Mathematics And Probability Theory For Political Science Research

The application of rigorous statistical methods is a core aspect of modern political science research. Moreover, many key contributions to political science and political economy are based on game theoretic modeling. In order to fully understand these statistical and game theoretic approaches, comprehensive knowledge of the underlying mathematical tools is essential. Therefore, this class introduces Ph.D. students to a number of related topics: (1) We begin with a quick introduction to the fundamentals of mathematics, including mathematical notation, functions, limits, and other basic topics. (2) Then we study calculus in one dimension, including differentiation, integration, and the identification of extrema. (3) Probability theory is an essential building block of statistical analysis, which is the reason for us to devote a significant amount of time to this topic. (4) The fourth topic is linear algebra, including systems of equations and Eigenvalues. (5) In addition, we also discuss key aspects of multivariate calculus, especially the optimization of multivariate functions. (6) In the final part of the course, building up on all previous insights, we consider the mathematical properties of regression analysis and maximum likelihood estimation. Knowledge of these tools will enable the students not only to better understand the application of statistical methods in modern research, including methods of maximum likelihood, but also to take advanced methodological courses.

By the end of the class, students will be able to:

• Understand the fundamental building blocks of mathematics, including mathematical notation, functions, sequences and series, and more.
• Describe the rules of calculus in one dimension with respect to differentiation, integration, and the evaluation of extrema.
• Elaborate on key components of probability theory, which entail different types of statistical distributions and probability functions, among others.
• Apply the tools of linear algebra to vectors and matrices, solve systems of equations, and find Eigenvalues of matrices.
• Combine several of the mathematical tools we have learned about to identify extrema in multivariate functions.
• Use their newly acquired knowledge of mathematics and probability theory to better understand regression analysis and maximum likelihood estimation in social research.

The class is primarily based on selections from the following books:

• Moore, W. H., & Siegel, D. A. (2013). A mathematics course for political and social research. Princeton University Press. [Moore & Siegel]
• Gailmard, S. (2014). Statistical modeling and inference for social science. Cambridge University Press. [Gailmard]
• Pitman, J. (1993). Probability. Springer. [Pitman]
• Cunningham, S. (2021). “Probability and Regression Review (Chapter 2)”, in: Causal Inference: The Mixtape. Yale University Press. [Cunningham]
• Pishro-Nik, H. (2014). Introduction to probability, statistics, and random processes. Kappa Research. [Pishro-Nik]

• 14 sessions of 3 hours each (42 hours total)
• Expected preparation and review time for each session: 5-7 hours (6 hours on average) (84 hours total)
• Expected additional time for ungraded take-home assignments: 8 hours for each (32 hours total)
• Expected time for final exam preparation: 20 hours

• Written in-class (sit-in) examination of 3 hours (in person)