Upper case (P) or lower case (p) to denote p-values and probabilities in frequentist and Bayesian statistics I am conducting a study in which I am reporting results of hypothesis tests conducted in a frequentist framework, and also some additional analyses conducted in a Bayesian framework.
Should I use upper case (capital) $P$ or lower case $p$ to denote $P$/$p$-values and Bayesian probabilities?


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*"p" for both frequentist and Bayesian analyses?

*"P" for both frequentist and Bayesian analyses?

*"p" for frequentist analyses and "P" for Bayesian analyses?

*or "P" for frequentist analyses and "p" for Bayesian analyses?
 A: Notationally: $P(\cdot)$ is a functional to denote probability of events. The same notation is used when speaking of probability in a frequentist framework (e.g. probability is a frequency of events observed in infinite replications of the universe, or counterfactual probability) as in a Bayesian framework (e.g. probability is a degree of belief).
The history about $p$-values seems to date back to De Moivre in the 18th century according to Wikipedia. Pearson used the capital $P$ to denote a measure of inconsistency of an observed set of data from a hypothesis which is to be tested using a test statistic which assumes a known distribution when that hypothesis is true. Modern usage has reverted to lower case $p$ more often than not, I find, because the $p$ value is not a random variable, a type of distinction which is also somewhat antiquated in modern probability theory. I think you may find for submitting statistical research that most journals use lowercase $p$ but there may be instances of $P$, the only recommendation is to agree on one usage and be consistent.
There is no $p$ for Bayesian statistics. Bayesian testing is a controversial subject, but all agree that the $p$-value should immediately be bucked when doing a Bayesian analysis. My personal preference is to report the results of a Bayesian analysis using credible intervals which provide a range of plausible (believable) effects for the parameter. Bayes factors can summarize tests in a manner similar to $p$-values, but $p$-values suck, why would you use them if you're doing a Bayesian analysis?
A: I favour $P$ for P-value partly because that usage goes back a long way and I like history of ideas, but perhaps more because $p$ is already overloaded: I often want $p$ to mean 


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*probability in general 

*a particular probability (e.g. in notation for binomial distributions) 

*the number of predictors or covariates. 
Contrary to that, a reminder or two in text that we are discussing p-values might reduce the ambiguity or puzzlement about what $p$ means. 
I've not picked up any hint that being frequentist or Bayesian makes any difference to preferred notation, but that's just an uninformative prior speaking. Evidence and argument on that detail is especially welcome. 
Notation in statistics (as generally in any subject with mathematical content) is a messy mixture of tradition, accident and logic. We have some guidelines, such as Greek for parameters and roman for statistics, but consistency is elusive. In directional statistics, for example, trigonometric conventions dominate and $\theta$ and $\phi$ are routinely names for variables. 
We would all benefit by agreeing on some better and more consistent notations: there is just the small detail of what those might be. 
Here is a case of two common notations. It would usually be futile to try to change anybody else's choice. At most we can carp if authors aren't consistent within publications we review or don't follow an arbitrary style standard for a journal. 
A: You are free to use any notation you want $p(X=x)$, $P(X=x)$, $\mathrm{P}(X=x)$, $\Pr(X=x)$ etc. I never heard about any formal rules about it. I have an impression that $p(X)$ is more often used when authors want to talk about probability in general and use catchall term for things like probabilities of events $p(X=x)$, probability density functions $p(x)$ etc., while uppercase $P$ or $\Pr$ is used more commonly when talking about events.
