous defense of the logic of Isaac Newton's calculus. Yet an argument he wrote before his death in 1761 would shape the course of history.

One important approach to learning Bayesian networks (BNs) from data uses a scoring metric to evaluate the fitness of any given candidate network for the data base, and applies a search procedure to explore the set of candidate networks. The most usual search methods are greedy hill climbing, either deterministic or stochastic, although other techniques have also been used. In this paper we propose a new algorithm for learning BNs based on a recently introduced metaheuristic, which has been successfully applied to solve a variety of combinatorial optimization problems: ant colony optimization (ACO). We describe all the elements necessary to tackle our learning problem using this metaheuristic, and experimentally compare the performance of our ACO-based algorithm with other algorithms used in the literature. The experimental work is carried out using three different domains: ALARM, INSURANCE and BOBLO.

Suppose that we wish to infer the value of a statistical parameter at a law from which we sample independent observations. Suppose that this parameter is smooth and that we can define two variation-independent, infinite-dimensional features of the law, its so called Q- and G-components (comp.), such that if we estimate them consistently at a fast enough product of rates, then we can build a confidence interval (CI) with a given asymptotic level based on a plain targeted minimum loss estimator (TMLE). The estimators of the Q- and G-comp. would typically be by products of machine learning algorithms. We focus on the case that the machine learning algorithm for the G-comp. is fine-tuned by a real-valued parameter h. Then, a plain TMLE with an h chosen by cross-validation would typically not lend itself to the construction of a CI, because the selection of h would trade-off its empirical bias with something akin to the empirical variance of the estimator of the G-comp. as opposed to that of the TMLE. A collaborative TMLE (C-TMLE) might, however, succeed in achieving the relevant trade-off. We construct a C-TMLE and show that, under high-level empirical processes conditions, and if there exists an oracle h that makes a bulky remainder term asymptotically Gaussian, then the C-TMLE is asymptotically Gaussian hence amenable to building a CI provided that its asymptotic variance can be estimated too. We illustrate the construction and main result with the inference of the average treatment effect, where the Q-comp. consists in a marginal law and a conditional expectation, and the G-comp. is a propensity score (a conditional probability). We also conduct a multi-faceted simulation study to investigate the empirical properties of the collaborative TMLE when the G-comp. is estimated by the LASSO. Here, h is the bound on the l1-norm of the candidate coefficients.

In these notes we describe heuristics to predict computational-to-statistical gaps in certain statistical problems. These are regimes in which the underlying statistical problem is information-theoretically possible although no efficient algorithm exists, rendering the problem essentially unsolvable for large instances. The methods we describe here are based on mature, albeit non-rigorous, tools from statistical physics. These notes are based on a lecture series given by the authors at the Courant Institute of Mathematical Sciences in New York City, on May 16th, 2017.

Variational Bayes (VB), also known as independent mean-field approximation, has become a popular method for Bayesian network inference in recent years. Its application is vast, e.g. in neural network, compressed sensing, clustering, etc. to name just a few. In this paper, the independence constraint in VB will be relaxed to a conditional constraint class, called copula in statistics. Since a joint probability distribution always belongs to a copula class, the novel copula VB (CVB) approximation is a generalized form of VB. Via information geometry, we will see that CVB algorithm iteratively projects the original joint distribution to a copula constraint space until it reaches a local minimum Kullback-Leibler (KL) divergence. By this way, all mean-field approximations, e.g. iterative VB, Expectation-Maximization (EM), Iterated Conditional Mode (ICM) and k-means algorithms, are special cases of CVB approximation. For a generic Bayesian network, an augmented hierarchy form of CVB will also be designed. While mean-field algorithms can only return a locally optimal approximation for a correlated network, the augmented CVB network, which is an optimally weighted average of a mixture of simpler network structures, can potentially achieve the globally optimal approximation for the first time. Via simulations of Gaussian mixture clustering, the classification's accuracy of CVB will be shown to be far superior to that of state-of-the-art VB, EM and k-means algorithms.

Well-established methods for the solution of stochastic partial differential equations (SPDEs) typically struggle in problems with high-dimensional inputs/outputs. Such difficulties are only amplified in large-scale applications where even a few tens of full-order model runs are impracticable. While dimensionality reduction can alleviate some of these issues, it is not known which and how many features of the (high-dimensional) input are actually predictive of the (high-dimensional) output. In this paper, we advocate a Bayesian formulation that is capable of performing simultaneous dimension and model-order reduction. It consists of a component that encodes the high-dimensional input into a low-dimensional set of feature functions by employing sparsity-enforcing priors and a decoding component that makes use of the solution of a coarse-grained model in order to reconstruct that of the full-order model. Both components are represented with latent variables in a probabilistic graphical model and are simultaneously trained using Stochastic Variational Inference methods. The model is capable of quantifying the predictive uncertainty due to the information loss that unavoidably takes place in any model-order/dimension reduction as well as the uncertainty arising from finite-sized training datasets. We demonstrate its capabilities in the context of random media where fine-scale fluctuations can give rise to random inputs with tens of thousands of variables. With a few tens of full-order model simulations, the proposed model is capable of identifying salient physical features and produce sharp predictions under different boundary conditions of the full output which itself consists of thousands of components.

The recent successes of AI have captured the wildest imagination of both the scientific communities and the general public. Robotics and AI amplify human potentials, increase productivity and are moving from simple reasoning towards human-like cognitive abilities. Current AI technologies are used in a set area of applications, ranging from healthcare, manufacturing, transport, energy, to financial services, banking, advertising, management consulting and government agencies. The global AI market is around 260 billion USD in 2016 and it is estimated to exceed 3 trillion by 2024. To understand the impact of AI, it is important to draw lessons from it's past successes and failures and this white paper provides a comprehensive explanation of the evolution of AI, its current status and future directions.

We propose a feed-forward inference method applicable to belief and neural networks. In a belief network, the method estimates an approximate factorized posterior of all hidden units given the input. In neural networks the method propagates uncertainty of the input through all the layers. In neural networks with injected noise, the method analytically takes into account uncertainties resulting from this noise. Such feed-forward analytic propagation is differentiable in parameters and can be trained end-to-end. Compared to standard NN, which can be viewed as propagating only the means, we propagate the mean and variance. The method can be useful in all scenarios that require knowledge of the neuron statistics, e.g. when dealing with uncertain inputs, considering sigmoid activations as probabilities of Bernoulli units, training the models regularized by injected noise (dropout) or estimating activation statistics over the dataset (as needed for normalization methods). In the experiments we show the possible utility of the method in all these tasks as well as its current limitations.

We introduce a Bayesian framework for inference with a supervised version of the Gaussian process latent variable model. The framework overcomes the high correlations between latent variables and hyperparameters by using an unbiased pseudo estimate for the marginal likelihood that approximately integrates over the latent variables. This is used to construct a Markov Chain to explore the posterior of the hyperparameters. We demonstrate the procedure on simulated and real examples, showing its ability to capture uncertainty and multimodality of the hyperparameters and improved uncertainty quantification in predictions when compared with variational inference.

Recently, the intervention calculus when the DAG is absent (IDA) method was developed to estimate lower bounds of causal effects from observational high-dimensional data. Originally it was introduced to assess the effect of baseline biomarkers which do not vary over time. However, in many clinical settings, measurements of biomarkers are repeated at fixed time points during treatment exposure and, therefore, this method need to be extended. The purpose of this paper is then to extend the first step of the IDA, the Peter Clarks (PC)-algorithm, to a time-dependent exposure in the context of a binary outcome. We generalised the PC-algorithm for taking into account the chronological order of repeated measurements of the exposure and propose to apply the IDA with our new version, the chronologically ordered PC-algorithm (COPC-algorithm). A simulation study has been performed before applying the method for estimating causal effects of time-dependent immunological biomarkers on toxicity, death and progression in patients with metastatic melanoma. The simulation study showed that the completed partially directed acyclic graphs (CPDAGs) obtained using COPC-algorithm were structurally closer to the true CPDAG than CPDAGs obtained using PC-algorithm. Also, causal effects were more accurate when they were estimated based on CPDAGs obtained using COPC-algorithm. Moreover, CPDAGs obtained by COPC-algorithm allowed removing non-chronologic arrows with a variable measured at a time t pointing to a variable measured at a time t' where t'< t. Bidirected edges were less present in CPDAGs obtained with the COPC-algorithm, supporting the fact that there was less variability in causal effects estimated from these CPDAGs. The COPC-algorithm provided CPDAGs that keep the chronological structure present in the data, thus allowed to estimate lower bounds of the causal effect of time-dependent biomarkers.