gam 0.3.125

//! GLM working-vector updates: the family-dispatched `update_glmvectors*`
//! entry points and the working-weight / Newton-curvature derivatives w.r.t.
//! `\eta`.

use super::*;
use crate::types::MIN_WEIGHT;

pub fn update_glmvectors(
    y: ArrayView1<f64>,
    eta: &Array1<f64>,
    inverse_link: &InverseLink,
    priorweights: ArrayView1<f64>,
    mu: &mut Array1<f64>,
    weights: &mut Array1<f64>,
    z: &mut Array1<f64>,
    derivatives: Option<WorkingDerivativeBuffersMut<'_>>,
) -> Result<(), EstimationError> {
    let link = inverse_link.link_function();

    // Fast vectorized path for pure logit (most common binomial link).
    // Avoids per-element function dispatch; structured for SIMD auto-vectorization.
    if matches!(link, LinkFunction::Logit)
        && inverse_link.mixture_state().is_none()
        && inverse_link.sas_state().is_none()
    {
        if let Some(mut derivs) = derivatives {
            let WorkingSlices {
                mu: mu_s,
                weights: weights_s,
                z: z_s,
            } = working_slices(mu, weights, z);
            let WorkingDerivSlices {
                c: c_s,
                d: d_s,
                dmu: dmu_s,
                d2: d2_s,
                d3: d3_s,
            } = working_deriv_slices(&mut derivs);
            mu_s.par_iter_mut()
                .zip(weights_s.par_iter_mut())
                .zip(z_s.par_iter_mut())
                .zip(c_s.par_iter_mut())
                .zip(d_s.par_iter_mut())
                .zip(dmu_s.par_iter_mut())
                .zip(d2_s.par_iter_mut())
                .zip(d3_s.par_iter_mut())
                .enumerate()
                .for_each(
                    |(i, (((((((mu_o, w_o), z_o), c_o), d_o), dmu_o), d2_o), d3_o))| {
                        let eta_raw = eta[i];
                        let eta_c = eta_raw.clamp(-ETA_CLAMP, ETA_CLAMP);
                        let jet = logit_inverse_link_jet5(eta_c);
                        let geom = bernoulli_logit_geometry_from_jet(
                            eta_raw,
                            eta_c,
                            y[i],
                            priorweights[i],
                            jet,
                            true,
                        );
                        *mu_o = geom.mu;
                        *w_o = geom.weight;
                        *z_o = geom.z;
                        *c_o = geom.c;
                        *d_o = geom.d;
                        *dmu_o = jet.d1;
                        *d2_o = jet.d2;
                        *d3_o = jet.d3;
                    },
                );
        } else {
            let WorkingSlices {
                mu: mu_s,
                weights: weights_s,
                z: z_s,
            } = working_slices(mu, weights, z);
            mu_s.par_iter_mut()
                .zip(weights_s.par_iter_mut())
                .zip(z_s.par_iter_mut())
                .enumerate()
                .for_each(|(i, ((mu_o, w_o), z_o))| {
                    let eta_raw = eta[i];
                    let eta_c = eta_raw.clamp(-ETA_CLAMP, ETA_CLAMP);
                    let jet = logit_inverse_link_jet5(eta_c);
                    let geom = bernoulli_logit_geometry_from_jet(
                        eta_raw,
                        eta_c,
                        y[i],
                        priorweights[i],
                        jet,
                        true,
                    );
                    *mu_o = geom.mu;
                    *w_o = geom.weight;
                    *z_o = geom.z;
                });
        }
        return Ok(());
    }

    match link {
        LinkFunction::Logit
        | LinkFunction::Probit
        | LinkFunction::CLogLog
        | LinkFunction::Sas
        | LinkFunction::BetaLogistic => {
            // On logit geometry, freeze higher η-derivatives in nonsmooth
            // regions so PIRLS and outer derivative code differentiate the
            // same piecewise-smooth surface.
            let zero_on_nonsmooth = matches!(link, LinkFunction::Logit);
            if let Some(mut derivs) = derivatives {
                let WorkingSlices {
                    mu: mu_s,
                    weights: weights_s,
                    z: z_s,
                } = working_slices(mu, weights, z);
                let WorkingDerivSlices {
                    c: c_s,
                    d: d_s,
                    dmu: dmu_s,
                    d2: d2_s,
                    d3: d3_s,
                } = working_deriv_slices(&mut derivs);
                mu_s.par_iter_mut()
                    .zip(weights_s.par_iter_mut())
                    .zip(z_s.par_iter_mut())
                    .zip(c_s.par_iter_mut())
                    .zip(d_s.par_iter_mut())
                    .zip(dmu_s.par_iter_mut())
                    .zip(d2_s.par_iter_mut())
                    .zip(d3_s.par_iter_mut())
                    .enumerate()
                    .try_for_each(
                        |(
                            i,
                            (((((((mu_o, w_o), z_o), c_o), d_o), dmu_o), d2_o), d3_o),
                        )|
                         -> Result<(), EstimationError> {
                            let eta_used = eta_for_observed_hessian_jet(inverse_link, eta[i]);
                            if matches!(link, LinkFunction::Logit) {
                                let jet = logit_inverse_link_jet5(eta_used);
                                let geom = bernoulli_logit_geometry_from_jet(
                                    eta[i],
                                    eta_used,
                                    y[i],
                                    priorweights[i],
                                    jet,
                                    zero_on_nonsmooth,
                                );
                                *mu_o = geom.mu;
                                *w_o = geom.weight;
                                *z_o = geom.z;
                                *c_o = geom.c;
                                *d_o = geom.d;
                                *dmu_o = jet.d1;
                                *d2_o = jet.d2;
                                *d3_o = jet.d3;
                            } else {
                                let jet = standard_inverse_link_jet(inverse_link, eta_used)?;
                                let geom = bernoulli_geometry_from_jet(
                                    eta[i],
                                    eta_used,
                                    y[i],
                                    priorweights[i],
                                    jet,
                                );
                                *mu_o = geom.mu;
                                *w_o = geom.weight;
                                *z_o = geom.z;
                                *c_o = geom.c;
                                *d_o = geom.d;
                                *dmu_o = jet.d1;
                                *d2_o = jet.d2;
                                *d3_o = jet.d3;
                            }
                            Ok(())
                        },
                    )?;
            } else {
                let WorkingSlices {
                    mu: mu_s,
                    weights: weights_s,
                    z: z_s,
                } = working_slices(mu, weights, z);
                mu_s.par_iter_mut()
                    .zip(weights_s.par_iter_mut())
                    .zip(z_s.par_iter_mut())
                    .enumerate()
                    .try_for_each(|(i, ((mu_o, w_o), z_o))| -> Result<(), EstimationError> {
                        let eta_used = eta_for_observed_hessian_jet(inverse_link, eta[i]);
                        if matches!(link, LinkFunction::Logit) {
                            let jet = logit_inverse_link_jet5(eta_used);
                            let geom = bernoulli_logit_geometry_from_jet(
                                eta[i],
                                eta_used,
                                y[i],
                                priorweights[i],
                                jet,
                                zero_on_nonsmooth,
                            );
                            *mu_o = geom.mu;
                            *w_o = geom.weight;
                            *z_o = geom.z;
                        } else {
                            let jet = standard_inverse_link_jet(inverse_link, eta_used)?;
                            let geom = bernoulli_geometry_from_jet(
                                eta[i],
                                eta_used,
                                y[i],
                                priorweights[i],
                                jet,
                            );
                            *mu_o = geom.mu;
                            *w_o = geom.weight;
                            *z_o = geom.z;
                        }
                        Ok(())
                    })?;
            }
            Ok(())
        }
        LinkFunction::Identity => {
            write_identityworking_state(y, eta, priorweights, mu, weights, z, derivatives);
            Ok(())
        }
        LinkFunction::Log => {
            write_poisson_log_working_state(y, eta, priorweights, mu, weights, z, derivatives);
            Ok(())
        }
    }
}

/// Family-dispatched GLM vector update helper.
#[inline]
pub fn update_glmvectors_by_family(
    y: ArrayView1<f64>,
    eta: &Array1<f64>,
    likelihood: &GlmLikelihoodSpec,
    priorweights: ArrayView1<f64>,
    mu: &mut Array1<f64>,
    weights: &mut Array1<f64>,
    z: &mut Array1<f64>,
) -> Result<(), EstimationError> {
    likelihood.irls_update(y, eta, priorweights, mu, weights, z, None, None)
}

pub(crate) fn integrated_inverse_link_from_family(
    spec: &LikelihoodSpec,
    mixture_link_state: Option<&MixtureLinkState>,
    sas_link_state: Option<&SasLinkState>,
) -> Result<InverseLink, EstimationError> {
    match (&spec.response, &spec.link) {
        (ResponseFamily::Binomial, InverseLink::Standard(StandardLink::Logit))
        | (ResponseFamily::Binomial, InverseLink::Standard(StandardLink::Probit))
        | (ResponseFamily::Binomial, InverseLink::Standard(StandardLink::CLogLog)) => {
            Ok(spec.link.clone())
        }
        (ResponseFamily::Binomial, InverseLink::Sas(_)) => {
            let state = sas_link_state.ok_or_else(|| {
                EstimationError::InvalidInput(
                    "Integrated BinomialSas update requires explicit SasLinkState".to_string(),
                )
            })?;
            Ok(InverseLink::Sas(*state))
        }
        (ResponseFamily::Binomial, InverseLink::BetaLogistic(_)) => {
            let state = sas_link_state.ok_or_else(|| {
                EstimationError::InvalidInput(
                    "Integrated BinomialBetaLogistic update requires explicit SasLinkState"
                        .to_string(),
                )
            })?;
            Ok(InverseLink::BetaLogistic(*state))
        }
        (ResponseFamily::Binomial, InverseLink::Mixture(_)) => {
            let state = mixture_link_state.ok_or_else(|| {
                EstimationError::InvalidInput(
                    "Integrated BinomialMixture update requires explicit MixtureLinkState"
                        .to_string(),
                )
            })?;
            Ok(InverseLink::Mixture(state.clone()))
        }
        _ => Err(EstimationError::InvalidInput(format!(
            "Integrated link-runtime update is not supported for likelihood (response={:?}, link={:?})",
            spec.response, spec.link
        ))),
    }
}

/// Updates Bernoulli-family GLM working vectors using an integrated
/// (uncertainty-aware) inverse-link runtime.
///
/// For the calibrator, we model:
///   μᵢ = E[σ(ηᵢ + ε)] where ε ~ N(0, SEᵢ²)
///
/// This integrates out uncertainty in the base prediction, giving a coherent
/// probabilistic treatment of measurement error. The effect is that steep
/// calibration adjustments are automatically attenuated when SE is high.
///
/// Uses the general IRLS formula (not canonical shortcut):
///   weight = prior × (dμ/dη)² / (μ(1-μ))
///   z = η + (y - μ) / (dμ/dη)
///
/// Derivation of the integrated quantities:
/// Let the uncertain latent predictor at row i be
///   eta_tilde_i = eta_i + eps_i,   eps_i ~ N(0, se_i^2).
/// Then the integrated mean used by PIRLS is
///   mu_i = E[g^{-1}(eta_tilde_i)].
/// Because the Gaussian family is a location family,
///   dmu_i / deta_i
///   = d/deta_i E[g^{-1}(eta_i + eps_i)]
///   = E[(g^{-1})'(eta_i + eps_i)].
/// That derivative is the exact object needed in the general GLM scoring update:
///   W_i = prior_i * (dmu_i/deta_i)^2 / Var(Y_i | mu_i),
///   z_i = eta_i + (y_i - mu_i) / (dmu_i/deta_i).
/// So any future exact link-specific replacement only needs to preserve the
/// contract
///   (eta_i, se_i) -> (mu_i, dmu_i/deta_i),
/// and the rest of the PIRLS machinery remains unchanged.
///
/// Why this matters for performance:
/// This helper runs inside the inner PIRLS loop, so any per-row integration cost
/// is multiplied by both the sample count and the number of IRLS iterations.
/// GHQ is robust, but it means repeated evaluation of quadrature nodes in a hot
/// path that can dominate calibrator or measurement-error fits.
///
/// Link-specific exact replacements:
/// - Probit:
///     E[Phi(eta + eps)] = Phi(eta / sqrt(1 + sigma^2))
///   exactly, with equally simple derivative. Integrated probit updates should
///   never need GHQ once they are routed through a dedicated family dispatch.
/// - Logit:
///   logistic-normal moments admit exact convergent Faddeeva / erfcx series,
///   which are the natural replacement for the GHQ calls below.
/// - Cloglog:
///   the mean is the complement of the lognormal Laplace transform and has
///   exact non-GHQ representations (Gamma / erfc / asymptotic series), which
///   is also relevant to survival transforms of the form exp(-exp(eta)).
///
/// This is the canonical integrated PIRLS update for binomial-style inverse
/// links. The runtime `InverseLink` carries the exact link state, so callers do
/// not have to thread `family + optional SAS/Mixture state` separately. Family
///-level integrated updates should reconstruct an `InverseLink` and delegate
/// here.
#[inline]
pub fn update_glmvectors_integrated_for_link(
    quadctx: &crate::quadrature::QuadratureContext,
    y: ArrayView1<f64>,
    eta: &Array1<f64>,
    se: ArrayView1<f64>,
    inverse_link: &InverseLink,
    priorweights: ArrayView1<f64>,
    mu: &mut Array1<f64>,
    weights: &mut Array1<f64>,
    z: &mut Array1<f64>,
    derivatives: Option<WorkingDerivativeBuffersMut<'_>>,
) -> Result<(), EstimationError> {
    let link = inverse_link.link_function();
    if !matches!(
        inverse_link,
        InverseLink::Standard(StandardLink::Logit)
            | InverseLink::Standard(StandardLink::Probit)
            | InverseLink::Standard(StandardLink::CLogLog)
            | InverseLink::LatentCLogLog(_)
            | InverseLink::Sas(_)
            | InverseLink::BetaLogistic(_)
            | InverseLink::Mixture(_)
    ) {
        crate::bail_invalid_estim!(
            "Integrated link-runtime update is not supported for inverse link {:?}",
            inverse_link
        );
    }
    if let Some(mut derivs) = derivatives {
        let WorkingSlices {
            mu: mu_s,
            weights: weights_s,
            z: z_s,
        } = working_slices(mu, weights, z);
        let WorkingDerivSlices {
            c: c_s,
            d: d_s,
            dmu: dmu_s,
            d2: d2_s,
            d3: d3_s,
        } = working_deriv_slices(&mut derivs);
        mu_s.par_iter_mut()
            .zip(weights_s.par_iter_mut())
            .zip(z_s.par_iter_mut())
            .zip(c_s.par_iter_mut())
            .zip(d_s.par_iter_mut())
            .zip(dmu_s.par_iter_mut())
            .zip(d2_s.par_iter_mut())
            .zip(d3_s.par_iter_mut())
            .enumerate()
            .try_for_each(
                |(i, (((((((mu_o, w_o), z_o), c_o), d_o), dmu_o), d2_o), d3_o))|
                 -> Result<(), EstimationError> {
                    let jet = if let InverseLink::LatentCLogLog(state) = inverse_link {
                        crate::quadrature::latent_cloglog_inverse_link_jet(
                            quadctx,
                            eta[i],
                            se[i].hypot(state.latent_sd),
                        )?
                    } else if matches!(inverse_link, InverseLink::Standard(StandardLink::Logit)) {
                        crate::quadrature::integrated_logit_inverse_link_jet_pirls(
                            quadctx, eta[i], se[i],
                        )?
                    } else {
                        crate::quadrature::integrated_inverse_link_jetwith_state(
                            quadctx,
                            link,
                            eta[i],
                            se[i],
                            inverse_link.mixture_state(),
                            inverse_link.sas_state(),
                        )?
                    };
                    let local_jet = MixtureInverseLinkJet {
                        mu: jet.mean,
                        d1: jet.d1,
                        d2: jet.d2,
                        d3: jet.d3,
                    };
                    let e = eta[i].clamp(-ETA_CLAMP, ETA_CLAMP);
                    let geom = bernoulli_geometry_from_jet(
                        eta[i],
                        e,
                        y[i],
                        priorweights[i],
                        local_jet,
                    );
                    *mu_o = geom.mu;
                    *w_o = geom.weight;
                    *z_o = geom.z;
                    *c_o = geom.c;
                    *d_o = geom.d;
                    *dmu_o = local_jet.d1;
                    *d2_o = local_jet.d2;
                    *d3_o = local_jet.d3;
                    Ok(())
                },
            )?;
    } else {
        let WorkingSlices {
            mu: mu_s,
            weights: weights_s,
            z: z_s,
        } = working_slices(mu, weights, z);
        mu_s.par_iter_mut()
            .zip(weights_s.par_iter_mut())
            .zip(z_s.par_iter_mut())
            .enumerate()
            .try_for_each(|(i, ((mu_o, w_o), z_o))| -> Result<(), EstimationError> {
                let jet = if let InverseLink::LatentCLogLog(state) = inverse_link {
                    crate::quadrature::latent_cloglog_inverse_link_jet(
                        quadctx,
                        eta[i],
                        se[i].hypot(state.latent_sd),
                    )?
                } else if matches!(inverse_link, InverseLink::Standard(StandardLink::Logit)) {
                    crate::quadrature::integrated_logit_inverse_link_jet_pirls(
                        quadctx, eta[i], se[i],
                    )?
                } else {
                    crate::quadrature::integrated_inverse_link_jetwith_state(
                        quadctx,
                        link,
                        eta[i],
                        se[i],
                        inverse_link.mixture_state(),
                        inverse_link.sas_state(),
                    )?
                };
                let local_jet = MixtureInverseLinkJet {
                    mu: jet.mean,
                    d1: jet.d1,
                    d2: jet.d2,
                    d3: jet.d3,
                };
                let e = eta[i].clamp(-ETA_CLAMP, ETA_CLAMP);
                let geom = bernoulli_geometry_from_jet(eta[i], e, y[i], priorweights[i], local_jet);
                *mu_o = geom.mu;
                *w_o = geom.weight;
                *z_o = geom.z;
                Ok(())
            })?;
    }
    Ok(())
}

/// Family-dispatched integrated GLM vector update helper.
///
/// This is the adapter from structural likelihood families onto the canonical
/// link-runtime implementation above. It keeps existing family-based call sites
/// working while making the `InverseLink` path authoritative.
///
/// This remains the intended dispatch point for eliminating GHQ link-by-link:
/// - `BinomialProbit` uses the exact Gaussian-probit convolution identity,
/// - `BinomialLogit` uses the best validated exact/special-function path and
///   otherwise falls back,
/// - `BinomialCLogLog` uses the plug-in / Taylor / Miles / Gamma ladder.
///
/// The important architectural point is that each family-specific exact path
/// only needs to provide:
///   1. the integrated mean
///        mu_i = E[g^{-1}(eta_i + eps_i)]
///   2. the integrated derivative
///        dmu_i / deta_i = E[(g^{-1})'(eta_i + eps_i)].
/// Once those are available, the general IRLS weight and working-response
/// formulas above remain unchanged. That makes this dispatch site the natural
/// place to swap GHQ out for exact link-specific mathematics without touching
/// the rest of the PIRLS update logic.
///
/// Keeping the dispatch here avoids contaminating the general PIRLS machinery
/// with link-specific special-function code and lets each family choose the
/// mathematically correct integration strategy.
#[inline]
pub fn update_glmvectors_integrated_by_family(
    quadctx: &crate::quadrature::QuadratureContext,
    y: ArrayView1<f64>,
    eta: &Array1<f64>,
    se: ArrayView1<f64>,
    spec: &LikelihoodSpec,
    priorweights: ArrayView1<f64>,
    mu: &mut Array1<f64>,
    weights: &mut Array1<f64>,
    z: &mut Array1<f64>,
    derivatives: Option<WorkingDerivativeBuffersMut<'_>>,
    mixture_link_state: Option<&MixtureLinkState>,
    sas_link_state: Option<&SasLinkState>,
) -> Result<(), EstimationError> {
    let inverse_link =
        integrated_inverse_link_from_family(spec, mixture_link_state, sas_link_state)?;
    update_glmvectors_integrated_for_link(
        quadctx,
        y,
        eta,
        se,
        &inverse_link,
        priorweights,
        mu,
        weights,
        z,
        derivatives,
    )
}

/// Compute first/second eta derivatives of the PIRLS working curvature W(eta),
/// consistent with the clamped working-geometry rules used by
/// `update_glmvectors`.
///
/// Math note:
/// - In the smooth interior (no clamps/floors active), `c[i]` and `d[i]` are
///   classical derivatives of the diagonal PIRLS curvature W_i(eta):
///     c_i = dW_i/dη_i,  d_i = d²W_i/dη_i².
/// - For canonical GLM families, these are the per-observation carriers of
///   higher likelihood derivatives (`-ℓ'''(η_i)` and `-ℓ''''(η_i)`) expressed
///   through the working-curvature map W(η).
/// - They are load-bearing in exact outer derivatives:
///   `c` enters dH/dρ (outer gradient), and `d` enters d²H/dρ² (outer Hessian).
/// - When hard clamps activate, the update map is piecewise and no longer C².
///   Setting c_i=d_i=0 is a practical subgradient-like choice to avoid unstable
///   explosive derivatives at the kink.
pub(crate) fn computeworkingweight_derivatives_from_eta(
    likelihood: &GlmLikelihoodSpec,
    inverse_link: &InverseLink,
    eta: &Array1<f64>,
    priorweights: ArrayView1<f64>,
) -> Result<
    (
        Array1<f64>,
        Array1<f64>,
        Array1<f64>,
        Array1<f64>,
        Array1<f64>,
    ),
    EstimationError,
> {
    let n = eta.len();
    let mut c = Array1::<f64>::zeros(n);
    let mut d = Array1::<f64>::zeros(n);
    let mut dmu_deta = Array1::<f64>::zeros(n);
    let mut d2mu_deta2 = Array1::<f64>::zeros(n);
    let mut d3mu_deta3 = Array1::<f64>::zeros(n);
    match &likelihood.spec.response {
        ResponseFamily::Gaussian => {
            dmu_deta.fill(1.0);
        }
        ResponseFamily::Poisson => {
            log_link_working_state::write_log_link_eta_curvature(
                &log_link_working_state::LogLinkRule {
                    weight: log_link_working_state::WorkingWeight::PoissonIdentity,
                    curvature: log_link_working_state::WorkingCurvature::Proportional {
                        c_ratio: 1.0,
                        d_ratio: 1.0,
                    },
                    floor_weight: true,
                    zero_mu_jet_on_clamp: false,
                },
                inverse_link,
                eta,
                priorweights,
                WorkingDerivativeBuffersMut {
                    c: &mut c,
                    d: &mut d,
                    dmu_deta: &mut dmu_deta,
                    d2mu_deta2: &mut d2mu_deta2,
                    d3mu_deta3: &mut d3mu_deta3,
                },
            )?;
        }
        ResponseFamily::Tweedie { p } => {
            let p = *p;
            let phi = fixed_glm_dispersion(likelihood);
            if !is_valid_tweedie_power(p) {
                crate::bail_invalid_estim!(
                    "Tweedie variance power must be finite and strictly between 1 and 2; got {p}",
                    p = p
                );
            }
            if !(phi.is_finite() && phi > 0.0) {
                crate::bail_invalid_estim!(
                    "Tweedie dispersion phi must be finite and > 0; got {phi}",
                    phi = phi
                );
            }
            let exponent = 2.0 - p;
            log_link_working_state::write_log_link_eta_curvature(
                &log_link_working_state::LogLinkRule {
                    weight: log_link_working_state::WorkingWeight::TweediePower { p, phi },
                    curvature: log_link_working_state::WorkingCurvature::Proportional {
                        c_ratio: exponent,
                        d_ratio: exponent * exponent,
                    },
                    floor_weight: true,
                    zero_mu_jet_on_clamp: true,
                },
                inverse_link,
                eta,
                priorweights,
                WorkingDerivativeBuffersMut {
                    c: &mut c,
                    d: &mut d,
                    dmu_deta: &mut dmu_deta,
                    d2mu_deta2: &mut d2mu_deta2,
                    d3mu_deta3: &mut d3mu_deta3,
                },
            )?;
        }
        ResponseFamily::NegativeBinomial { theta, .. } => {
            let theta = *theta;
            if !valid_negbin_theta(theta) {
                crate::bail_invalid_estim!(
                    "negative-binomial theta must be finite and > 0; got {theta}",
                    theta = theta
                );
            }
            log_link_working_state::write_log_link_eta_curvature(
                &log_link_working_state::LogLinkRule {
                    weight: log_link_working_state::WorkingWeight::NegativeBinomial { theta },
                    curvature: log_link_working_state::WorkingCurvature::NegativeBinomial { theta },
                    floor_weight: true,
                    zero_mu_jet_on_clamp: false,
                },
                inverse_link,
                eta,
                priorweights,
                WorkingDerivativeBuffersMut {
                    c: &mut c,
                    d: &mut d,
                    dmu_deta: &mut dmu_deta,
                    d2mu_deta2: &mut d2mu_deta2,
                    d3mu_deta3: &mut d3mu_deta3,
                },
            )?;
        }
        ResponseFamily::Beta { phi } => {
            let phi = *phi;
            if !valid_beta_phi(phi) {
                crate::bail_invalid_estim!("beta-regression phi must be finite and > 0; got {phi}");
            }
            let c_s = c.as_slice_mut().expect("c must be contiguous");
            let d_s = d.as_slice_mut().expect("d must be contiguous");
            let dmu_s = dmu_deta
                .as_slice_mut()
                .expect("dmu_deta must be contiguous");
            let d2_s = d2mu_deta2
                .as_slice_mut()
                .expect("d2mu_deta2 must be contiguous");
            let d3_s = d3mu_deta3
                .as_slice_mut()
                .expect("d3mu_deta3 must be contiguous");
            c_s.par_iter_mut()
                .zip(d_s.par_iter_mut())
                .zip(dmu_s.par_iter_mut())
                .zip(d2_s.par_iter_mut())
                .zip(d3_s.par_iter_mut())
                .enumerate()
                .for_each(|(i, ((((c_o, d_o), dmu_o), d2_o), d3_o))| {
                    let eta_raw = eta[i];
                    let eta_i = eta_raw.clamp(-ETA_CLAMP, ETA_CLAMP);
                    let jet = logit_inverse_link_jet5(eta_i);
                    let mu_i = safe_beta_mu(jet.mu);
                    let q = (mu_i * (1.0 - mu_i)).max(BETA_MU_EPS);
                    let a = (mu_i * phi).max(BETA_MU_EPS);
                    let b = ((1.0 - mu_i) * phi).max(BETA_MU_EPS);
                    let trigamma_sum = trigamma(a) + trigamma(b);
                    let prior_weight = priorweights[i].max(0.0);
                    let raw_weight = prior_weight * q * q * phi * phi * trigamma_sum;
                    let floor_active = raw_weight > 0.0 && raw_weight <= MIN_WEIGHT;
                    if floor_active || eta_raw != eta_i {
                        *c_o = 0.0;
                        *d_o = 0.0;
                    } else {
                        let (c_i, d_i) = beta_logit_working_curvature_eta_derivatives(
                            prior_weight,
                            phi,
                            mu_i,
                            q,
                            a,
                            b,
                            trigamma_sum,
                        );
                        *c_o = c_i;
                        *d_o = d_i;
                    }
                    *dmu_o = q;
                    *d2_o = q * (1.0 - 2.0 * mu_i);
                    *d3_o = q * (1.0 - 6.0 * q);
                });
        }
        ResponseFamily::Gamma => {
            // The Gamma log-link Fisher weight is independent of η, so the
            // working-curvature carriers `c`/`d` vanish identically (the kernel
            // returns `(0, 0)`); only the link jet is written here.
            log_link_working_state::write_log_link_eta_curvature(
                &log_link_working_state::LogLinkRule {
                    weight: log_link_working_state::WorkingWeight::Constant { factor: 1.0 },
                    curvature: log_link_working_state::WorkingCurvature::Proportional {
                        c_ratio: 0.0,
                        d_ratio: 0.0,
                    },
                    floor_weight: false,
                    zero_mu_jet_on_clamp: false,
                },
                inverse_link,
                eta,
                priorweights,
                WorkingDerivativeBuffersMut {
                    c: &mut c,
                    d: &mut d,
                    dmu_deta: &mut dmu_deta,
                    d2mu_deta2: &mut d2mu_deta2,
                    d3mu_deta3: &mut d3mu_deta3,
                },
            )?;
        }
        ResponseFamily::Binomial => {
            let link = inverse_link.link_function();
            // On logit geometry, freeze higher η-derivatives in nonsmooth
            // regions so PIRLS and outer derivative code differentiate the
            // same piecewise-smooth surface.
            let zero_on_nonsmooth = matches!(link, LinkFunction::Logit);
            // Five independent per-row writes: same parallelization shape as
            // `update_glmvectors` above. Note the `jet.mu` argument is reused
            // here as the response (matching the original serial code) — this
            // is the score-derivative path where y is replaced by mu so the
            // (y - mu) residual term vanishes by construction.
            let c_s = c.as_slice_mut().expect("c must be contiguous");
            let d_s = d.as_slice_mut().expect("d must be contiguous");
            let dmu_s = dmu_deta
                .as_slice_mut()
                .expect("dmu_deta must be contiguous");
            let d2_s = d2mu_deta2
                .as_slice_mut()
                .expect("d2mu_deta2 must be contiguous");
            let d3_s = d3mu_deta3
                .as_slice_mut()
                .expect("d3mu_deta3 must be contiguous");
            c_s.par_iter_mut()
                .zip(d_s.par_iter_mut())
                .zip(dmu_s.par_iter_mut())
                .zip(d2_s.par_iter_mut())
                .zip(d3_s.par_iter_mut())
                .enumerate()
                .try_for_each(
                    |(i, ((((c_o, d_o), dmu_o), d2_o), d3_o))| -> Result<(), EstimationError> {
                        let eta_used = match link {
                            LinkFunction::Logit => eta[i].clamp(-ETA_CLAMP, ETA_CLAMP),
                            LinkFunction::Probit
                            | LinkFunction::CLogLog
                            | LinkFunction::Sas
                            | LinkFunction::BetaLogistic => eta[i].clamp(-30.0, 30.0),
                            LinkFunction::Log => eta[i].clamp(-ETA_CLAMP, ETA_CLAMP),
                            LinkFunction::Identity => eta[i],
                        };
                        if matches!(link, LinkFunction::Logit) {
                            let jet = logit_inverse_link_jet5(eta_used);
                            let geom = bernoulli_logit_geometry_from_jet(
                                eta[i],
                                eta_used,
                                jet.mu,
                                priorweights[i],
                                jet,
                                zero_on_nonsmooth,
                            );
                            *c_o = geom.c;
                            *d_o = geom.d;
                            *dmu_o = jet.d1;
                            *d2_o = jet.d2;
                            *d3_o = jet.d3;
                        } else {
                            let jet = standard_inverse_link_jet(inverse_link, eta_used)?;
                            let geom = bernoulli_geometry_from_jet(
                                eta[i],
                                eta_used,
                                jet.mu,
                                priorweights[i],
                                jet,
                            );
                            *c_o = geom.c;
                            *d_o = geom.d;
                            *dmu_o = jet.d1;
                            *d2_o = jet.d2;
                            *d3_o = jet.d3;
                        }
                        Ok(())
                    },
                )?;
        }
        ResponseFamily::RoystonParmar => {
            crate::bail_invalid_estim!(
                "RoystonParmar is survival-specific and not a GLM IRLS family"
            );
        }
    }
    Ok((c, d, dmu_deta, d2mu_deta2, d3mu_deta3))
}

// General noncanonical observed-information weight corrections
//
// For an exponential-dispersion family with noncanonical link g, where
// h(η) = g⁻¹(η) is the inverse link and μ = h(η):
//
// Notation (all evaluated at a single observation):
//   h₁ = h'(η),  h₂ = h''(η),  h₃ = h'''(η),  h₄ = h''''(η)
//   V  = V(μ),   V₁ = V'(μ),   V₂ = V''(μ),    V₃ = V'''(μ)
//   φ  = dispersion parameter
//   pw = prior weight for this observation
//
// Fisher (expected) weight and its first two η-derivatives:
//   w_F = h₁² / (φV)
//   c_F = (2 h₁ h₂ V − h₁³ V₁) / (φ V²)
//   d_F = ∂c_F/∂η   (derived below)
//
// The observed weight subtracts a (y−μ)-dependent correction:
//   B   = (h₂ V − h₁² V₁) / (φ V²)
//   w_obs = w_F − (y−μ) · B
//
// First η-derivative of B:
//   B_η = (h₃ V² − 3 h₁ h₂ V V₁ − h₁³ V V₂ + 2 h₁³ V₁²) / (φ V³)
//
// Observed c (∂w_obs/∂η):
//   c_obs = c_F + h₁·B − (y−μ)·B_η
//
// Second η-derivative of B:
//   B_ηη = ∂B_η/∂η  (full expression in code below)
//
// Observed d (∂²w_obs/∂η²):
//   d_obs = d_F + h₂·B + 2 h₁·B_η − (y−μ)·B_ηη
//
// This function unifies all per-link hardcoded c/d computations: given the
// inverse-link jet (h₁…h₄) and the variance-function jet (V…V₃), it returns
// (w_obs, c_obs, d_obs) without any family- or link-specific dispatch.