Multi-chain `generate_quantities` writes the constrained parameters into the csv

[avehtari]

The fix and issue text were assisted by Claude

Summary:

gq_writer has two write_gq_values overloads. The std::vector one
drops the first num_constrained_params_ values before writing; the
Eigen one does not. Only the multi-chain path of standalone_generate
uses the Eigen overload, so generate_quantities num_chains=N writes
each row with the constrained parameters prepended, while the header
lists only the generated quantities.

src/stan/services/util/gq_writer.hpp, Eigen overload:

model.write_array(rng, draw, values, false, true, &ss);
...
sample_writer_(values);                     // parameters + generated quantities

versus the std::vector overload:

std::vector<double> gq_values(values.begin() + num_constrained_params_,
                              values.end());
sample_writer_(gq_values);                  // generated quantities only

The result is a CSV whose rows have num_params more fields than the
header has names, so any positional reader mis-assigns every column.

Only generate_quantities num_chains=N with N > 1 in a single process
is affected, which in practice means the CmdStan command line. CmdStanR
and CmdStanPy both launch one process per chain, so each process runs
with num_chains = 1 and is delegated to the single-chain overload;
RStan's standalone_gqs() calls that overload directly. All three
therefore take the std::vector path and are unaffected.

Reproducible Steps:

gq_rng.stan:

parameters {
  real theta;
}
model {
  theta ~ std_normal();
}
generated quantities {
  real u = uniform_rng(0, 1);
}

./gq_rng sample num_warmup=200 num_samples=4 output file=fit.csv random seed=1
cp fit.csv fit_1.csv; cp fit.csv fit_2.csv
./gq_rng generate_quantities fitted_params=fit.csv num_chains=2 output file=gqmc.csv random seed=999

Current Output:

gqmc_1.csv — one column name, two values per row; the leading value is
theta, a parameter:

u
-1.871546, 0.89641628
-0.60661605, 0.63008851
-0.36586517, 0.17265139
-0.96383407, 0.36561701

Expected Output:

u
0.89641628
0.63008851
0.17265139
0.36561701

Additional Information:

Fix is to slice in the Eigen overload as the std::vector one does:

plain_type_t<EigVec> gq_values
    = values.tail(values.size() - num_constrained_params_);
sample_writer_(gq_values);

Note this changes the output of generate_quantities num_chains=N by
removing columns, so it deserves a changelog entry for anyone whose
parser adapted to the current behaviour.

Minor, separate: the two paths also disagree on the separator. The
Eigen writer emits ", " where the std::vector writer emits ",".
Not addressed here.

The regression test asserts that each row of a two-chain run has as
many comma-separated fields as the header has names.
It fails on develop and passes with the fix. PR #3402 changes the rng seed
behavior which changes the output, and after #3402 is merged, the test for this
needs to be updated.

Current Version:

v2.39.0 (also reproduced on develop: cmdstan b6c5aa2, stan
d39d26ba7).