Truncation in target +

Modeling
1

Hi,

The code below has 2 ways of writing out a model: one with the target+= syntax and the simplified syntax.

Is the “- normal_lccdf( 0 | 0, 10);” always needed when using the target+= syntax? And it is never needed when using the simplified syntax?


data{
vector[N] y;
}

parameters{
  real a;
  real<lower=0> sigma;
} 
//traget+=
model{
  target += normal_lpdf(a | 0,4);
  target += normal_lpdf(sigma | 0,10) - normal_lccdf( 0 | 0, 10);
  target += normal_lpdf(y | a , sigma);
} 
//simplified 
model{
  a ~ normal(0,4);
  sigma ~ normal(0,10);
  y ~ normal(a, sigma);
}

Thank you!

2

Correct parameter estimation requires only the posterior density up to an unknown multiplicative normalizing constant, which means that constant multiplicative terms (additive terms on the log scale) can be dropped. [Aside: it’s not an accident that Stan works with the gradient of the log density and dropping these additive terms doesn’t affect the gradient]. Thus, the normal_lccdf term can in general be dropped regardless of whether you’re using ~ or target +=. However, there are specific applications where it matters to retain the normalizing constants. The main ones that I can think of are when you intend to work with Bayes factors or when you are computing terms that are to be used as mixture components. Importantly, the ~ notation already drops other relevant normalizing constants for computational efficiency. y ~ Distribution(theta) means target += distribution_lupdf(y | theta). I cannot think of any reason why it would ever be relevant or useful to add the normal_lccdf term back in when you are already dropping normalizing constants via the ~ syntax.

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