Varying upper bound without lower bound

Modeling
,
2

Yes, that works when all Us are positive.
Another idea is to have upper=0 bound and add U to the raw values.

data {
  int N;
  vector[N] U;  // upper bounds
  ...
parameters {
  vector<upper=0>[N] alpha_raw;
  ...
transformed parameters {
  vector[N] alpha = U + alpha_raw;
}

Also, vectorized bounds work in the latest CmdStan (but not yet in RStan)

data {
  int N;
  vector[N] U;  // upper bounds
  ...
parameters {
  vector<upper=U>[N] alpha; // upper=_ is the same shape as alpha
  ...
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