math [DRAFT/RFC] Gaussian Copula Cholesky LPDF

Summary

Opening this as a draft PR for feedback on the signature/approach.

For the Gaussian copula (and other copula families), the user needs to provide both the y variable and a means of transforming that variable to the unit-scale.

The method I've gone with is to require a tuple of the same length as y, where each element is itself a tuple - with a functor for computing the LCDF as the first element, and any additional args as the remaining.

For example, if the user wanted to model the correlation between a Gamma(2, 1) variable and a Exp(2) variable, they would pass the tuple-of-tuples:

std::make_tuple(
    std::make_tuple(gamma_lcdf, 2, 1),
    std::make_tuple(exp_lcdf, 2)
)

So the final signature would be:

real gaussian_copula_cholesky_lpdf(vector | tuple-of-tuples, cholesky_factor_corr)

The current framework only supports continuous outcomes, but once we settle on a good approach I can expand to discrete outcomes by requiring that a uniform parameter for data-augmentation is also included in the tuple for that outcome.

Tests

Basic prim and mix are added, but the fwd components of the mix tests are currently failing

Side Effects

Extended a few of the utilities (size, vector_seq_view) for tuples

Release notes

Replace this text with a short note on what will change if this pull request is merged in which case this will be included in the release notes.

Checklist

[x] 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)
[x] the code is written in idiomatic C++ and changes are documented in the doxygen
[x] the new changes are tested

Jun 15 '25 12:06 andrjohns

@spinkney would you have a look at the general signature/idea here? Not 100% that is the way to go, but I'm not sure of a better alternative

Jun 15 '25 12:06 andrjohns

Name	Old Result	New Result	Ratio	Performance change( 1 - new / old )
arma/arma.stan	0.32	0.31	1.05	4.39% faster
low_dim_corr_gauss/low_dim_corr_gauss.stan	0.01	0.01	0.97	-3.29% slower
gp_regr/gen_gp_data.stan	0.03	0.02	1.12	10.83% faster
gp_regr/gp_regr.stan	0.1	0.09	1.04	3.45% faster
sir/sir.stan	72.93	71.01	1.03	2.64% faster
irt_2pl/irt_2pl.stan	4.21	4.17	1.01	0.83% faster
eight_schools/eight_schools.stan	0.06	0.06	1.01	1.01% faster
pkpd/sim_one_comp_mm_elim_abs.stan	0.25	0.25	1.0	0.22% faster
pkpd/one_comp_mm_elim_abs.stan	20.26	19.77	1.02	2.43% faster
garch/garch.stan	0.45	0.43	1.05	4.65% faster
low_dim_gauss_mix/low_dim_gauss_mix.stan	2.89	2.75	1.05	4.71% faster
arK/arK.stan	1.86	1.81	1.03	3.02% faster
gp_pois_regr/gp_pois_regr.stan	2.91	2.88	1.01	0.87% faster
low_dim_gauss_mix_collapse/low_dim_gauss_mix_collapse.stan	9.4	8.93	1.05	5.0% faster
performance.compilation	185.4	188.99	0.98	-1.94% slower
Mean result: 1.0277131269411455

Jenkins Console Log Blue Ocean Commit hash: ae79d13bb5a669624fb639d9bb15aef05c4c6f00

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: 3259.121 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 Reg file data sampling: Not affected 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; BHI 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

Jun 16 '25 20:06 stan-buildbot

@andrjohns thanks for getting to this!

In the signature

real gaussian_copula_cholesky_lpdf(vector | tuple-of-tuples, cholesky_factor_corr)

Does this imply that it is looped over for each observation, that is for each $n \in N$ we have a tuple-of-tuples? I believe we also want to support an array vectors for the first argument.

@spinkney would you have a look at the general signature/idea here? Not 100% that is the way to go, but I'm not sure of a better alternative

Yea, I see that there's a few different options with different pros and cons. I like the current implementation but I think it would be nice to have additional signatures. Here's a few questions and options:

Can we pass in an empty functor to signal that this variate is already on the Gaussian scale? Or is there another way to accomplish this?
Do we also want to support passing in the lpdf as an optional arugment? I just feel like we're already looping over the lcdf so can accomplish this in one pass too.
I would like to be able to choose or have the option to input a user defined inverse normal cdf function. Users may prefer to have a fast approximation or use a custom algorithm. It would be nice to support this but it's not necessary for this PR to go through.

To make this a bit more concrete:

data {
  int<lower=0> N;
  int<lower=1>  K;            // K different marginals
  array[K] int<lower=0> P;    // number of parameters for each k marginal
  array[N] vector[K] y;       // continuous outcome and an array of vector
}
transformed data {
  int P_total = to_int(sum(to_vector(P)));
}
parameters {
  vector[P_total] theta_free;
  cholesky_factor_corr[K] L;
}
model {
  tuple( tuple(real, ...), ..., tuple(real, ...) ) marginal_lcdf_tuples =  
   tuple(
     tuple(k1_lcdf, theta_free[1], theta_free[2], ...),
     tuple(k2_lcdf, theta_free[p], theta_free[p + 1], ...),
     ....,
     tuple(K_lcdf, ..., theta_free[P_total])
   );
 
  // Can the above also be any container for the parameters?
  //  such as
  //
  //  tuple( tuple(real, ...), ..., tuple(real, ...) ) marginal_tuples =  
  //   tuple(
  //     tuple(k1_lcdf, vector),
  //     tuple(k2_lcdf, array[] real),
  //     ....,
  //     tuple(K_lcdf, ..., matrix)
  //   );
  //
 
  // Why not also have another input for the lpdfs?
  //
  // tuple( tuple(real, ...), ..., tuple(real, ...) ) marginal_lpdf_tuples =  
  //  tuple(
  //    tuple(k1_lpdf, theta_free[1], theta_free[2], ...),
  //    tuple(k2_lpdf, theta_free[p], theta_free[p + 1], ...),
  //    ....,
  //    tuple(K_lpdf, ..., theta_free[P_total])
  //  );
  //

  y ~ gaussian_copula_cholesky(marginal_lcdf_tuples | L);
  
  // or optionally
  // y ~ 
  //  gaussian_copula_cholesky(
  //   marginal_lcdf_tuples | marginal_lpdf_tuples, L
  // );
  //
}

Jun 26 '25 11:06 spinkney

Just typing that out was tiresome and made me think another way

// lcdf functors
 tuple( real, real, ... ) marginal_lcdf_tuples =  
  tuple( k1_lcdf, k2_lcdf, ..., K_lcdf );
 
// lpdf functors, same K number of objects in the tuple as marginal_lcdf_tuples
  tuple( real, real, ... ) marginal_lpdf_tuples =  
   tuple( k1_lpdf, k2_lpdf, ..., K_lpdf ); 

// parameters
// I'm assuming that the containers in this tuple would match the
// allowable parameters in their corresponding lcdf/lpdf functor signatures
  tuple( params_k1, ..., params_K) marginal_params_tuple =  
   tuple( k1_params, ..., K_params );

 y ~ gaussian_copula_cholesky(
         marginal_lcdf_tuples | 
         marginal_params_tuple,        
         marginal_lpdf_tuples, // optional, if omitted assumed you are doing this somewhere else
         L);

Jun 26 '25 11:06 spinkney

Is it legal to have a user defined _lcdf/_lpdf input into the signature as functor? I'm not up-to-date with the use of functors in Stan functions.

@WardBrian @SteveBronder

Jun 26 '25 12:06 spinkney

Yes, we currently allow lpdfs in reduce_sum, and I don’t think cdfs would require any different code generation than what we already have

Jun 26 '25 12:06 WardBrian