Cruising The Simplex: Hamiltonian Monte Carlo and the Dirichlet Distribution

Due to its constrained support, the Dirichlet distribution is uniquely suited to many applications. The constraints that make it powerful, however, can also hinder practical implementations, particularly those utilizing Markov Chain Monte Carlo (MCMC) techniques such as Hamiltonian Monte Carlo. I demonstrate a series of transformations that reshape the canonical Dirichlet distribution into a form much more amenable to MCMC algorithms.Comment: 5 pages, 0 figure

Nested Sampling with Constrained Hamiltonian Monte Carlo

Nested sampling is a powerful approach to Bayesian inference ultimately limited by the computationally demanding task of sampling from a heavily constrained probability distribution. An effective algorithm in its own right, Hamiltonian Monte Carlo is readily adapted to efficiently sample from any smooth, constrained distribution. Utilizing this constrained Hamiltonian Monte Carlo, I introduce a general implementation of the nested sampling algorithm.Comment: 15 pages, 4 figure

The Design, Implementation, and Assessment of a New Capstone Course Aimed at Science Education Majors

This paper rationalizes the selection of the concept of energy as the central theme of a new capstone course aimed at science education majors. It describes the goals of the course and the activities that preceded the course design and led to the selection of the topics, of the educational materials, and of the teaching methodologies. It presents a sequential description of the manner in which the conceptual knowledge of energy was to be developed. The specific experiments, interactive demonstrations and other educational materials utilized for the conceptual development of the concept of energy in context are described and referenced. The course objectives are described, as well as the instruments utilized to assess student learning. It also presents the activities utilized to assess the course, in addition to the modifications made to the course syllabus based on this assessment