This document serves to complement our website which was developed with the aim of exposing the students to Gaussian Processes (GPs). GPs are non-parametric Bayesian regression models that are largely used by statisticians and geospatial data scientists for modeling spatial data. Several open source libraries spanning from Matlab [1], Python [2], R [3] etc., are already available for simple plug-and-use. The objective of this handout and in turn the website was to allow the users to develop stand-alone GPs in Python by relying on minimal external dependencies. To this end, we only use the default python modules and assist the users in developing their own GPs from scratch giving them an in-depth knowledge of what goes on under the hood. The module covers GP inference using maximum likelihood estimation (MLE) and gives examples of 1D (dummy) spatial data.

We consider the problem of inferring the directed, causal graph from observational data, assuming no hidden confounders. We take an information theoretic approach, and make three main contributions. First, we show how through algorithmic information theory we can obtain SCI, a highly robust, effective and computationally efficient test for conditional independence---and show it outperforms the state of the art when applied in constraint-based inference methods such as stable PC. Second, building upon on SCI, we show how to tell apart the parents and children of a given node based on the algorithmic Markov condition. We give the Climb algorithm to efficiently discover the directed, causal Markov blanket---and show it is at least as accurate as inferring the global network, while being much more efficient. Last, but not least, we detail how we can use the Climb score to direct those edges that state of the art causal discovery algorithms based on PC or GES leave undirected---and show this improves their precision, recall and F1 scores by up to 20%.

This paper proposes a systematic framework to design a classification model that yields a classifier which optimizes a utility function based on prior knowledge. Specifically, as the data size grows, we prove that the produced classifier asymptotically converges to the optimal classifier, an extended version of the Bayes rule, which maximizes the utility function. Therefore, we provide a meaningful theoretical interpretation for modeling with the knowledge incorporated. Our knowledge incorporation method allows domain experts to guide the classifier towards correctly classifying data that they think to be more significant.

We study the effect of the quality and quantity of side information on the recovery of a hidden community of size $K=o(n)$ in a graph of size $n$. Side information for each node in the graph is modeled by a random vector with the following features: either the dimension of the vector is allowed to vary with $n$, while log-likelihood ratio (LLR) of each component with respect to the node label is fixed, or the LLR is allowed to vary and the vector dimension is fixed. These two models represent the variation in quality and quantity of side information. Under maximum likelihood detection, we calculate tight necessary and sufficient conditions for exact recovery of the labels. We demonstrate how side information needs to evolve with $n$ in terms of either its quantity, or quality, to improve the exact recovery threshold. A similar set of results are obtained for weak recovery. Under belief propagation, tight necessary and sufficient conditions for weak recovery are calculated when the LLRs are constant, and sufficient conditions when the LLRs vary with $n$. Moreover, we design and analyze a local voting procedure using side information that can achieve exact recovery when applied after belief propagation. The results for belief propagation are validated via simulations on finite synthetic data-sets, showing that the asymptotic results of this paper can also shed light on the performance at finite $n$.

Efficient distributed numerical word representation models (word embeddings) combined with modern machine learning algorithms have recently yielded considerable improvement on automatic document classification tasks. However, the effectiveness of such techniques has not been assessed for the hierarchical text classification (HTC) yet. This study investigates the application of those models and algorithms on this specific problem by means of experimentation and analysis. We trained classification models with prominent machine learning algorithm implementations---fastText, XGBoost, SVM, and Keras' CNN---and noticeable word embeddings generation methods---GloVe, word2vec, and fastText---with publicly available data and evaluated them with measures specifically appropriate for the hierarchical context. FastText achieved an ${}_{LCA}F_1$ of 0.893 on a single-labeled version of the RCV1 dataset. An analysis indicates that using word embeddings and its flavors is a very promising approach for HTC.

We analyze the value-loading problem. This is the problem of encoding moral values into an AI agent interacting with a complex environment. Like many before, we argue that this is both a major concern and an extremely challenging problem. Solving it will likely require years, if not decades, of multidisciplinary work by teams of top scientists and experts. Given how uncertain the timeline of human-level AI research is, we thus argue that a pragmatic partial solution should be designed as soon as possible. To this end, we propose a preliminary research program. This roadmap identifies several key steps. We hope that this will allow scholars, engineers and decision-makers to better grasp the upcoming difficulties, and to foresee how they can best contribute to the global effort.

Bayesian Inference (BI) uses the Bayes' posterior whereas Logical Bayesian Inference (LBI) uses the truth function or membership function as the inference tool. LBI was proposed because BI was not compatible with the classical Bayes' prediction and didn't use logical probability and hence couldn't express semantic meaning. In LBI, statistical probability and logical probability are strictly distinguished, used at the same time, and linked by the third kind of Bayes' Theorem. The Shannon channel consists of a set of transition probability functions whereas the semantic channel consists of a set of truth functions. When a sample is large enough, we can directly derive the semantic channel from Shannon's channel. Otherwise, we can use parameters to construct truth functions and use the Maximum Semantic Information (MSI) criterion to optimize the truth functions. The MSI criterion is equivalent to the Maximum Likelihood (ML) criterion, and compatible with the Regularized Least Square (RLS) criterion. By matching the two channels one with another, we can obtain the Channels' Matching (CM) algorithm. This algorithm can improve multi-label classifications, maximum likelihood estimations (including unseen instance classifications), and mixture models. In comparison with BI, LBI 1) uses the prior P(X) of X instead of that of Y or {\theta} and fits cases where the source P(X) changes, 2) can be used to solve the denotations of labels, and 3) is more compatible with the classical Bayes' prediction and likelihood method. LBI also provides a confirmation measure between -1 and 1 for induction.

Consider the case that one observes a single time-series, where at each time t one observes a data record O(t) involving treatment nodes A(t), possible covariates L(t) and an outcome node Y(t). The data record at time t carries information for an (potentially causal) effect of the treatment A(t) on the outcome Y(t), in the context defined by a fixed dimensional summary measure Co(t). We are concerned with defining causal effects that can be consistently estimated, with valid inference, for sequentially randomized experiments without further assumptions. More generally, we consider the case when the (possibly causal) effects can be estimated in a double robust manner, analogue to double robust estimation of effects in the i.i.d. causal inference literature. We propose a general class of averages of conditional (context-specific) causal parameters that can be estimated in a double robust manner, therefore fully utilizing the sequential randomization. We propose a targeted maximum likelihood estimator (TMLE) of these causal parameters, and present a general theorem establishing the asymptotic consistency and normality of the TMLE. We extend our general framework to a number of typically studied causal target parameters, including a sequentially adaptive design within a single unit that learns the optimal treatment rule for the unit over time. Our work opens up robust statistical inference for causal questions based on observing a single time-series on a particular unit.

Sequence-to-Sequence models were introduced to tackle many real-life problems like machine translation, summarization, image captioning, etc. The standard optimization algorithms are mainly based on example-to-example matching like maximum likelihood estimation, which is known to suffer from data sparsity problem. Here we present an alternate view to explain sequence-to-sequence learning as a distribution matching problem, where each source or target example is viewed to represent a local latent distribution in the source or target domain. Then, we interpret sequence-to-sequence learning as learning a transductive model to transform the source local latent distributions to match their corresponding target distributions. In our framework, we approximate both the source and target latent distributions with recurrent neural networks (augmenter). During training, the parallel augmenters learn to better approximate the local latent distributions, while the sequence prediction model learns to minimize the KL-divergence of the transformed source distributions and the approximated target distributions. This algorithm can alleviate the data sparsity issues in sequence learning by locally augmenting more unseen data pairs and increasing the model's robustness. Experiments conducted on machine translation and image captioning consistently demonstrate the superiority of our proposed algorithm over the other competing algorithms.

We present a novel framework that enables efficient probabilistic inference in large-scale scientific models by allowing the execution of existing domain-specific simulators as probabilistic programs, resulting in highly interpretable posterior inference. Our framework is general purpose and scalable, and is based on a cross-platform probabilistic execution protocol through which an inference engine can control simulators in a language-agnostic way. We demonstrate the technique in particle physics, on a scientifically accurate simulation of the tau lepton decay, which is a key ingredient in establishing the properties of the Higgs boson. High-energy physics has a rich set of simulators based on quantum field theory and the interaction of particles in matter. We show how to use probabilistic programming to perform Bayesian inference in these existing simulator codebases directly, in particular conditioning on observable outputs from a simulated particle detector to directly produce an interpretable posterior distribution over decay pathways. Inference efficiency is achieved via inference compilation where a deep recurrent neural network is trained to parameterize proposal distributions and control the stochastic simulator in a sequential importance sampling scheme, at a fraction of the computational cost of Markov chain Monte Carlo sampling.