[SteveBronder]

Part 2 of 3 of the split of #3409. Stacked on #3409 (base stable-inc-beta). Part 3, #3490, is stacked on this one.

Summary

Adds a reverse-mode grad_reg_inc_beta. For var arguments the values are computed in double by the prim overload. The reverse pass computes the a and b adjoints directly as H v, where H is the Hessian of I_z(a, b) in (a, b), from the power series or the continued fraction, instead of taping through the loop. The z adjoint is closed form. A one-sided request computes only H_aa or H_bb unless the cross term is needed. rev/fun/inc_beta.hpp is rewritten with make_callback_var. The GRAD_RIB overload of prim grad_reg_inc_beta now excludes var arguments, which go to the new rev overload. In #3409 alone it accepts every scalar type.

The algorithm, the accuracy study and the benchmarks are in the description of #3409.

Files

• stan/math/rev/fun/grad_reg_inc_beta.hpp (new)
• stan/math/rev/fun/inc_beta.hpp
• stan/math/rev/fun.hpp
• stan/math/prim/fun/grad_reg_inc_beta.hpp: the require_all_not_var_t on the GRAD_RIB overload
• test/unit/math/rev/fun/inc_beta_test.cpp (new): shape gradients of inc_beta with var arguments at the defect points of More stable incomplete beta gradient roots #3409.
• test/unit/math/mix/fun/grad_reg_inc_beta_test.cpp, test/unit/math/mix/fun/inc_beta_test.cpp: mix tests, with the upper-bound test started from an empty autodiff stack.

Release notes

See #3409.

Checklist

• Copyright holder: Aalto University

  The copyright holder is typically you or your assignee, such as a university or company. By submitting this pull request, the copyright holder is agreeing to the license the submitted work under the following licenses:
  - Code: BSD 3-clause (https://opensource.org/licenses/BSD-3-Clause)
  - Documentation: CC-BY 4.0 (https://creativecommons.org/licenses/by/4.0/)

• the basic tests are passing

• the code is written in idiomatic C++ and changes are documented in the doxygen

• the new changes are tested