# Mixing probabilities with probability densities in Bayesian inference

I just noticed that in TrueSkill they use this implementation of Bayes theorem:

$\mathcal p(\mathbf s|\mathbf r, A) = \frac { \mathcal P(\mathbf r|\mathbf s, A) \; \mathcal p(\mathbf s) }{ \mathcal P(\mathbf r|A) }$

What puzzles me here is the mixing of probailities ($\mathcal P$) with probability densities ($\mathcal p$) in the formulation, as in my admittedly far form exhaustive readings on Bayes Theorem to date I only see Probabilities mentioned.

To wit, I wonder if anyone can recommend something to read on the use of probability densities and or a mix of probability densities and probabilities in Bayesian inference.

$\frac { \mathcal p(\mathbf s|\mathbf r, A) } { \mathcal p(\mathbf s) } = \frac { \mathcal P(\mathbf r|\mathbf s, A) }{ \mathcal P(\mathbf r|A) }$
Which has elegant symmetry to it and note that $\mathcal p$ is in fact the derivative of $\mathcal P$ (did I mention shades of L'Hopital's?).