What Does Stan Calculate in case of Inconsistent Modeling Assumptions

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:

target += 0.5 * normal_lpdf(x | t, 1);
target += 0.5 * normal_lpdf(x | square(t), 1);

Then X would be normally distributed with parameters \mu^\ast = \frac{t + t^2}{2} and \sigma^\ast = 1. This is called geometric pooling.

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