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### How Bayesian Machine Learning Works

Classical statistics is said to follow the frequentist approach because it interprets probability as the relative frequency of an event over the long run that is, after observing many trials. In the context of probabilities, an event is a combination of one or more elementary outcomes of an experiment, such as any of six equal results in rolls of two dice or an asset price dropping by 10 percent or more on a given day.

### Bayesian Networks. Or: How I Learned to Stop Worrying and Love Probability

The tragedy happened to the AirFrance 447 more than 10 years ago, in 2009. The flight took off in Rio de Janeiro and was planned to land in Paris. It suddenly disappeared in the middle of the Atlantic ocean without any warning. Immediately, rescuers reached the zone and what they found were just some wreckage and corpse. All 228 people onboard died in the crash.

### Statistics with R

Offered by Duke University. In this Specialization, you will learn to analyze and visualize data in R and create reproducible data analysis reports, demonstrate a conceptual understanding of the unified nature of statistical inference, perform frequentist and Bayesian statistical inference and modeling to understand natural phenomena and make data-based decisions, communicate statistical results correctly, effectively, and in context without relying on statistical jargon, critique data-based claims and evaluated data-based decisions, and wrangle and visualize data with R packages for data analysis. You will produce a portfolio of data analysis projects from the Specialization that demonstrates mastery of statistical data analysis from exploratory analysis to inference to modeling, suitable for applying for statistical analysis or data scientist positions.

### Representation Learning from Limited Educational Data with Crowdsourced Labels

Representation learning has been proven to play an important role in the unprecedented success of machine learning models in numerous tasks, such as machine translation, face recognition and recommendation. The majority of existing representation learning approaches often require a large number of consistent and noise-free labels. However, due to various reasons such as budget constraints and privacy concerns, labels are very limited in many real-world scenarios. Directly applying standard representation learning approaches on small labeled data sets will easily run into over-fitting problems and lead to sub-optimal solutions. Even worse, in some domains such as education, the limited labels are usually annotated by multiple workers with diverse expertise, which yields noises and inconsistency in such crowdsourcing settings. In this paper, we propose a novel framework which aims to learn effective representations from limited data with crowdsourced labels. Specifically, we design a grouping based deep neural network to learn embeddings from a limited number of training samples and present a Bayesian confidence estimator to capture the inconsistency among crowdsourced labels. Furthermore, to expedite the training process, we develop a hard example selection procedure to adaptively pick up training examples that are misclassified by the model. Extensive experiments conducted on three real-world data sets demonstrate the superiority of our framework on learning representations from limited data with crowdsourced labels, comparing with various state-of-the-art baselines. In addition, we provide a comprehensive analysis on each of the main components of our proposed framework and also introduce the promising results it achieved in our real production to fully understand the proposed framework.

### Joint introduction to Gaussian Processes and Relevance Vector Machines with Connections to Kalman filtering and other Kernel Smoothers

The expressive power of Bayesian kernel-based methods has led them to become an important tool across many different facets of artificial intelligence, and useful to a plethora of modern application domains, providing both power and interpretability via uncertainty analysis. This article introduces and discusses two methods which straddle the areas of probabilistic Bayesian schemes and kernel methods for regression: Gaussian Processes and Relevance Vector Machines. Our focus is on developing a common framework with which to view these methods, via intermediate methods a probabilistic version of the well-known kernel ridge regression, and drawing connections among them, via dual formulations, and discussion of their application in the context of major tasks: regression, smoothing, interpolation, and filtering. Overall, we provide understanding of the mathematical concepts behind these models, and we summarize and discuss in depth different interpretations and highlight the relationship to other methods, such as linear kernel smoothers, Kalman filtering and Fourier approximations. Throughout, we provide numerous figures to promote understanding, and we make numerous recommendations to practitioners. Benefits and drawbacks of the different techniques are highlighted. To our knowledge, this is the most in-depth study of its kind to date focused on these two methods, and will be relevant to theoretical understanding and practitioners throughout the domains of data-science, signal processing, machine learning, and artificial intelligence in general.

### Identifying Causal Effects via Context-specific Independence Relations

Causal effect identification considers whether an interventional probability distribution can be uniquely determined from a passively observed distribution in a given causal structure. If the generating system induces context-specific independence (CSI) relations, the existing identification procedures and criteria based on do-calculus are inherently incomplete. We show that deciding causal effect non-identifiability is NP-hard in the presence of CSIs. Motivated by this, we design a calculus and an automated search procedure for identifying causal effects in the presence of CSIs. The approach is provably sound and it includes standard do-calculus as a special case. With the approach we can obtain identifying formulas that were unobtainable previously, and demonstrate that a small number of CSI-relations may be sufficient to turn a previously non-identifiable instance to identifiable.

### Epidemic mitigation by statistical inference from contact tracing data

Contact-tracing is an essential tool in order to mitigate the impact of pandemic such as the COVID-19. In order to achieve efficient and scalable contact-tracing in real time, digital devices can play an important role. While a lot of attention has been paid to analyzing the privacy and ethical risks of the associated mobile applications, so far much less research has been devoted to optimizing their performance and assessing their impact on the mitigation of the epidemic. We develop Bayesian inference methods to estimate the risk that an individual is infected. This inference is based on the list of his recent contacts and their own risk levels, as well as personal information such as results of tests or presence of syndromes. We propose to use probabilistic risk estimation in order to optimize testing and quarantining strategies for the control of an epidemic. Our results show that in some range of epidemic spreading (typically when the manual tracing of all contacts of infected people becomes practically impossible, but before the fraction of infected people reaches the scale where a lockdown becomes unavoidable), this inference of individuals at risk could be an efficient way to mitigate the epidemic. Our approaches translate into fully distributed algorithms that only require communication between individuals who have recently been in contact. Such communication may be encrypted and anonymized and thus compatible with privacy preserving standards. We conclude that probabilistic risk estimation is capable to enhance performance of digital contact tracing and should be considered in the currently developed mobile applications. Identifying, calling, testing, and if needed quarantining the recent contacts of an individual who has just been tested positive is the standard route for limiting the transmission of a highly contagious virus.

### Group Fairness by Probabilistic Modeling with Latent Fair Decisions

Machine learning systems are increasingly being used to make impactful decisions such as loan applications and criminal justice risk assessments, and as such, ensuring fairness of these systems is critical. This is often challenging as the labels in the data are biased. This paper studies learning fair probability distributions from biased data by explicitly modeling a latent variable that represents a hidden, unbiased label. In particular, we aim to achieve demographic parity by enforcing certain independencies in the learned model. We also show that group fairness guarantees are meaningful only if the distribution used to provide those guarantees indeed captures the real-world data. In order to closely model the data distribution, we employ probabilistic circuits, an expressive and tractable probabilistic model, and propose an algorithm to learn them from incomplete data. We evaluate our approach on a synthetic dataset in which observed labels indeed come from fair labels but with added bias, and demonstrate that the fair labels are successfully retrieved. Moreover, we show on real-world datasets that our approach not only is a better model than existing methods of how the data was generated but also achieves competitive accuracy.

### Use of Bayesian Network characteristics to link project management maturity and risk of project overcost

The project management field has the imperative to increase the project probability of success. Experts have developed several project management maturity models to assets and improve the project outcome. However, the current literature lacks of models allowing correlating the measured maturity and the expected probability of success. This paper uses the characteristics of Bayesian networks to formalize experts' knowledge and to extract knowledge from a project overcost database. It develops a method to estimate the impact of project management maturity on the risk of project overcost. A general framework is presented. An industrial case is used to illustrate the application of the method.

### Modeling human visual search: A combined Bayesian searcher and saliency map approach for eye movement guidance in natural scenes

Finding objects is essential for almost any daily-life visual task. Saliency models have been useful to predict fixation locations in natural images, but are static, i.e., they provide no information about the time-sequence of fixations. Nowadays, one of the biggest challenges in the field is to go beyond saliency maps to predict a sequence of fixations related to a visual task, such as searching for a given target. Bayesian observer models have been proposed for this task, as they represent visual search as an active sampling process. Nevertheless, they were mostly evaluated on artificial images, and how they adapt to natural images remains largely unexplored. Here, we propose a unified Bayesian model for visual search guided by saliency maps as prior information. We validated our model with a visual search experiment in natural scenes recording eye movements. We show that, although state-of-the-art saliency models perform well in predicting the first two fixations in a visual search task, their performance degrades to chance afterward. This suggests that saliency maps alone are good to model bottom-up first impressions, but are not enough to explain the scanpaths when top-down task information is critical. Thus, we propose to use them as priors of Bayesian searchers. This approach leads to a behavior very similar to humans for the whole scanpath, both in the percentage of target found as a function of the fixation rank and the scanpath similarity, reproducing the entire sequence of eye movements.