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:

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