Are the two ways of writing models equivalent? I have a maybe silly question in my head for days: are the following two models equivalent (assume there is only one observation)?


*

*Observation $Y\sim N(\mu, 1)$ and $\mu\sim N(0,2)$.

*Observation $Y=\mu+r$, where $\mu\sim N(0,2)$ and $r\sim N(0,1)$.
I can argue from two sides:


*

*They are the same: because from the perspective of generating observation $Y$, model 1 implies the same as the second one.

*They are different: because if we see model 2 alone, we cannot tell which is the parameter and which is the residual, i.e., we do not know how the likelihood of observation is defined.
Do you think they are the same or different? What is wrong with the above two arguments?
 A: The distribution of $Y$ is the same in either case. 
There is no likelihood to be defined, because there are no unknown parameters. Unless you mean to imply by your notation that $\mu$ in the first case, and $\mu$ and $r$ in the second case are parameters that have prior distributions (rather than exactly known distributions). So there is no difference in that respect.
The main difference is that you have an additional latent variable (that may or may not be observable). In practice, there are examples like this such as the negative binomial distribution (2 parameters), which can also be specified as a Poisson distribution with a gamma distributed random effect, in which case the number of parameters is 2+number of experimental units. However, when you conduct using either negative binomial regression or the corresponding Poisson random effects model, the results for estimates, standard errors etc. will be the same. However, one parameterization is often more computationally convenient.
