I'm currently trying to learn Bayesian Statistics but I keep losing time trying to figure out what exactly is meant by notation. Could someone answer the following for me?
Let's say $X \sim N(\mu,\sigma^2)$
(1) I'm trying to calculate a posterior distribution $p(\mu\mid X) \propto p(X\mid\mu)p(\mu)$. So my understanding is that $p(\mu\mid X)$ is the probability distribution of the parameter $\mu$ given the data $X$. What then does $p(\mu\mid X,\sigma^2)$ mean exactly? My guess is that it is the probability distribution of the parameter $\mu$ given the data $X$ and assuming that $\sigma^2$ is fixed. Is that correct?
(2) Following up from (1), for $p(\mu\mid X,\sigma^2)$, what does the posterior function transform into? Is it $p(\mu\mid X,\sigma^2) \propto p(X\mid\mu,\sigma^2)p(\mu,\sigma^2)$? If it is different, how does the likelihood function really change? My understanding is that the likelihood function is based on the way the data are distributed and not the parameters conditioned on. Is there a difference.
(3) Similar question regarding the prior. If we are told that $p(\mu) \propto 1$, is there a difference between $p(\mu)$ and $p(\mu,\sigma^2)$?
Any clarification would be greatly appreciated!