[WIP] Imbedded Laplace Approximation

[SteveBronder]

Summary

Code for the embedded laplace approximation. The tests are all passing but there is a few things still needed

• Update docs for all functions (@charlesm93 I added the basic arguments for a lot of the functions but could use your expertise to give the definitions for each argument)
• Add corresponding rngs for each specialized laplace approximation (and tests)
• Add tests for negative binomial

The file for laplace have been added to the mix folder since it uses higher order auto diff.

The current signature for the generalized laplace looks like the following in C++

inline auto laplace_marginal_lpdf(LFun&& L_f, LArgs&& l_args,
                                  const Theta0& theta_0, CovarFun&& K_f,
                                  std::ostream* msgs, Args&&... args)

which will translate to stan like the following

target += laplace_marginal_tol_lpdf(
  likelihood_functor,  make_tuple(data_arg1, data_arg2), 
  eta, theta_init,
  covariance_function, 
  1e-6, 10, 2, 1, 5, // tuning args 
  covar_fun_arg1, covar_fun_arg2);

Note that the first tuple used for the likelihood arguments must be data.

Instead of using a tuple for the first functor's inputs and variadic arguments for the covariance functors arguments I would rather have them both be tuples like the following

target += laplace_marginal_tol_lpdf(
  likelihood_functor, make_tuple(data_arg1, data_arg2), 
  covariance_function,  make_tuple(covar_fun_arg1, covar_fun_arg2)
  theta_init,
  eta,
  1e-6, 10, 2, 1, 5);

I think this is nice because it makes it makes the tolerance parameters always sit at the end and both functors have the same input scheme for their arguments. Does anyone have thoughts on this

---

Other additions related to this PR

• A filter_map function that applies a conditionally applies a lambda f to each input of a tuple given a type_trait i.e. the following code would print "fp detected" twice and increment the double elements of the tuple by 1.

std::tuple<double, int, double> tup{1.0, 2, 3.0};
std::tuple<double, int, double> tup2 = filter_map<std::is_floating_point>([](auto x) { 
  std::cout << "fp detected" << std::endl;
  return x + 1;
  }, tup);

The test_ad suite now has a compile time option for only running the tests with only prim and reverse mode with a new boolean template parameter to expect_ad. This is needed to use laplace with the test framework as the laplace impl here does not work with higer order autodiff (since it needs higher order autodiff)

Tests

Since the tests all seem very related I kept them in their own folder, is that alright? Or should I distribute them across the test folders like normal? While this PR is WIP I'm going to leave them in the same folder and if we don't want that then we can move them before we merge

./runTests.py -j20 ./test/unit/math/mix/laplace/ 

Side Effects

Release notes

Checklist

• Copyright holder: Simon's Foundation

  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

[stan-buildbot]

---

Name	Old Result	New Result	Ratio	Performance change( 1 - new / old )
arma/arma.stan	0.32	0.31	1.06	5.61% faster
low_dim_corr_gauss/low_dim_corr_gauss.stan	0.01	0.01	1.09	8.38% faster
gp_regr/gen_gp_data.stan	0.03	0.02	1.1	8.97% faster
gp_regr/gp_regr.stan	0.1	0.09	1.05	4.49% faster
sir/sir.stan	69.01	69.8	0.99	-1.14% slower
irt_2pl/irt_2pl.stan	4.14	4.31	0.96	-4.15% slower
eight_schools/eight_schools.stan	0.06	0.06	1.02	1.93% faster
pkpd/sim_one_comp_mm_elim_abs.stan	0.25	0.25	0.99	-1.33% slower
pkpd/one_comp_mm_elim_abs.stan	19.51	19.39	1.01	0.59% faster
garch/garch.stan	0.44	0.41	1.07	6.74% faster
low_dim_gauss_mix/low_dim_gauss_mix.stan	2.71	2.61	1.04	3.66% faster
arK/arK.stan	1.8	1.71	1.05	4.83% faster
gp_pois_regr/gp_pois_regr.stan	2.86	2.93	0.97	-2.68% slower
low_dim_gauss_mix_collapse/low_dim_gauss_mix_collapse.stan	8.81	8.35	1.05	5.21% faster
performance.compilation	182.99	184.36	0.99	-0.75% slower
Mean result: 1.0293461908769272

---

Jenkins Console Log
Blue Ocean
Commit hash: 4f8838bd96b1d1590209d845f484e5867c6c54a0

---

Machine information

No LSB modules are available. Distributor ID: Ubuntu Description: Ubuntu 20.04.3 LTS Release: 20.04 Codename: focal

CPU:
Architecture: x86_64
CPU op-mode(s): 32-bit, 64-bit
Byte Order: Little Endian
Address sizes: 46 bits physical, 48 bits virtual
CPU(s): 80
On-line CPU(s) list: 0-79
Thread(s) per core: 2
Core(s) per socket: 20
Socket(s): 2
NUMA node(s): 2
Vendor ID: GenuineIntel
CPU family: 6
Model: 85
Model name: Intel(R) Xeon(R) Gold 6148 CPU @ 2.40GHz
Stepping: 4
CPU MHz: 2400.000
CPU max MHz: 3700.0000
CPU min MHz: 1000.0000
BogoMIPS: 4800.00
Virtualization: VT-x
L1d cache: 1.3 MiB
L1i cache: 1.3 MiB
L2 cache: 40 MiB
L3 cache: 55 MiB
NUMA node0 CPU(s): 0,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78
NUMA node1 CPU(s): 1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41,43,45,47,49,51,53,55,57,59,61,63,65,67,69,71,73,75,77,79
Vulnerability Gather data sampling: Mitigation; Microcode
Vulnerability Itlb multihit: KVM: Mitigation: VMX disabled
Vulnerability L1tf: Mitigation; PTE Inversion; VMX conditional cache flushes, SMT vulnerable
Vulnerability Mds: Mitigation; Clear CPU buffers; SMT vulnerable
Vulnerability Meltdown: Mitigation; PTI
Vulnerability Mmio stale data: Mitigation; Clear CPU buffers; SMT vulnerable
Vulnerability Retbleed: Mitigation; IBRS
Vulnerability Spec rstack overflow: Not affected
Vulnerability Spec store bypass: Mitigation; Speculative Store Bypass disabled via prctl
Vulnerability Spectre v1: Mitigation; usercopy/swapgs barriers and __user pointer sanitization
Vulnerability Spectre v2: Mitigation; IBRS, IBPB conditional, STIBP conditional, RSB filling, PBRSB-eIBRS Not affected
Vulnerability Srbds: Not affected
Vulnerability Tsx async abort: Mitigation; Clear CPU buffers; SMT vulnerable
Flags: fpu vme de pse tsc msr pae mce cx8 apic sep mtrr pge mca cmov pat pse36 clflush dts acpi mmx fxsr sse sse2 ss ht tm pbe syscall nx pdpe1gb rdtscp lm constant_tsc art arch_perfmon pebs bts rep_good nopl xtopology nonstop_tsc cpuid aperfmperf pni pclmulqdq dtes64 monitor ds_cpl vmx smx est tm2 ssse3 sdbg fma cx16 xtpr pdcm pcid dca sse4_1 sse4_2 x2apic movbe popcnt tsc_deadline_timer aes xsave avx f16c rdrand lahf_lm abm 3dnowprefetch cpuid_fault epb cat_l3 cdp_l3 invpcid_single pti intel_ppin ssbd mba ibrs ibpb stibp tpr_shadow vnmi flexpriority ept vpid ept_ad fsgsbase tsc_adjust bmi1 hle avx2 smep bmi2 erms invpcid rtm cqm mpx rdt_a avx512f avx512dq rdseed adx smap clflushopt clwb intel_pt avx512cd avx512bw avx512vl xsaveopt xsavec xgetbv1 xsaves cqm_llc cqm_occup_llc cqm_mbm_total cqm_mbm_local dtherm ida arat pln pts hwp hwp_act_window hwp_epp hwp_pkg_req pku ospke md_clear flush_l1d arch_capabilities

G++:
g++ (Ubuntu 9.4.0-1ubuntu1~20.04) 9.4.0
Copyright (C) 2019 Free Software Foundation, Inc.
This is free software; see the source for copying conditions. There is NO
warranty; not even for MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.

Clang:
clang version 10.0.0-4ubuntu1
Target: x86_64-pc-linux-gnu
Thread model: posix
InstalledDir: /usr/bin

[stan-buildbot]

---

Name	Old Result	New Result	Ratio	Performance change( 1 - new / old )
arma/arma.stan	0.37	0.36	1.02	2.4% faster
low_dim_corr_gauss/low_dim_corr_gauss.stan	0.01	0.01	1.14	12.11% faster
gp_regr/gen_gp_data.stan	0.03	0.03	1.07	6.17% faster
gp_regr/gp_regr.stan	0.11	0.09	1.16	14.02% faster
sir/sir.stan	69.59	70.33	0.99	-1.07% slower
irt_2pl/irt_2pl.stan	4.15	4.27	0.97	-2.84% slower
eight_schools/eight_schools.stan	0.06	0.06	1.0	0.31% faster
pkpd/sim_one_comp_mm_elim_abs.stan	0.25	0.26	0.99	-0.88% slower
pkpd/one_comp_mm_elim_abs.stan	19.53	19.53	1.0	0.0% slower
garch/garch.stan	0.44	0.44	1.01	0.55% faster
low_dim_gauss_mix/low_dim_gauss_mix.stan	2.73	2.61	1.05	4.62% faster
arK/arK.stan	1.88	1.72	1.09	8.53% faster
gp_pois_regr/gp_pois_regr.stan	2.86	2.69	1.06	5.95% faster
low_dim_gauss_mix_collapse/low_dim_gauss_mix_collapse.stan	8.84	8.42	1.05	4.76% faster
performance.compilation	181.51	181.26	1.0	0.14% faster
Mean result: 1.0406311165858395

---

Jenkins Console Log
Blue Ocean
Commit hash: e8ede7f282cc62475acff2b39429148836b76093

---

Machine information

No LSB modules are available. Distributor ID: Ubuntu Description: Ubuntu 20.04.3 LTS Release: 20.04 Codename: focal

CPU:
Architecture: x86_64
CPU op-mode(s): 32-bit, 64-bit
Byte Order: Little Endian
Address sizes: 46 bits physical, 48 bits virtual
CPU(s): 80
On-line CPU(s) list: 0-79
Thread(s) per core: 2
Core(s) per socket: 20
Socket(s): 2
NUMA node(s): 2
Vendor ID: GenuineIntel
CPU family: 6
Model: 85
Model name: Intel(R) Xeon(R) Gold 6148 CPU @ 2.40GHz
Stepping: 4
CPU MHz: 2400.000
CPU max MHz: 3700.0000
CPU min MHz: 1000.0000
BogoMIPS: 4800.00
Virtualization: VT-x
L1d cache: 1.3 MiB
L1i cache: 1.3 MiB
L2 cache: 40 MiB
L3 cache: 55 MiB
NUMA node0 CPU(s): 0,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78
NUMA node1 CPU(s): 1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41,43,45,47,49,51,53,55,57,59,61,63,65,67,69,71,73,75,77,79
Vulnerability Gather data sampling: Mitigation; Microcode
Vulnerability Itlb multihit: KVM: Mitigation: VMX disabled
Vulnerability L1tf: Mitigation; PTE Inversion; VMX conditional cache flushes, SMT vulnerable
Vulnerability Mds: Mitigation; Clear CPU buffers; SMT vulnerable
Vulnerability Meltdown: Mitigation; PTI
Vulnerability Mmio stale data: Mitigation; Clear CPU buffers; SMT vulnerable
Vulnerability Retbleed: Mitigation; IBRS
Vulnerability Spec rstack overflow: Not affected
Vulnerability Spec store bypass: Mitigation; Speculative Store Bypass disabled via prctl
Vulnerability Spectre v1: Mitigation; usercopy/swapgs barriers and __user pointer sanitization
Vulnerability Spectre v2: Mitigation; IBRS, IBPB conditional, STIBP conditional, RSB filling, PBRSB-eIBRS Not affected
Vulnerability Srbds: Not affected
Vulnerability Tsx async abort: Mitigation; Clear CPU buffers; SMT vulnerable
Flags: fpu vme de pse tsc msr pae mce cx8 apic sep mtrr pge mca cmov pat pse36 clflush dts acpi mmx fxsr sse sse2 ss ht tm pbe syscall nx pdpe1gb rdtscp lm constant_tsc art arch_perfmon pebs bts rep_good nopl xtopology nonstop_tsc cpuid aperfmperf pni pclmulqdq dtes64 monitor ds_cpl vmx smx est tm2 ssse3 sdbg fma cx16 xtpr pdcm pcid dca sse4_1 sse4_2 x2apic movbe popcnt tsc_deadline_timer aes xsave avx f16c rdrand lahf_lm abm 3dnowprefetch cpuid_fault epb cat_l3 cdp_l3 invpcid_single pti intel_ppin ssbd mba ibrs ibpb stibp tpr_shadow vnmi flexpriority ept vpid ept_ad fsgsbase tsc_adjust bmi1 hle avx2 smep bmi2 erms invpcid rtm cqm mpx rdt_a avx512f avx512dq rdseed adx smap clflushopt clwb intel_pt avx512cd avx512bw avx512vl xsaveopt xsavec xgetbv1 xsaves cqm_llc cqm_occup_llc cqm_mbm_total cqm_mbm_local dtherm ida arat pln pts hwp hwp_act_window hwp_epp hwp_pkg_req pku ospke md_clear flush_l1d arch_capabilities

G++:
g++ (Ubuntu 9.4.0-1ubuntu1~20.04) 9.4.0
Copyright (C) 2019 Free Software Foundation, Inc.
This is free software; see the source for copying conditions. There is NO
warranty; not even for MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.

Clang:
clang version 10.0.0-4ubuntu1
Target: x86_64-pc-linux-gnu
Thread model: posix
InstalledDir: /usr/bin