Different notation for Bayes' prior and posterior distributions

Question

Bayes' rule is given by:

$$P(\theta|X) = \frac{P(X|\theta)P(\theta)}{P(X)}$$

Where $X$ are observations and $\theta$ is some model parameter. I would like to use an alternate notation to more strongly differentiate between the prior $P(\theta)$ and posterior $P(\theta|X)$ distributions. Is it appropriate to write:

$$P(\theta_\text{post}|X) = \frac{P(X|\theta_\text{prior})P(\theta_\text{prior})}{P(X)}$$

Can it be said that the posterior and prior describe the distributions of two different random variables namely $\theta_\text{post}$ and $\theta_\text{prior}$ respectively? Or are the prior and posterior different distributions of the same random variable $\theta$? So perhaps we should write:

$$P_\text{post}(\theta|X) = \frac{P(X|\theta)P_\text{prior}(\theta)}{P(X)}$$

Equally how should one denote the prior distribution?

$$\theta_\text{prior} \sim N(0,1)$$

Or:

$$P_\text{prior}(\theta) \sim N(0,1)$$

Answer 1

If you want to distinguish them, you can use subscripts on the probability mass (or density) functions directly (as you have done in your second example). For simplicity, this is usually written using conditional notation, i.e. $$ P_{\theta|X}(\theta|X) = \frac{P_{X|\theta}(X|\theta)P_\theta(\theta)}{P_X(X)} $$ This has the advantage that you can distinguish $P_X(1)$ vs $P_\theta(2)$ and $P_{X|\theta}(1|2)$ vs $P_{\theta|X}(2|1)$. But if this distinction is unnecessary, then the notation just appears repetitive because the subscript is identical to the content between the parentheses. Thus, the subscripts are typically dropped.

To the second question $$ \theta \sim N(0,1)$$ is appropriate because the random variable $\theta$ has a standard normal distribution. Rather than the probability density function for the random variable $\theta$ having a standard normal distribution.

Answer 2

$\theta$ is the same random variable in both the posterior and the prior. The difference is that in the posterior you are conditioning on the data. It's your understanding of the values $\theta$ can take after you've considered the data whereas the prior is your understanding before considering the data. The $|X$ is already distinguishes the two so you don't need a subscript like $\theta_{post}$ and $\theta_{prior}$! It would be both redundant and confusing.

I also strongly suggest you do not drop the $|X$ part as in $P_{post}(\theta)$. In Bayesian statisics, it's very important to remember what information you're conditioning on. Leaving the $|X$ in there will emphasis this.

Answer 3

$P_r(\theta)$ is usually used to denote the prior distribution of $\theta$. Besides, as the posterior distribution of $\theta$ is a conditional distribution given $\theta$, no other notation is needed to distinguish it.