10.3 Sampling Notation

The notation

 y ~ normal(mu, sigma);

provides the same (proportional) contribution to the model log density as the explicit target density increment,

 target += normal_lpdf(y | mu, sigma);

In both cases, the effect is to add terms to the target log density. The only difference is that the example with the sampling (~) notation drops all additive constants in the log density; the constants are not necessary for any of Stan’s sampling, approximation, or optimization algorithms.