Influence modelling and learning between dynamic bayesian networks using score-based structure learning

Abstract

Although partially observable stochastic processes are ubiquitous in many fields of science, little work has been devoted to discovering and analysing the means by which several such processes may interact to influence each other. In this thesis we extend probabilistic structure learning between random variables to the context of temporal models which represent partially observable stochastic processes. Learning an influence structure and distribution between processes can be useful for density estimation and knowledge discovery. A common approach to structure learning, in observable data, is score-based structure learning, where we search for the most suitable structure by using a scoring metric to value structural configurations relative to the data. Most popular structure scores are variations on the likelihood score which calculates the probability of the data given a potential structure. In observable data, the decomposability of the likelihood score, which is the ability to represent the score as a sum of family scores, allows for efficient learning procedures and significant computational saving. However, in incomplete data (either by latent variables or missing samples), the likelihood score is not decomposable and we have to perform inference to evaluate it. This forces us to use non-linear optimisation techniques to optimise the likelihood function. Furthermore, local changes to the network can affect other parts of the network, which makes learning with incomplete data all the more difficult. We define two general types of influence scenarios: direct influence and delayed influence which can be used to define influence around richly structured spaces; consisting of multiple processes that are interrelated in various ways. We will see that although it is possible to capture both types of influence in a single complex model by using a setting of the parameters, complex representations run into fragmentation issues. This is handled by extending the language of dynamic Bayesian networks to allow us to construct single compact models that capture the properties of a system’s dynamics, and produce influence distributions dynamically. The novelty and intuition of our approach is to learn the optimal influence structure in layers. We firstly learn a set of independent temporal models, and thereafter, optimise a structure score over possible structural configurations between these temporal models. Since the search for the optimal structure is done using complete data we can take advantage of efficient learning procedures from the structure learning literature. We provide the following contributions: we (a) introduce the notion of influence between temporal models; (b) extend traditional structure scores for random variables to structure scores for temporal models; (c) provide a complete algorithm to recover the influence structure between temporal models; (d) provide a notion of structural assembles to relate temporal models for types of influence; and finally, (e) provide empirical evidence for the effectiveness of our method with respect to generative ground-truth distributions. The presented results emphasise the trade-off between likelihood of an influence structure to the ground-truth and the computational complexity to express it. Depending on the availability of samples we might choose different learning methods to express influence relations between processes. On one hand, when given too few samples, we may choose to learn a sparse structure using tree-based structure learning or even using no influence structure at all. On the other hand, when given an abundant number of samples, we can use penalty-based procedures that achieve rich meaningful representations using local search techniques. Once we consider high-level representations of dynamic influence between temporal models, we open the door to very rich and expressive representations which emphasise the importance of knowledge discovery and density estimation in the temporal setting.