Dependency Diagrams and Graph-Constrained Correlation Dynamics: New Systems for Probabilistic Graphical Modeling
My PhD thesis focused on two novel approaches in probabilistic graphical models. Each of these is described on its own page. Each page also includes links to download software packages for the Mathematica programming langauge, and example notebooks.

Dependency Diagrams extend factor graphs (a superset of Bayesian networks and Markov random fields). Dependency diagrams add to factor graphs the power to represent indexing, gating, and hard constraints. This new formalism makes the modeling of systems with unknown or variable structures explicit and straightforward. The dependency diagram framework also enables the automatic translation of diagrams that represent models into diagrams that represent Markov chain Monte Carlo sampling and inference algorithms, and the subsequent autogeneration of runnable code. I used Dependency Diagrams to implement a package that, among other things, automatically generates runnable source code for the process of performing Metropolis-Hastings sampling on arbitrary distributions.

Graph-Constrained Correlation Dynamics formalizes a method of representing probability distributions that evolve continuously in time according to the chemical master equation. This is accomplished by combining a Markov Random Field, which represents the instantaneous probability of the system, with a set of ordinary differential equations on the parameters of the MRF. My research included the development of two methods to optimize the fit between a GCCD model and the corresponding reaction network. I have implemented a software package, linked above, which leverages Dependency Diagrams to optimize GCCD models.

A pdf copy of my thesis is available here.