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Disease Models, Part I: Graphical Models

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Medical Imaging Informatics

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.

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Notes

  1. 1.

    We follow the standard notation of capitalizing random variables and of using lower case letters for outcomes. Bold characters symbolize sets or vectors of variables.

  2. 2.

    Bayesian networks are also referred to as Bayesian belief networks (BBNs), or beliefnetworks. We use all three terms interchangeably throughout this book.

  3. 3.

    These and other types of Bayesian queries are covered in further detail in Chapter 9.

  4. 4.

    For causal models, described later in this chapter, this process is called inductive causal inference (also known as causal discovery), which is learning the causal graph from data.

  5. 5.

    Briefly, the EM algorithm consists of two parts: the E-step, wherein the missing data are estimated using the conditional expectation, based on the observed data and the current estimate of the model parameters; and the M-step, where the likelihood function is maximized assuming the missing data are known (the estimated data from the E-step being used in lieu of the actual missing data).

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  1. School of Public Health, Harvard University, Harvard, USA

    Ilya Shpitser PhD

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  1. Medical Imaging Informatics Group, University of California, Los Angeles, Westwood Blvd. 924 , Los Angeles, 90024, U.S.A.

    Alex A.T. Bui

  2. Medical Imaging Informatics Group, University of California, Los Angeles, Westwood Blvd. 924 , Los Angeles, 90024, U.S.A.

    Ricky K. Taira

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Shpitser, I. (2010). Disease Models, Part I: Graphical Models. In: Bui, A., Taira, R. (eds) Medical Imaging Informatics. Springer, Boston, MA. https://doi.org/10.1007/978-1-4419-0385-3_8

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