Beta-CDF derivative

The problem

The regularised incomplete beta I_x(α, β) is the Beta CDF, and SpecialFunctions.beta_inc computes it accurately across every regime. Its ChainRule, though, leaves the partials with respect to the shape parameters α and β unimplemented. A Beta CDF differentiated in its shape therefore has no derivative on any ChainRules-based backend, and the forward-mode Dual and Enzyme paths hit the same wall — the beta_inc analogue of the Gamma-CDF derivative gap.

This also blocks any distribution whose CDF routes through beta_inc, not just Beta itself — Student-t and LogLogistic among them. The Student-t case is covered here, in _t_cdf and its companions; see Student-t on the hooks page.

The fix

_beta_cdf(α, β, x) evaluates I_x(α, β) on the primal path through beta_inc, and the extensions supply the derivative analytically:

• _rib_value_and_partials computes the missing shape partials by term-by-term differentiation of the continued-fraction form Boik & Robinson-Cox (1998) derive for I_x(p, q), evaluated with the standard modified Lentz-style recurrence, and ported from the reference C implementation in Caner Türkmen's betaincder.

• _beta_cdf_value_and_partials returns the primal and the (dα, dβ, dx) partials from one helper, applying the reflection symmetry I_x(α,β) = 1 - I_{1-x}(β,α) when x > α/(α+β) (the continued fraction is only valid on the other side), so the ChainRules rule, the ForwardDiff Dual methods, and the Enzyme rule share the same formulas.

• EpiAwareADToolsChainRulesCoreExt defines the reverse-mode rrule and forward-mode frule; EpiAwareADToolsReverseDiffExt and EpiAwareADToolsMooncakeExt lift these into their backends; EpiAwareADToolsForwardDiffExt adds Dual methods directly; and EpiAwareADToolsEnzymeExt adds a direct Enzyme rule.

_t_cdf(ν, x) is a second consumer of the same machinery rather than a fix of its own. A Student-t CDF is the regularised incomplete beta at ν / (ν + x^2), so _t_cdf composes over _beta_cdf and every backend rule listed above carries through the composition unchanged; no per-backend Student-t rule exists, and the AD suite sweeps the whole matrix to check that. Two details are its own. It always evaluates the smaller of the two tails, so logcdf stays accurate arbitrarily deep into either tail instead of reconstructing a small number as 1 - F. And at x == 0 the beta argument sits at 1, where _beta_cdf's x-partial diverges while the inner derivative is exactly zero, so the chain rule asks for 0 * Inf; the guard there returns the constant 1/2 plus a term linear in x carrying the finite true partial. That guard tests primal rather than the argument itself, since ForwardDiff's == and iszero compare a Dual's partials as well and a seeded zero therefore tests unequal to 0.

ad_safe.jl's pdf_ad_safe(::Beta) override is a separate, narrower fix in the same family: it routes around a confirmed-wrong Enzyme rule for LogExpFunctions.xlog1py's first argument (the stock pdf(::Beta) path) by computing the density directly from log/log1p, rather than reaching for _beta_cdf.

This machinery is internal. Consumers reach it through the AD-safe hooks; the Internal API page carries the full docstrings.

Upstream target

The whole family stands in for a differentiable beta_inc. There is no tracking issue open against SpecialFunctions.jl for the shape-parameter partials at the time of writing (unlike gamma_inc's issue #531). Once beta_inc carries a complete ChainRule, _beta_cdf and its rules are deleted and the hooks route through the stock CDF.