Inferential Statistics and Bayesian Statistics for Materials Science
Inferential statistics and Bayesian statistics provide a mathematical framework for drawing scientific inferences about a population from limited data. In this series, you will learn both theory and Python implementation in pairs, from classical inferential statistics such as point estimation, interval estimation, and hypothesis testing, to Bayesian inference, MCMC, and hierarchical Bayesian models. We cover a rich set of practical applications in materials science, including quality control, experimental data analysis, process optimization, and Bayesian optimization. Through implementations using SciPy, statsmodels, and PyMC3, you can gain a deep understanding of both the theory and practice of statistical inference.
An understanding of the fundamentals of probability theory (random variables, probability distributions, expectation, variance) is required. Learning is possible with basic knowledge of Python and the fundamentals of NumPy/Matplotlib. Knowledge of linear algebra (matrix operations) will allow for a deeper understanding.
Learn the concept of point estimation, the properties of unbiased and consistent estimators, the theory and implementation of maximum likelihood estimation (MLE), the method of moments, evaluation criteria for estimators (unbiasedness, efficiency, consistency), and the Cramér-Rao lower bound.
Learn the concept and interpretation of confidence intervals, confidence intervals for the population mean and variance in normal populations, applications of the t-distribution, chi-square distribution, and F-distribution, large-sample theory and asymptotic confidence intervals, and the bootstrap method.
Learn the framework of hypothesis testing, null and alternative hypotheses, Type I and Type II errors, interpretation of p-values, various testing methods (z-test, t-test, chi-square test, F-test), power analysis, and the multiple comparison problem.
Learn Bayes' theorem and the concepts of prior and posterior distributions, the properties of conjugate priors, Markov Chain Monte Carlo (MCMC) methods, the Metropolis-Hastings method, Gibbs sampling, and the implementation of Bayesian inference using PyMC3.
Learn the structure of hierarchical Bayesian models and hyperparameters, Bayesian linear regression, Bayesian logistic regression, Bayes factors and model selection, applications to quality control, and hyperparameter tuning via Bayesian optimization.