An Introduction to Probability Theory and Stochastic Processes with Applications to Biology

Abstract

Biology is becoming increasingly quantitative and mathematical modeling is now an integral part of biological research. In many biological processes, ranging from gene-expression to evolution, randomness plays an important role that can only be understood using stochastic models. This course will provide the students with a theoretical foundation for developing such stochastic models and analyzing

Objective

The aim of this course is to introduce certain topics in Probability Theory and Stochastic Processes that have been specifically selected with an eye on biological applications. This course will teach students the tools and techniques for modeling and analyzing random phenomena. Throughout the course, several biological applications will be discussed and students will be encouraged to do additional reading based on their research interests.

Content

The first half of the course will cover the basics of Probability Theory while the second half will delve into the theory of Stochastic Processes. Below is the list of topics that will be covered in the course. 1. The mathematical representation of random phenomena: The probability space, properties of the probability measure, Independence of events, Conditional probability and Bayes formula, applications to parameter inference. 2. Random Variables and their distributions: Discrete and continuous random variables, Expectation and Variance, Important Examples of Random Variables, Independent random variables and their sums, Conditional Distribution and Conditional Expectation, Markov and Chebyshev inequalities. Law of total variation, estimation of intrinsic and extrinsic noise in biological systems. 3. Convergence of Random Variables: Modes of convergence, Laws of large numbers, the central limit theorem, the law of the iterated logarithm, Applications to the analysis of cell population data. 4. Generating functions and their applications: Definition and important examples, Random Walks, Branching processes, Coalescent processes, Modeling epidemic processes and stem-cell differentiation. 5. Markov chains: Transition functions and related computations, Classification of states and classification of chains. Concepts of recurrence, transience, irreducibility and periodicity, Stationary distributions, Continuous time Markov Chain model of a biochemical reaction network. 6. Stochastic Processes: Existence and Construction, Stationary Processes, Renewal Processes, The Wiener Process, The Ergodic Theorem, Leveraging experimental techniques in Biology. 7. Introduction to the theory of Martingales: Basic definitions, Martingale differences and Hoeffding's inequality, Martingale Convergence Theorem, Crossings and convergence, Stopping times and the optional sampling theorem, Doob's maximal inequalities, Applications to the analysis of stochastic biochemical reaction networks.

Resources