Notation for probability density function in Bayesian context

Question

The Bayes theorem is often quoted as, $$P(\theta|X) = \frac{P(X|\theta)P(\theta)}{P(X)}.$$ In my use case, I'm dealing with Gaussian continuous variables. So, by $P(X|\theta)$ I'm referring to the sum of the negative log-likelihood in my optimisation. However, my PhD supervisor said $P$ typically refers to probability which only makes sense for discrete variables. For PDF, I should be using $f$ (e.g., $\int_x f(x)dx$). This is where the confusion comes in, I've never seen anyone write (for example, I'm doing some Bayesian stuff that leads to a t-distribution): $$f(x|\theta)=St(x;\gamma,\nu,\alpha),$$ but I sometimes see small $p$ in other papers. What's the correct notation? $P$, $p$ or $f$?

Answer 1

I believe that there is no right answer: I think it is a matter of preference. For example, my professor had no problem when I used $p$ notation.

Having studied in some statistics books, I must admit that I often found the $p$ notation. I will give you some examples of books where it is used:

• Gaussian Processes for Machine Learning: an important reference in Gaussian processes, basically every source that talks about GP cites this book. You can find some $p$ notation in 5.4
• Bayesian Data Analysis: a reference text for Bayesian statistics. For example check chapter 21
• Machine Learning: A Probabilistic Perspective: a kind of bible for machine learning. You can find some $p$ in chapter 15

It may not be the full answer to the question, but I hope it will be useful!