Notation of parts of Bayes’s Theorem

The symbol $P$ to denote probability is a bit overloaded. To help aid in notation, we will use the following conventions going forward in the class.

• Probability densities describing measured data are denoted with $f$.

• Probability densities describing parameter values, hypotheses, or other non-measured quantities, are denoted with $g$.

• A set of parameters for a given model are denoted $\theta$.

So, if we were to write down Bayes’s theorem for a parameter estimation problem, it would be

$$\begin{aligned} g(\theta \mid y) = \frac{f(y\mid \theta)\,g(\theta)}{f(y)}. \end{aligned}$$

Probabilities written with a $g$ denote the prior or posterior, and those with an $f$ denote the likelihood or evidence.