Gradient of cdf of t distribution crashes Julia

[droodman]

Getting the gradient with respect to the shape parameter of the cumulative distribution function of the t distribution is a hard problem, I think because it involves the incomplete beta function. ForwardDiff can't do, but at least produces an error message. For me it is crashing Zygote:

               _
   _       _ _(_)_     |  Documentation: https://docs.julialang.org
  (_)     | (_) (_)    |
   _ _   _| |_  __ _   |  Type "?" for help, "]?" for Pkg help.
  | | | | | | |/ _` |  |
  | | |_| | | | (_| |  |  Version 1.12.4 (2026-01-06)
 _/ |\__'_|_|_|\__'_|  |  Official https://julialang.org release
|__/                   |

(@v1.12) pkg> activate @temp
  Activating new project at `~/.julia/environments/temp`

(@temp) pkg> add Distributions, Zygote
   Resolving package versions...
    Updating `~/.julia/environments/temp/Project.toml`
  [31c24e10] + Distributions v0.25.123
  [e88e6eb3] + Zygote v0.7.10
    Updating `~/.julia/environments/temp/Manifest.toml`
  [621f4979] + AbstractFFTs v1.5.0
  [79e6a3ab] + Adapt v4.4.0
  [66dad0bd] + AliasTables v1.1.3
  [082447d4] + ChainRules v1.72.6
  [d360d2e6] + ChainRulesCore v1.26.0
  [bbf7d656] + CommonSubexpressions v0.3.1
  [34da2185] + Compat v4.18.1
  [187b0558] + ConstructionBase v1.6.0
  [9a962f9c] + DataAPI v1.16.0
  [864edb3b] + DataStructures v0.19.3
  [e2d170a0] + DataValueInterfaces v1.0.0
  [163ba53b] + DiffResults v1.1.0
  [b552c78f] + DiffRules v1.15.1
  [31c24e10] + Distributions v0.25.123
  [ffbed154] + DocStringExtensions v0.9.5
  [1a297f60] + FillArrays v1.16.0
  [f6369f11] + ForwardDiff v1.3.1
  [46192b85] + GPUArraysCore v0.2.0
  [34004b35] + HypergeometricFunctions v0.3.28
  [7869d1d1] + IRTools v0.4.15
  [92d709cd] + IrrationalConstants v0.2.6
  [82899510] + IteratorInterfaceExtensions v1.0.0
  [692b3bcd] + JLLWrappers v1.7.1
  [2ab3a3ac] + LogExpFunctions v0.3.29
  [1914dd2f] + MacroTools v0.5.16
  [e1d29d7a] + Missings v1.2.0
  [77ba4419] + NaNMath v1.1.3
  [bac558e1] + OrderedCollections v1.8.1
  [90014a1f] + PDMats v0.11.37
  [aea7be01] + PrecompileTools v1.3.3
  [21216c6a] + Preferences v1.5.1
  [43287f4e] + PtrArrays v1.3.0
  [1fd47b50] + QuadGK v2.11.2
  [c1ae055f] + RealDot v0.1.0
  [189a3867] + Reexport v1.2.2
  [ae029012] + Requires v1.3.1
  [79098fc4] + Rmath v0.9.0
  [a2af1166] + SortingAlgorithms v1.2.2
  [dc90abb0] + SparseInverseSubset v0.1.2
  [276daf66] + SpecialFunctions v2.6.1
  [1e83bf80] + StaticArraysCore v1.4.4
  [10745b16] + Statistics v1.11.1
  [82ae8749] + StatsAPI v1.8.0
  [2913bbd2] + StatsBase v0.34.10
  [4c63d2b9] + StatsFuns v1.5.2
  [09ab397b] + StructArrays v0.7.2
  [3783bdb8] + TableTraits v1.0.1
  [bd369af6] + Tables v1.12.1
  [e88e6eb3] + Zygote v0.7.10
  [700de1a5] + ZygoteRules v0.2.7
  [efe28fd5] + OpenSpecFun_jll v0.5.6+0
  [f50d1b31] + Rmath_jll v0.5.1+0
  [56f22d72] + Artifacts v1.11.0
  [2a0f44e3] + Base64 v1.11.0
  [ade2ca70] + Dates v1.11.0
  [8ba89e20] + Distributed v1.11.0
  [b77e0a4c] + InteractiveUtils v1.11.0
  [ac6e5ff7] + JuliaSyntaxHighlighting v1.12.0
  [8f399da3] + Libdl v1.11.0
  [37e2e46d] + LinearAlgebra v1.12.0
  [d6f4376e] + Markdown v1.11.0
  [de0858da] + Printf v1.11.0
  [9a3f8284] + Random v1.11.0
  [ea8e919c] + SHA v0.7.0
  [9e88b42a] + Serialization v1.11.0
  [6462fe0b] + Sockets v1.11.0
  [2f01184e] + SparseArrays v1.12.0
  [f489334b] + StyledStrings v1.11.0
  [4607b0f0] + SuiteSparse
  [fa267f1f] + TOML v1.0.3
  [cf7118a7] + UUIDs v1.11.0
  [4ec0a83e] + Unicode v1.11.0
  [e66e0078] + CompilerSupportLibraries_jll v1.3.0+1
  [4536629a] + OpenBLAS_jll v0.3.29+0
  [05823500] + OpenLibm_jll v0.8.7+0
  [bea87d4a] + SuiteSparse_jll v7.8.3+2
  [8e850b90] + libblastrampoline_jll v5.15.0+0
Precompiling packages finished.
  1 dependency successfully precompiled in 11 seconds. 85 already precompiled.

julia> using Zygote, Distributions

julia> Zygote.gradient(x->cdf(TDist(x),2.), 2.)

[94187] signal 11 (2): Segmentation fault: 11
in expression starting at REPL[5]:1
getType at /Users/julia/.julia/scratchspaces/a66863c6-20e8-4ff4-8a62-49f30b1f605e/agent-cache/default-honeycrisp-XG3Q6T6R70.0/build/default-honeycrisp-XG3Q6T6R70-0/julialang/julia-release-1-dot-12/usr/include/llvm/IR/Value.h:255 [inlined]
emit_unboxed_coercion at /Users/julia/.julia/scratchspaces/a66863c6-20e8-4ff4-8a62-49f30b1f605e/agent-cache/default-honeycrisp-XG3Q6T6R70.0/build/default-honeycrisp-XG3Q6T6R70-0/julialang/julia-release-1-dot-12/src/./intrinsics.cpp:394 [inlined]
emit_unbox at /Users/julia/.julia/scratchspaces/a66863c6-20e8-4ff4-8a62-49f30b1f605e/agent-cache/default-honeycrisp-XG3Q6T6R70.0/build/default-honeycrisp-XG3Q6T6R70-0/julialang/julia-release-1-dot-12/src/./intrinsics.cpp:458
emit_condition at /Users/julia/.julia/scratchspaces/a66863c6-20e8-4ff4-8a62-49f30b1f605e/agent-cache/default-honeycrisp-XG3Q6T6R70.0/build/default-honeycrisp-XG3Q6T6R70-0/julialang/julia-release-1-dot-12/src/codegen.cpp:6065
emit_condition at /Users/julia/.julia/scratchspaces/a66863c6-20e8-4ff4-8a62-49f30b1f605e/agent-cache/default-honeycrisp-XG3Q6T6R70.0/build/default-honeycrisp-XG3Q6T6R70-0/julialang/julia-release-1-dot-12/src/codegen.cpp:6078 [inlined]
emit_function at /Users/julia/.julia/scratchspaces/a66863c6-20e8-4ff4-8a62-49f30b1f605e/agent-cache/default-honeycrisp-XG3Q6T6R70.0/build/default-honeycrisp-XG3Q6T6R70-0/julialang/julia-release-1-dot-12/src/codegen.cpp:9270
jl_emit_code at /Users/julia/.julia/scratchspaces/a66863c6-20e8-4ff4-8a62-49f30b1f605e/agent-cache/default-honeycrisp-XG3Q6T6R70.0/build/default-honeycrisp-XG3Q6T6R70-0/julialang/julia-release-1-dot-12/src/codegen.cpp:9726
jl_emit_codeinst at /Users/julia/.julia/scratchspaces/a66863c6-20e8-4ff4-8a62-49f30b1f605e/agent-cache/default-honeycrisp-XG3Q6T6R70.0/build/default-honeycrisp-XG3Q6T6R70-0/julialang/julia-release-1-dot-12/src/codegen.cpp:9797
jl_emit_codeinst_to_jit_impl at /Users/julia/.julia/scratchspaces/a66863c6-20e8-4ff4-8a62-49f30b1f605e/agent-cache/default-honeycrisp-XG3Q6T6R70.0/build/default-honeycrisp-XG3Q6T6R70-0/julialang/julia-release-1-dot-12/src/jitlayers.cpp:770
jl_add_codeinst_to_jit at /Users/julia/.julia/scratchspaces/a66863c6-20e8-4ff4-8a62-49f30b1f605e/agent-cache/default-honeycrisp-XG3Q6T6R70.0/build/default-honeycrisp-XG3Q6T6R70-0/julialang/julia-release-1-dot-12/src/gf.c:3372
add_codeinsts_to_jit! at ./../usr/share/julia/Compiler/src/typeinfer.jl:1429
typeinf_ext_toplevel at ./../usr/share/julia/Compiler/src/typeinfer.jl:1436 [inlined]
typeinf_ext_toplevel at ./../usr/share/julia/Compiler/src/typeinfer.jl:1444
jfptr_typeinf_ext_toplevel_83167.1 at /Users/davidroodman/.julia/juliaup/julia-1.12.4+0.aarch64.apple.darwin14/lib/julia/sys.dylib (unknown line)
jl_apply at /Users/julia/.julia/scratchspaces/a66863c6-20e8-4ff4-8a62-49f30b1f605e/agent-cache/default-honeycrisp-XG3Q6T6R70.0/build/default-honeycrisp-XG3Q6T6R70-0/julialang/julia-release-1-dot-12/src/./julia.h:2391 [inlined]
jl_type_infer at /Users/julia/.julia/scratchspaces/a66863c6-20e8-4ff4-8a62-49f30b1f605e/agent-cache/default-honeycrisp-XG3Q6T6R70.0/build/default-honeycrisp-XG3Q6T6R70-0/julialang/julia-release-1-dot-12/src/gf.c:462
jl_compile_method_internal at /Users/julia/.julia/scratchspaces/a66863c6-20e8-4ff4-8a62-49f30b1f605e/agent-cache/default-honeycrisp-XG3Q6T6R70.0/build/default-honeycrisp-XG3Q6T6R70-0/julialang/julia-release-1-dot-12/src/gf.c:3512
_jl_invoke at /Users/julia/.julia/scratchspaces/a66863c6-20e8-4ff4-8a62-49f30b1f605e/agent-cache/default-honeycrisp-XG3Q6T6R70.0/build/default-honeycrisp-XG3Q6T6R70-0/julialang/julia-release-1-dot-12/src/gf.c:4002 [inlined]
ijl_invoke at /Users/julia/.julia/scratchspaces/a66863c6-20e8-4ff4-8a62-49f30b1f605e/agent-cache/default-honeycrisp-XG3Q6T6R70.0/build/default-honeycrisp-XG3Q6T6R70-0/julialang/julia-release-1-dot-12/src/gf.c:4017
macro expansion at /Users/davidroodman/.julia/packages/Zygote/55SqB/src/compiler/interface2.jl (unknown line) [inlined]
_pullback at /Users/davidroodman/.julia/packages/Zygote/55SqB/src/compiler/interface2.jl:81 [inlined]
betaccdf at /Users/davidroodman/.julia/packages/StatsFuns/Gkkox/src/distrs/beta.jl:43 [inlined]
_pullback at /Users/davidroodman/.julia/packages/Zygote/55SqB/src/compiler/interface2.jl:0
fdistccdf at /Users/davidroodman/.julia/packages/StatsFuns/Gkkox/src/distrs/fdist.jl:18 [inlined]
tdistcdf at /Users/davidroodman/.julia/packages/StatsFuns/Gkkox/src/distrs/tdist.jl:0 [inlined]
_pullback at /Users/davidroodman/.julia/packages/Zygote/55SqB/src/compiler/interface2.jl:0
getindex at ./tuple.jl:33 [inlined]
#5 at ./REPL[5]:1 [inlined]
_pullback at /Users/davidroodman/.julia/packages/Zygote/55SqB/src/compiler/interface2.jl:0
unknown function (ip: 0x15493c2f7) at (unknown file)
pullback at /Users/davidroodman/.julia/packages/Zygote/55SqB/src/compiler/interface.jl:96
pullback at /Users/davidroodman/.julia/packages/Zygote/55SqB/src/compiler/interface.jl:94 [inlined]
gradient at /Users/davidroodman/.julia/packages/Zygote/55SqB/src/compiler/interface.jl:153
unknown function (ip: 0x13b03c1ff) at (unknown file)
jl_apply at /Users/julia/.julia/scratchspaces/a66863c6-20e8-4ff4-8a62-49f30b1f605e/agent-cache/default-honeycrisp-XG3Q6T6R70.0/build/default-honeycrisp-XG3Q6T6R70-0/julialang/julia-release-1-dot-12/src/./julia.h:2391 [inlined]
do_call at /Users/julia/.julia/scratchspaces/a66863c6-20e8-4ff4-8a62-49f30b1f605e/agent-cache/default-honeycrisp-XG3Q6T6R70.0/build/default-honeycrisp-XG3Q6T6R70-0/julialang/julia-release-1-dot-12/src/interpreter.c:123
eval_stmt_value at /Users/julia/.julia/scratchspaces/a66863c6-20e8-4ff4-8a62-49f30b1f605e/agent-cache/default-honeycrisp-XG3Q6T6R70.0/build/default-honeycrisp-XG3Q6T6R70-0/julialang/julia-release-1-dot-12/src/interpreter.c:194
eval_body at /Users/julia/.julia/scratchspaces/a66863c6-20e8-4ff4-8a62-49f30b1f605e/agent-cache/default-honeycrisp-XG3Q6T6R70.0/build/default-honeycrisp-XG3Q6T6R70-0/julialang/julia-release-1-dot-12/src/interpreter.c:689
jl_interpret_toplevel_thunk at /Users/julia/.julia/scratchspaces/a66863c6-20e8-4ff4-8a62-49f30b1f605e/agent-cache/default-honeycrisp-XG3Q6T6R70.0/build/default-honeycrisp-XG3Q6T6R70-0/julialang/julia-release-1-dot-12/src/interpreter.c:898
jl_toplevel_eval_flex at /Users/julia/.julia/scratchspaces/a66863c6-20e8-4ff4-8a62-49f30b1f605e/agent-cache/default-honeycrisp-XG3Q6T6R70.0/build/default-honeycrisp-XG3Q6T6R70-0/julialang/julia-release-1-dot-12/src/toplevel.c:1035
__repl_entry_eval_expanded_with_loc at /Users/julia/.julia/scratchspaces/a66863c6-20e8-4ff4-8a62-49f30b1f605e/agent-cache/default-honeycrisp-XG3Q6T6R70.0/build/default-honeycrisp-XG3Q6T6R70-0/julialang/julia-release-1-dot-12/usr/share/julia/stdlib/v1.12/REPL/src/REPL.jl:301
jl_apply at /Users/julia/.julia/scratchspaces/a66863c6-20e8-4ff4-8a62-49f30b1f605e/agent-cache/default-honeycrisp-XG3Q6T6R70.0/build/default-honeycrisp-XG3Q6T6R70-0/julialang/julia-release-1-dot-12/src/./julia.h:2391 [inlined]
jl_f_invokelatest at /Users/julia/.julia/scratchspaces/a66863c6-20e8-4ff4-8a62-49f30b1f605e/agent-cache/default-honeycrisp-XG3Q6T6R70.0/build/default-honeycrisp-XG3Q6T6R70-0/julialang/julia-release-1-dot-12/src/builtins.c:881
toplevel_eval_with_hooks at /Users/julia/.julia/scratchspaces/a66863c6-20e8-4ff4-8a62-49f30b1f605e/agent-cache/default-honeycrisp-XG3Q6T6R70.0/build/default-honeycrisp-XG3Q6T6R70-0/julialang/julia-release-1-dot-12/usr/share/julia/stdlib/v1.12/REPL/src/REPL.jl:308
toplevel_eval_with_hooks at /Users/julia/.julia/scratchspaces/a66863c6-20e8-4ff4-8a62-49f30b1f605e/agent-cache/default-honeycrisp-XG3Q6T6R70.0/build/default-honeycrisp-XG3Q6T6R70-0/julialang/julia-release-1-dot-12/usr/share/julia/stdlib/v1.12/REPL/src/REPL.jl:312
toplevel_eval_with_hooks at /Users/julia/.julia/scratchspaces/a66863c6-20e8-4ff4-8a62-49f30b1f605e/agent-cache/default-honeycrisp-XG3Q6T6R70.0/build/default-honeycrisp-XG3Q6T6R70-0/julialang/julia-release-1-dot-12/usr/share/julia/stdlib/v1.12/REPL/src/REPL.jl:305 [inlined]
eval_user_input at /Users/julia/.julia/scratchspaces/a66863c6-20e8-4ff4-8a62-49f30b1f605e/agent-cache/default-honeycrisp-XG3Q6T6R70.0/build/default-honeycrisp-XG3Q6T6R70-0/julialang/julia-release-1-dot-12/usr/share/julia/stdlib/v1.12/REPL/src/REPL.jl:330
repl_backend_loop at /Users/julia/.julia/scratchspaces/a66863c6-20e8-4ff4-8a62-49f30b1f605e/agent-cache/default-honeycrisp-XG3Q6T6R70.0/build/default-honeycrisp-XG3Q6T6R70-0/julialang/julia-release-1-dot-12/usr/share/julia/stdlib/v1.12/REPL/src/REPL.jl:452
#start_repl_backend#41 at /Users/julia/.julia/scratchspaces/a66863c6-20e8-4ff4-8a62-49f30b1f605e/agent-cache/default-honeycrisp-XG3Q6T6R70.0/build/default-honeycrisp-XG3Q6T6R70-0/julialang/julia-release-1-dot-12/usr/share/julia/stdlib/v1.12/REPL/src/REPL.jl:427
start_repl_backend at /Users/julia/.julia/scratchspaces/a66863c6-20e8-4ff4-8a62-49f30b1f605e/agent-cache/default-honeycrisp-XG3Q6T6R70.0/build/default-honeycrisp-XG3Q6T6R70-0/julialang/julia-release-1-dot-12/usr/share/julia/stdlib/v1.12/REPL/src/REPL.jl:424 [inlined]
#run_repl#50 at /Users/julia/.julia/scratchspaces/a66863c6-20e8-4ff4-8a62-49f30b1f605e/agent-cache/default-honeycrisp-XG3Q6T6R70.0/build/default-honeycrisp-XG3Q6T6R70-0/julialang/julia-release-1-dot-12/usr/share/julia/stdlib/v1.12/REPL/src/REPL.jl:653
run_repl at /Users/julia/.julia/scratchspaces/a66863c6-20e8-4ff4-8a62-49f30b1f605e/agent-cache/default-honeycrisp-XG3Q6T6R70.0/build/default-honeycrisp-XG3Q6T6R70-0/julialang/julia-release-1-dot-12/usr/share/julia/stdlib/v1.12/REPL/src/REPL.jl:639
jfptr_run_repl_18516.1 at /Users/davidroodman/.julia/juliaup/julia-1.12.4+0.aarch64.apple.darwin14/share/julia/compiled/v1.12/REPL/u0gqU_vMi9E.dylib (unknown line)
run_std_repl at ./client.jl:478
jfptr_run_std_repl_28137.1 at /Users/davidroodman/.julia/juliaup/julia-1.12.4+0.aarch64.apple.darwin14/lib/julia/sys.dylib (unknown line)
jl_apply at /Users/julia/.julia/scratchspaces/a66863c6-20e8-4ff4-8a62-49f30b1f605e/agent-cache/default-honeycrisp-XG3Q6T6R70.0/build/default-honeycrisp-XG3Q6T6R70-0/julialang/julia-release-1-dot-12/src/./julia.h:2391 [inlined]
jl_f_invokelatest at /Users/julia/.julia/scratchspaces/a66863c6-20e8-4ff4-8a62-49f30b1f605e/agent-cache/default-honeycrisp-XG3Q6T6R70.0/build/default-honeycrisp-XG3Q6T6R70-0/julialang/julia-release-1-dot-12/src/builtins.c:881
run_main_repl at ./client.jl:499
repl_main at ./client.jl:586 [inlined]
_start at ./client.jl:561
jfptr__start_70416.1 at /Users/davidroodman/.julia/juliaup/julia-1.12.4+0.aarch64.apple.darwin14/lib/julia/sys.dylib (unknown line)
jl_apply at /Users/julia/.julia/scratchspaces/a66863c6-20e8-4ff4-8a62-49f30b1f605e/agent-cache/default-honeycrisp-XG3Q6T6R70.0/build/default-honeycrisp-XG3Q6T6R70-0/julialang/julia-release-1-dot-12/src/./julia.h:2391 [inlined]
true_main at /Users/julia/.julia/scratchspaces/a66863c6-20e8-4ff4-8a62-49f30b1f605e/agent-cache/default-honeycrisp-XG3Q6T6R70.0/build/default-honeycrisp-XG3Q6T6R70-0/julialang/julia-release-1-dot-12/src/jlapi.c:971
jl_repl_entrypoint at /Users/julia/.julia/scratchspaces/a66863c6-20e8-4ff4-8a62-49f30b1f605e/agent-cache/default-honeycrisp-XG3Q6T6R70.0/build/default-honeycrisp-XG3Q6T6R70-0/julialang/julia-release-1-dot-12/src/jlapi.c:1139
Allocations: 461796515 (Pool: 461794383; Big: 2132); GC: 283
zsh: segmentation fault  julia

[mcabbott]

For what it's worth, this runs on Julia 1.11 (after compiling for a while) but does not seem to be correct:

julia> using Zygote, Distributions

julia> (cdf(TDist(2.01),2.) - cdf(TDist(2.0),2.) )/0.01
0.03171902653607006

julia> (cdf(TDist(2.0001),2.) - cdf(TDist(2.0),2.) )/0.0001
0.03185638729030771

julia> Zygote.gradient(x->cdf(TDist(x),2.), 2.)
(0.012462187893996536,)

julia> VERSION
v"1.11.8"

julia> using Plots

julia> plot(x->cdf(TDist(x),2.), 1, 3)

julia> plot!(x -> 0.0124621878*(x-2) + cdf(TDist(2.0),2.), 1, 3)

[droodman]

BTW, a chain rule is being added for beta_inc now to SpecialFunctions.jl, so this specific problem may go away. Of course if it's crashing Julia, that still strongly suggests there's a bug in Zygote. I'm using SpecialFunctions v2.6.1.