Introduction to Bayesian statistics for Crystallography

Bayesian statistics are a great way to combine all our available information about some quantity of interest "x". They can include our prior knowledge about "x" as well as any measurements that have information about "x". We use Bayesian statistics in everyday life all the time without thinking about it. If we see a cup containing a yellow-brown liquid on a dinner table in China we might deduce it is a cup of tea because (1) our prior knowledge is that tea is often found in this context and (2) our observation of a yellow-brown liquid is consistent with this hypothesis and inconsistent with most others. If we made a similar observation in another context (a yellow-brown liquid in a beaker in a chemistry laboratory) we would draw a very different conclusion. This lecture will cover the basics of Bayesian statistics, including familiar probability distributions, the Bayesian view of making measurements, Bayes' rule, and marginalization. Simple examples will be given to illustrate how to apply Bayesian statistics. A simple approach for applying the Bayesian approach to nearly any measurement problem will be presented.