What is happening in dealing with parameter supported on [a,\infty) in stan(.)?

General
1

The following is the HMC implemented in stan(.) if I choose to use “HMC” not “NUTS”

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The above sampler is for the case of \theta that is assumed to be real (or vector of real numbers).

Interestingly, it seems that stan(.) can also easily sample from a parameter \theta which is, saying, supported on [a,\infty) for some a>0.

What stan do for this??
Certainly, the Gaussian proposal does not work here.

Do stan(.) transform the \theta, via logarithm, appropriately???

2

Yes! When you declare the following

real<lower=0> theta;

Stan will automatically sample \log \theta and include the appropriate density adjustment term to get the equivalent density so that samples of e^{\log \theta} are distributed just as how \theta would be if you sampled on the original coordinate space.

3

Thanks a lot!
Would you please give me some reference for that ?
I want to know more mathematically detail about this.

4

See https://betanalpha.github.io/assets/case_studies/stan_intro.html#62_constraint_implementation_in_the_parameters_block.