unit_vector transform should work with zero-length unconstrained

[bob-carpenter]

Description

Unit vector transforms should work with a zero-length unconstrained vector.

Example

The size-1 unit vector is just a constant [1]'. This should be the result of transforming the unconstrained 0-vector []'.

Expected Output

Stan programs that use:

parameters {
  unit_vector[1] alpha;
}

should lead to alpha == [1]'.

Current Version:

v4.1.0

[PracticalMetal]

Hi, I would like to work on this. Can you suggest me ways to get started?

[nhuurre]

The expected change is adding if (size(y)==0) branches in stan/math/prim/fun/unit_vector_constrain.hpp and a test case in test/unit/math/prim/fun/unit_vector_constrain.hpp.

However, I think the issue is wrong. As currently implemented, the constraining transform does not change the size of the vector. (That's not ideal but we need some hack to match the constrained and unconstrained topologies.) So size-0 unconstrained vector should result not in a size-1 but a size-0 unit vector. And there is no such thing.

Furthermore, the current implementation handles size-1 vectors correctly. Keep in mind that [1]' is not the only size-1 unit vector, there's also [-1]'.
The example program compiles but when you run it you get

Exception: Found dimension size one in unit vector declaration. One-dimensional unit vector is discrete but the target distribution must be continuous. variable=alpha; dimension size expression=1 (in 'example.stan', line 2, column 14 to column 15)

[bob-carpenter]

Thanks for the correction, @nhuure---you're absolutely right.

It's too bad we have to waste a parameter on it, but I suppose we have to for uniformity. It'll just be a single normally distributed parameter that always maps to [1]'. There's very little overhead in an additional standard normal component of the density.