Module probability 

Functions

beta_quantile

Quantile (inverse CDF) of a Beta distribution with shape parameters a > 0 and b > 0 at probability p: the value x in [0, 1] with I_x(a, b) = p, where I is the regularized incomplete beta.

cone_boundary_log_factor

Log of the standardized one-sided truncated-Gaussian boundary factor for the constrained-LAML cone correction (gam#2306 §4).

cone_boundary_log_factor_and_derivatives

cone_boundary_log_factor together with its exact partial derivatives in the standardized arguments — the pieces the outer ρ-gradient chains through (μ̃, h̃, s)(ρ) (gam#2306 §4 “the g-factors differentiate in closed form”). With ξ = b − a and the Mills ratio λ(ξ) = φ(ξ)/Φ(ξ):

erfcx_nonnegative

Scaled complementary error function erfcx(x) = exp(x²) · erfc(x), specialized to the closed domain x ∈ [0, +∞].

log1mexp_positive

Computes log(1 - exp(-a)) for a >= 0 without cancellation.

normal_cdf

Standard normal CDF Phi(x) evaluated via the exact special-function identity

normal_logcdf

Numerically stable ln Φ(x) for the standard normal CDF. For x ≥ 0, evaluates ln(1 - 0.5 erfc(x/sqrt(2))) with ln_1p, retaining the small negative result after Φ(x) itself rounds to one. For x < 0, rewrites ln Φ(x) = −u² + ln(½·erfcx(u)), u = −x/√2, which preserves digits throughout the representable left tail without a probability floor. Returns the corresponding IEEE limit at infinities and propagates NaN.

normal_logcdf_derivatives

Stable value and first four derivatives of ln Φ(x).

normal_logsf

Numerically stable ln(1 − Φ(x)) = ln Φ(−x) for the standard normal survival function. Delegates to normal_logcdf(-x) so the deep-right tail benefits from the same erfcx-based representation.

normal_pdf

Standard normal PDF phi(x).

signed_log_sum_exp

Numerically stable signed log-sum-exp. Given pairs (log|aⱼ|, sign(aⱼ)) (with signs[j] ∈ {−1, 0, +1}), returns (log|S|, sign(S)) for S = Σⱼ signs[j]·exp(log_mags[j]). Positive and negative magnitudes are reduced separately with the standard log-sum-exp trick (subtract the max, sum, log, add back); the two partial sums are then combined via log(|p − n|) = max(log p, log n) + log1mexp(|log p − log n|), preserving accuracy even when p ≈ n (catastrophic cancellation regime). When all signs are zero or all magnitudes are −∞, returns (NEG_INFINITY, 0.0).

signed_probit_logcdf_and_mills_ratio

Joint evaluation of ln Φ(x) and the Mills-ratio analogue φ(x) / Φ(x), signed for the symmetric branch. Used by the latent probit families where the inverse-link gradient needs the ratio and the likelihood needs the log-CDF on the same x; computing both in one call shares the erfcx evaluation that dominates the cost in the deep tail.

standard_normal_quantile

Standard normal quantile Φ⁻¹(p) using Acklam’s rational approximation.

standard_normal_quantile_from_log_cdf

Standard normal quantile from log_p = ln Φ(x).