Incomplete Beta Function Inverse

[andrjohns]

Summary

This PR adds the inverse of the incomplete beta function, will be needed for some of the quantile/icdf functions to come.

Tests

Individual rev/fwd/mix tests have been added, as the gradients use the F32 hypergeometric series and the autodiff finite-differencing tests are not stable.

Side Effects

The F32 templating has been relaxed to allow each of the inputs to be different types

Release notes

Add the incomplete beta function inverse

Checklist

• Math issue Incomplete Beta Inverse #2513

• Copyright holder: Andrew Johnson

  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

  • unit tests pass (to run, use: ./runTests.py test/unit)
  • header checks pass, (make test-headers)
  • dependencies checks pass, (make test-math-dependencies)
  • docs build, (make doxygen)
  • code passes the built in C++ standards checks (make cpplint)

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

• the new changes are tested

[spinkney]

Wow, so fast, thanks @andrjohns!

Do you mind if I put in the latex formulas for the partials in the rev file documentation?

[andrjohns]

Ah good point forgot about those! Yeah feel free to add them, otherwise I can do it tonight/tomorrow

[andrjohns]

@serban-nicusor-toptal it looks like the downstream tests for this are failing due to some odd git errors, would you mind having a look when you get a minute?

[serban-nicusor-toptal]

Hey, checking!

Edit: Should be fine now.

[andrjohns]

Thanks for sorting this so quickly!

[stan-buildbot]

---

Name	Old Result	New Result	Ratio	Performance change( 1 - new / old )
gp_pois_regr/gp_pois_regr.stan	3.52	3.51	1.0	0.28% faster
low_dim_corr_gauss/low_dim_corr_gauss.stan	0.02	0.02	1.04	4.02% faster
eight_schools/eight_schools.stan	0.09	0.09	0.99	-0.92% slower
gp_regr/gp_regr.stan	0.14	0.14	1.0	-0.3% slower
irt_2pl/irt_2pl.stan	5.78	5.73	1.01	1.01% faster
performance.compilation	93.32	91.94	1.01	1.48% faster
low_dim_gauss_mix_collapse/low_dim_gauss_mix_collapse.stan	8.15	8.09	1.01	0.76% faster
pkpd/one_comp_mm_elim_abs.stan	31.42	31.71	0.99	-0.92% slower
sir/sir.stan	119.43	118.71	1.01	0.61% faster
gp_regr/gen_gp_data.stan	0.03	0.04	0.94	-5.94% slower
low_dim_gauss_mix/low_dim_gauss_mix.stan	3.0	2.96	1.01	1.16% faster
pkpd/sim_one_comp_mm_elim_abs.stan	0.39	0.39	1.0	0.38% faster
arK/arK.stan	2.07	2.09	0.99	-0.85% slower
arma/arma.stan	0.23	0.23	0.99	-0.6% slower
garch/garch.stan	0.59	0.63	0.94	-6.75% slower
Mean result: 0.996282562232

---

Jenkins Console Log
Blue Ocean
Commit hash: 085f8ea

---

Machine information

ProductName: Mac OS X ProductVersion: 10.11.6 BuildVersion: 15G22010

CPU:
Intel(R) Xeon(R) CPU E5-1680 v2 @ 3.00GHz

G++:
Configured with: --prefix=/Applications/Xcode.app/Contents/Developer/usr --with-gxx-include-dir=/usr/include/c++/4.2.1
Apple LLVM version 7.0.2 (clang-700.1.81)
Target: x86_64-apple-darwin15.6.0
Thread model: posix

Clang:
Apple LLVM version 7.0.2 (clang-700.1.81)
Target: x86_64-apple-darwin15.6.0
Thread model: posix

[spinkney]

I don't feel comfortable reviewing the cpp but the derivatives and tests look good to me. Maybe @SteveBronder can take a look?

[stan-buildbot]

---

Name	Old Result	New Result	Ratio	Performance change( 1 - new / old )
gp_pois_regr/gp_pois_regr.stan	3.62	3.55	1.02	1.92% faster
low_dim_corr_gauss/low_dim_corr_gauss.stan	0.02	0.02	1.01	0.8% faster
eight_schools/eight_schools.stan	0.09	0.09	1.01	0.71% faster
gp_regr/gp_regr.stan	0.14	0.14	0.98	-2.3% slower
irt_2pl/irt_2pl.stan	5.77	5.71	1.01	0.92% faster
performance.compilation	93.4	91.94	1.02	1.55% faster
low_dim_gauss_mix_collapse/low_dim_gauss_mix_collapse.stan	8.04	8.12	0.99	-0.99% slower
pkpd/one_comp_mm_elim_abs.stan	30.77	32.9	0.94	-6.9% slower
sir/sir.stan	120.99	121.56	1.0	-0.47% slower
gp_regr/gen_gp_data.stan	0.03	0.04	0.99	-1.23% slower
low_dim_gauss_mix/low_dim_gauss_mix.stan	3.01	2.97	1.01	1.29% faster
pkpd/sim_one_comp_mm_elim_abs.stan	0.38	0.39	0.98	-2.34% slower
arK/arK.stan	2.07	2.11	0.98	-2.14% slower
arma/arma.stan	0.23	0.23	1.0	-0.38% slower
garch/garch.stan	0.58	0.61	0.95	-4.96% slower
Mean result: 0.990959384528

---

Jenkins Console Log
Blue Ocean
Commit hash: 5803eea

---

Machine information

ProductName: Mac OS X ProductVersion: 10.11.6 BuildVersion: 15G22010

CPU:
Intel(R) Xeon(R) CPU E5-1680 v2 @ 3.00GHz

G++:
Configured with: --prefix=/Applications/Xcode.app/Contents/Developer/usr --with-gxx-include-dir=/usr/include/c++/4.2.1
Apple LLVM version 7.0.2 (clang-700.1.81)
Target: x86_64-apple-darwin15.6.0
Thread model: posix

Clang:
Apple LLVM version 7.0.2 (clang-700.1.81)
Target: x86_64-apple-darwin15.6.0
Thread model: posix

[spinkney]

will it be possible to get this in to the 2.29 release? We can then do student_t copulas

[andrjohns]

@serban-nicusor-toptal looks like the jenkins for this one is having trouble with clang-format, if you wouldn't mind having a look when you've got a minute:

git checkout -b feature/ibeta_inv
Switched to a new branch 'feature/ibeta_inv'
+ clang-format --version
/home/jenkins/workspace/Stan_Math_PR-2637@tmp/durable-c616a978/script.sh: line 4: clang-format: command not found
+ xargs -n20 -P8 clang-format -i
+ find stan test -name '*.hpp' -o -name '*.cpp'
xargs: clang-format: No such file or directory
++ git diff
+ [[ '' != '' ]]

[rok-cesnovar]

Can you try merging in the latest develop, there were quite a few changes in the Jenkinsfile and test machine since then. Hopefully that will help.

[andrjohns]

Can you try merging in the latest develop, there were quite a few changes in the Jenkinsfile and test machine since then. Hopefully that will help.

Ah thought I had but it hadn't triggered a new CI run for some reason

[andrjohns]

@rok-cesnovar or @SteveBronder would either of you have a minute to review this one? @spinkney has signed off on the math, so it's just the c++ to be checked

[rok-cesnovar]

I can do it today. We wont be able to merge after we do the patch release (stan-dev/cmdstan#1086) which should be in a day or two.

[andrjohns]

Thanks! There's no rush at all on this, so just whenever you get the time

[rok-cesnovar]

Looks great. I only have one question in two places.

[andrjohns]

@rok-cesnovar This should be all ready for another look

[rok-cesnovar]

Good to go! Thanks Andrew!

[andrjohns]

Thanks!