Disease Models, Part I: Graphical Models

Abstract

Scientists building models of the world by necessity abstract away features not directly relevant to their line of inquiry. Furthermore, complete knowledge of relevant features is not generally possible. The mathematical formalism that has proven to be the most successful at simultaneously abstracting the irrelevant, while effectively summarizing incomplete knowledge, is probability theory. First studied in the context of analyzing games of chance, probability theory has flowered into a mature mathematical discipline today whose tools, methods, and concepts permeate statistics, engineering, and social and empirical sciences. A key insight, discovered multiple times independently during the 20th century, but refined, generalized, and popularized by computer scientists, is that there is a close link between probabilities and graphs. This link allows numerical, quantitative relationships such as conditional independence found in the study of probability to be expressed in a visual, qualitative way using the language of graphs. As human intuitions are more readily brought to bear in visual rather than algebraic and computational settings, graphs aid human comprehension in complex probabilistic domains. This connection between probabilities and graphs has other advantages as well - for instance the magnitude of computational resources needed to reason about a particular probabilistic domain can be read from a graph representing this domain. Finally, graphs provide a concise and intuitive language for reasoning about causes and effects. In this chapter, we explore the basic laws of probability, the relationship between probability and causation, the way in which graphs can be used to reason about probabilistic and causal models, and finally how such graphical models can be learned from data. The application of these graphs to formalize observations and knowledge about disease are provided.