So it’s perfectly valid Stan code to do things like:

x ~ normal(t, 1);
x ~ normal(t^2, 1);

Like you said though, it’s confusing what that means in terms of statistics.

That says that x comes from a distribution of which the density is proportional to the product of two normal densities (where the parameter is connected somehow). Presumably it’s some sorta distribution.

Rhat only checks that estimates within a chain (the front vs. the back) are comparable and between chain estimates are comparable as well. It isn’t made to tell you if your model is good or bad.

Probably the way to to do that is make predictions. To make predictions with this model though you’d need to figure out what the product of those two distributions are so you can draw random numbers from it. Maybe it’s another normal with a weird mean? But I do not know.

This problem has been bothering me a while. Now I am relieved. Thank you.

I think that it looks tough. :-'D

The convolution of two Gaussians is also Gaussian, namely N_{m+m',v+v'}(x)=\int_{-\infty}^{\infty} N_{m,v}(y)N_{m',v'}(x-y)dy, with simple algebraic relations between their mean and variance. But in the above code, it is not a convolution but a product of Gaussians, so I am not sure but I guess the product of two Gaussian is not so.

I found this answer that the product of two Gaussian random variables is distributed, in general, as a linear combination of two Chi-square random variables: https://math.stackexchange.com/a/397716

This is good info. In this case, however, one does not have two random variables. That would be the case if you had X \sim \text{normal}(t, 1) and Y \sim \text{normal}(t^2, 1) and you were interested in Z = XY. Here one is assuming a density on X such that f_X(x) = \phi(x; t, 1)\phi(x; t^2, 1), where \phi(\cdot; \mu, \sigma) is the normal density with parameters \mu and \sigma. In @Jean_Billie’s post above f_X is a weird density. If you were to write: