About this course: This is the second of a two-course sequence introducing the fundamentals of Bayesian statistics. It builds on the course Bayesian Statistics: From Concept to Data Analysis, which introduces Bayesian methods through use of simple conjugate models. Real-world data often require more sophisticated models to reach realistic conclusions. This course aims to expand our "Bayesian toolbox" with more general models, and computational techniques to fit them. In particular, we will introduce Markov chain Monte Carlo (MCMC) methods, which allow sampling from posterior distributions that have no analytical solution.

About this course: This course describes Bayesian statistics, in which one's inferences about parameters or hypotheses are updated as evidence accumulates. You will learn to use Bayes' rule to transform prior probabilities into posterior probabilities, and be introduced to the underlying theory and perspective of the Bayesian paradigm. The course will apply Bayesian methods to several practical problems, to show end-to-end Bayesian analyses that move from framing the question to building models to eliciting prior probabilities to implementing in R (free statistical software) the final posterior distribution. Additionally, the course will introduce credible regions, Bayesian comparisons of means and proportions, Bayesian regression and inference using multiple models, and discussion of Bayesian prediction. We assume learners in this course have background knowledge equivalent to what is covered in the earlier three courses in this specialization: "Introduction to Probability and Data," "Inferential Statistics," and "Linear Regression and Modeling."

Bayesian Computational Analyses with R is an introductory course on the use and implementation of Bayesian modeling using R software. The Bayesian approach is an alternative to the "frequentist" approach where one simply takes a sample of data and makes inferences about the likely parameters of the population. In contrast, the Bayesian approach uses both likelihood functions and a sample of observed data (the'prior') to estimate the most likely values and distributions for the estimated population parameters (the'posterior'). The course is useful to anyone who wishes to learn about Bayesian concepts and is suited to both novice and intermediate Bayesian students and Bayesian practitioners. It is both a practical, "hands-on" course with many examples using R scripts and software, and is conceptual, as the course explains the Bayesian concepts. All materials, software, R scripts, slides, exercises and solutions are included with the course materials. It is helpful to have some grounding in basic inferential statistics and probability theory. No experience with R is necessary, although it is also helpful.

Ahmed, Umair Z. (Indian Institute of Technology Kanpur) | Gulwani, Sumit (Microsoft Research Redmond) | Karkare, Amey (Indian Institute of Technology Kanpur)

Natural deduction, which is a method for establishing validity of propositional type arguments, helps develop important reasoning skills and is thus a key ingredient in a course on introductory logic. We present two core components, namely solution generation and practice problem generation, for enabling computer-aided education for this important subject domain. The key enabling technology is use of an offline-computed data-structure called Universal Proof Graph (UPG) that encodes all possible applications of inference rules over all small propositions abstracted using their bitvector-based truth-table representation. This allows an efficient forward search for solution generation. More interestingly, this allows generating fresh practice problems that have given solution characteristics by performing a backward search in UPG. We obtained around 300 natural deduction problems from various textbooks. Our solution generation procedure can solve many more problems than the traditional forward-chaining based procedure, while our problem generation procedure can efficiently generate several variants with desired characteristics.

Labutov, Igor (Carnegie Mellon University) | Studer, Christoph (Cornell University)

Generation and evaluation of crowdsourced content is commonly treated as two separate processes, performed at different times and by two distinct groups of people: content creators and content assessors. As a result, most crowdsourcing tasks follow this template: one group of workers generates content and another group of workers evaluates it. In an educational setting, for example, content creators are traditionally students that submit open-response answers to assignments (e.g., a short answer, a circuit diagram, or a formula) and content assessors are instructors that grade these submissions. Despite the considerable success of peer-grading in massive open online courses (MOOCs), the process of test-taking and grading are still treated as two distinct tasks which typically occur at different times, and require an additional overhead of grader training and incentivization. Inspired by this problem in the context of education, we propose a general crowdsourcing framework that fuses open-response test-taking (content generation) and assessment into a single, streamlined process that appears to students in the form of an explicit test, but where everyone also acts as an implicit grader. The advantages offered by our framework include: a common incentive mechanism for both the creation and evaluation of content, and a probabilistic model that jointly models the processes of contribution and evaluation, facilitating efficient estimation of the quality of the contributions and the competency of the contributors. We demonstrate the effectiveness and limits of our framework via simulations and a real-world user study.