[avehtari]

Fixes #3329.

This PR improves wiener_lpdf (5- and 7-parameter), wiener_lcdf_unnorm and wiener_lccdf_unnorm (4- and 7-parameter).

This PR was assisted by Claude

Summary

This PR fixes the underflow of #3329, and other accuracy defects of the Wiener functions that I found while I tested a fix for it. It is an alternative to #3435 (comparison: #3435 (comment)).

Functions and files (in stan/math/prim/prob; each function has an optional last argument for the precision):

short name	Stan function	file	computes with
none	wiener_lpdf(y | alpha, tau, beta, delta)	wiener_lpdf.hpp	its own series; not changed, because it is accurate (300 points against mpmath: value error at most 3.6e-9, gradients 5.2e-8)
5-parameter lpdf	wiener_lpdf(y | a, t0, w, v, sv)	wiener5_lpdf.hpp	series of the density and its gradients
7-parameter lpdf	wiener_lpdf(y | a, t0, w, v, sv, sw, st0)	wiener_full_lpdf.hpp	hcubature integrals of the 5-parameter density over sw and st0; the 5-parameter lpdf if sw = st0 = 0
4-parameter lcdf	wiener_lcdf_unnorm(y, a, t0, w, v)	wiener4_lcdf_unnorm.hpp	series of the cdf and its gradients, logMill
4-parameter lccdf	wiener_lccdf_unnorm(y, a, t0, w, v)	wiener4_lccdf_unnorm.hpp	P - F from the 4-parameter cdf, or the large-time series of P - F (new); P is the probability of the upper boundary
7-parameter lcdf	wiener_lcdf_unnorm(y, a, t0, w, v, sv, sw, st0)	wiener_full_lcdf_unnorm.hpp	hcubature integrals of the 4-parameter cdf over sv, sw and st0; the 4-parameter lcdf if sv = sw = st0 = 0
7-parameter lccdf	wiener_lccdf_unnorm(y, a, t0, w, v, sv, sw, st0)	wiener_full_lccdf_unnorm.hpp	the same integrals of the 4-parameter ccdf; the 4-parameter lccdf if sv = sw = st0 = 0

Because of these calls, fixes in the 5-parameter functions also reach the 7-parameter lpdf, and fixes in the 4-parameter cdf also reach the 4-parameter lccdf and the 7-parameter lcdf and lccdf (table below).

7-parameter lpdf, #3329. The integrands are 5-parameter densities on the natural scale, which underflow to 0 below a log density of about -708. This PR scales them per observation by exp(-c), with c the largest 5-parameter log density on a 3 x 3 grid of the integration box (17 x 17 if the scaled integral is outside [1e-100, 1e100]). The log density is c + log(integral); the partials are ratios of integrals, so c cancels, and the series tolerances are relative to exp(c). wiener5_density and wiener5_grad_* get an optional log_shift argument (default 0). All autodiff types use the same code path with the existing hcubature.

hcubature (used only by the three 7-parameter functions). Two defects: the loop condition used the error and value of the first box, so once that box failed the test, the loop ran to the evaluation limit; and a split box stayed in the list with error 0 and could be split and counted again. The integration now stops at the requested tolerance, which makes the 7-parameter lpdf 11 to 14 times faster when st0 > 0.

d/dsw and d/dst0 of the 7-parameter lpdf.

• The d/dsw integral over the st0 variable used the limits of the sw variable: for (y - t0)/st0 < 0.004, d/dsw was -1/sw.
• With sw > 0 and st0 = 0, d/dsw used an absolute series tolerance (relative errors up to 2e3 where the density is small).
• The end-point forms $\partial \log p / \partial\,\mathrm{sw} = I_\mathrm{sw} / I - 1/\mathrm{sw}$ and $\partial \log p / \partial\,\mathrm{st0} = f / (\mathrm{st0}\, I) - 1/\mathrm{st0}$ ($I$ the density integral, $I_\mathrm{sw}$ and $f$ the integrals at the ends of the sw and st0 ranges) multiply the errors of two integrations by 1/sw (1/st0); in develop the d/dsw error at sw = 1e-8 is up to 1.1. When both sw and st0 are non-zero and sw (st0) is below 1e-3, this PR differentiates under the integral instead (the idea of the fallback in Fix full Wiener density underflow #3435): with $x_w, x_t \in [0, 1]$, $\partial \log p / \partial\,\mathrm{sw} = \int (x_w - \tfrac{1}{2})\, \partial_w p_5 \, dx \,/ \int p_5 \, dx$, with the integrand folded at $x_w = \tfrac{1}{2}$ so that its odd part cancels, and $\partial \log p / \partial\,\mathrm{st0} = -\int x_t\, \partial_t p_5 \, dx \,/ \int p_5 \, dx$. Above 1e-3 the end-point forms stay: their error is about 3e-10/sw at the default precision, and the score integrals cost one more 2-D integral.
• The density integral uses a relative tolerance of at most 0.9e-6: the log density error at the default precision goes from 1.7e-6 to 1.0e-7.

5-parameter lpdf. Three faults on the path for small densities, with value errors up to 1e-2 at ordinary points: assign_err took its argument by value, so the re-estimate in estimate_with_err_check used the old tolerance; with LogResult == false the check compared |log density| with the log tolerance and never fired; and the gradient series of t, a and w used an absolute tolerance.

4-parameter lcdf. The small-time series of the derivatives with respect to a and w had sign errors: gradient errors up to 1.6e-4, and up to 3e-3 in the 7-parameter lcdf and lccdf. logMill now uses erfcx for x >= 0.

4-parameter lccdf. P - F loses all digits where F is close to P (-inf at 8 of 300 ordinary points). Where F >= P/2 the function now uses the large-time series of P - F and its derivatives.

7-parameter lcdf and lccdf. The integration tolerance is at most 0.9e-6, and the d/dt series tolerance is relative to the value. The lccdf now includes the trials whose non-decision time ends after y; they contribute P, and without them the log ccdf was off by up to 1.4 where st0 > y - t0. The lccdf integrates the 4-parameter ccdf, so it is finite down to a ccdf of about 1e-308; develop formed P - F and returned -inf below about 1e-16 of P (for example at log ccdf -39, -120 and -299). Where the value integral underflows to 0, the value is -inf and the gradient integrals are skipped.

Fixes in shared code reach more than one function:

fix	4-p lcdf	4-p lccdf	5-p lpdf	7-p lpdf	7-p lcdf	7-p lccdf
hcubature: loop condition, split box counted twice				x	x	x
assign_err by reference	x	x	x	x	x	x
estimate_with_err_check with LogResult == false	x		x	(x)	(x)
tolerance of the 5-parameter gradient series (t, a, w)			x	(x)
logMill with erfcx	x	x			x	x
sign errors in the 4-parameter cdf gradients (a, w)	x	x			x	x
4-parameter ccdf from the large-time series of P - F		x				x
log shift of the integrand (#3329)				x
d/dsw: integration limits, series tolerance				x
d/dsw, d/dst0 by differentiation under the integral (small sw, st0)				x
integration tolerance at most 0.9e-6				x	x	x
d/dt series tolerance relative to the value					x	x
guards: no NaN where the integrand weight is 0; -inf without gradient integrals where the value underflows					x	x
lccdf: mass of the trials with t0 + τ > y						x

(x): only through the call to the 5-parameter lpdf (sw = st0 = 0) or the 4-parameter lcdf (sv = sw = st0 = 0).

Accuracy, develop -> this PR (largest error over the test points; log scale for values, relative to max(|ref|, 1) for gradients; default precision):

function	value	gradients
4-parameter lcdf	1.1e-8 -> 1.1e-8	1.6e-4 -> 9.5e-9
4-parameter lccdf	-inf -> 1.4e-14	-inf -> 2.9e-9
5-parameter lpdf	1.1e-2 -> 4.6e-12	1.8e-2 -> 2.1e-14
7-parameter lpdf, ordinary	1.7e-6 -> 1.0e-7	2.3e-5 -> 2.1e-5
7-parameter lpdf, #3329 point and scan	-inf -> 4.8e-11	d/dsw -1e8 -> 5.3e-10
7-parameter lpdf, sw or st0 = 1e-8		1.1 -> 3.3e-9
7-parameter lcdf, 1-2 integration dims	3.1e-7 -> 3.1e-7	2.9e-3 -> 3.9e-7
7-parameter lccdf, 1-2 integration dims	-inf, 1.3 -> 8.4e-9	6.3e2 -> 2.7e-8

References: mpmath (50 digits) for the 4- and 5-parameter functions; for the 7-parameter functions, my own adaptive Gauss-Kronrod integration of the 5-parameter density (not hcubature), with gradients under the integral or by finite differences. Test points: sets of 300 random points (a 0.5-3, v -5..5 or -3..3, w 0.2-0.8, sv 0-1.5, sw up to 0.8 of its range, st0 up to 0.3, y - t0 0.05-3); the #3329 scan, 16 points with y - t0 from 3e-6 to 3e-5; 240 points with sw or st0 from 1e-8 to 0.1.

Run time with gradients, this PR / develop: 4-parameter lcdf 0.89, lccdf 0.62; 5-parameter lpdf 1.09-1.13; 7-parameter lpdf 0.07 (sw and st0), 0.09 (st0 only), 1.05 (sw only); 7-parameter lcdf 0.04-0.67, lccdf 0.06-0.87.

Not fixed (follow-up PR): the 7-parameter lcdf and lccdf integrate natural-scale values and return -inf below a cdf or ccdf of about 1e-308 (the #3329 mechanism; for the lccdf this PR moves the limit from about 1e-16 of P to 1e-308). With sv, sw and st0 all non-zero, the 3-dimensional integrals reach the evaluation limit of 6000 (errors up to 4.3e-5, as in develop).

Diff against develop (43bc568): code 7 files, +542/−149 (most in wiener4_lccdf_unnorm.hpp +227/−49 and wiener_full_lpdf.hpp +162/−35); tests 6 files, +454/−10 (4 new files, 198 lines; hcubature_test.cpp +47; wiener_full_test.cpp +209/−10).

Tests

Each new test fails with the develop versions of the 7 headers:

• hcubature_test: a step integrand, where develop counts split boxes twice (99.5 instead of 0.5); sqrt(x) and a 2-D Gaussian, which must stop at the tolerance.
• wiener4_lcdf_unnorm_test (prim: logMill; rev: gradients), wiener4_lccdf_unnorm_test (small ccdf) and wiener5_lpdf_test (small densities), against mpmath.
• rev wiener_full_test: the wiener_lpdf full model returns -inf for valid-support inputs in variability branches #3329 point and scan (value, d/dy, d/dst0, d/dsw); d/dsw with st0 > y - t0; the 7-parameter lccdf with st0 > y - t0, with a small ccdf, and at log ccdf -120 and -299 (develop: -inf); the 7-parameter lcdf and lccdf below the smallest double (-inf, without the gradient integrals).

Changed test: the third point of mathRevCorrectValues_wiener_lccdf_unnorm came from WienR, which computes cdf(inf) - cdf and loses digits there (the ccdf is 1.6e-14 of P): WienR -33.8123, mpmath -33.7956003. The row now has the mpmath values.

All Wiener prim, rev and mix suites and the hcubature suite pass.

Side Effects

None.

Release notes

Improved the accuracy and speed of wiener_lpdf (5- and 7-parameter), wiener_lcdf_unnorm and wiener_lccdf_unnorm (4- and 7-parameter).

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

  • 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 )
stat_comp_benchmarks/benchmarks/gp_regr/gen_gp_data.stan	0.04	0.04	0.96	-4.1% slower
stat_comp_benchmarks/benchmarks/gp_regr/gp_regr.stan	0.16	0.17	0.99	-0.73% slower
stat_comp_benchmarks/benchmarks/garch/garch.stan	0.85	0.89	0.96	-4.27% slower
stat_comp_benchmarks/benchmarks/low_dim_corr_gauss/low_dim_corr_gauss.stan	0.01	0.02	0.98	-2.13% slower
stat_comp_benchmarks/benchmarks/eight_schools/eight_schools.stan	0.11	0.11	1.0	-0.1% slower
stat_comp_benchmarks/benchmarks/low_dim_gauss_mix/low_dim_gauss_mix.stan	8.03	8.02	1.0	0.18% faster
stat_comp_benchmarks/benchmarks/gp_pois_regr/gp_pois_regr.stan	3.8	3.75	1.01	1.37% faster
stat_comp_benchmarks/benchmarks/pkpd/one_comp_mm_elim_abs.stan	33.01	32.53	1.01	1.44% faster
stat_comp_benchmarks/benchmarks/pkpd/sim_one_comp_mm_elim_abs.stan	0.38	0.37	1.02	2.17% faster
stat_comp_benchmarks/benchmarks/irt_2pl/irt_2pl.stan	6.92	6.93	1.0	-0.02% slower
stat_comp_benchmarks/benchmarks/sir/sir.stan	136.43	138.35	0.99	-1.41% slower
stat_comp_benchmarks/benchmarks/arK/arK.stan	3.56	3.41	1.05	4.36% faster
stat_comp_benchmarks/benchmarks/arma/arma.stan	0.61	0.61	1.0	0.17% faster
stat_comp_benchmarks/benchmarks/low_dim_gauss_mix_collapse/low_dim_gauss_mix_collapse.stan	25.89	25.83	1.0	0.23% faster
performance.compilation	404.47	401.84	1.01	0.65% faster
Mean result: 0.9990045408676703

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Machine information

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Release:	20.04
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CPU:

Architecture:                            x86_64
CPU op-mode(s):                          32-bit, 64-bit
Byte Order:                              Little Endian
Address sizes:                           52 bits physical, 57 bits virtual
CPU(s):                                  192
On-line CPU(s) list:                     0-191
Thread(s) per core:                      2
Core(s) per socket:                      48
Socket(s):                               2
NUMA node(s):                            2
Vendor ID:                               AuthenticAMD
CPU family:                              25
Model:                                   17
Model name:                              AMD EPYC 9474F 48-Core Processor
Stepping:                                1
Frequency boost:                         enabled
CPU MHz:                                 1496.938
CPU max MHz:                             4114.4229
CPU min MHz:                             1500.0000
BogoMIPS:                                7189.39
Virtualization:                          AMD-V
L1d cache:                               3 MiB
L1i cache:                               3 MiB
L2 cache:                                96 MiB
L3 cache:                                512 MiB
NUMA node0 CPU(s):                       0-47,96-143
NUMA node1 CPU(s):                       48-95,144-191
Vulnerability Gather data sampling:      Not affected
Vulnerability Indirect target selection: Not affected
Vulnerability Itlb multihit:             Not affected
Vulnerability L1tf:                      Not affected
Vulnerability Mds:                       Not affected
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Vulnerability Mmio stale data:           Not affected
Vulnerability Old microcode:             Not affected
Vulnerability Reg file data sampling:    Not affected
Vulnerability Retbleed:                  Not affected
Vulnerability Spec rstack overflow:      Mitigation; Safe RET
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; Enhanced / Automatic IBRS; IBPB conditional; STIBP always-on; PBRSB-eIBRS Not affected; BHI Not affected
Vulnerability Srbds:                     Not affected
Vulnerability Tsa:                       Mitigation; Clear CPU buffers
Vulnerability Tsx async abort:           Not affected
Vulnerability Vmscape:                   Mitigation; IBPB before exit to userspace
Flags:                                   fpu vme de pse tsc msr pae mce cx8 apic sep mtrr pge mca cmov pat pse36 clflush mmx fxsr sse sse2 ht syscall nx mmxext fxsr_opt pdpe1gb rdtscp lm constant_tsc rep_good amd_lbr_v2 nopl xtopology nonstop_tsc cpuid extd_apicid aperfmperf rapl pni pclmulqdq monitor ssse3 fma cx16 pcid sse4_1 sse4_2 x2apic movbe popcnt aes xsave avx f16c rdrand lahf_lm cmp_legacy svm extapic cr8_legacy abm sse4a misalignsse 3dnowprefetch osvw ibs skinit wdt tce topoext perfctr_core perfctr_nb bpext perfctr_llc mwaitx cpb cat_l3 cdp_l3 hw_pstate ssbd mba perfmon_v2 ibrs ibpb stibp ibrs_enhanced vmmcall fsgsbase bmi1 avx2 smep bmi2 erms invpcid cqm rdt_a avx512f avx512dq rdseed adx smap avx512ifma clflushopt clwb avx512cd sha_ni avx512bw avx512vl xsaveopt xsavec xgetbv1 xsaves cqm_llc cqm_occup_llc cqm_mbm_total cqm_mbm_local user_shstk avx512_bf16 clzero irperf xsaveerptr rdpru wbnoinvd amd_ppin cppc arat npt lbrv svm_lock nrip_save tsc_scale vmcb_clean flushbyasid decodeassists pausefilter pfthreshold avic v_vmsave_vmload vgif x2avic v_spec_ctrl vnmi avx512vbmi umip pku ospke avx512_vbmi2 gfni vaes vpclmulqdq avx512_vnni avx512_bitalg avx512_vpopcntdq la57 rdpid overflow_recov succor smca fsrm flush_l1d debug_swap

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