super stable gamma_lccdf

stan/math/prim/fun/log_gamma_q_dgamma.hpp

@@ -0,0 +1,156 @@
+#ifndef STAN_MATH_PRIM_FUN_LOG_GAMMA_Q_DGAMMA_HPP
+#define STAN_MATH_PRIM_FUN_LOG_GAMMA_Q_DGAMMA_HPP
+
+#include <stan/math/prim/meta.hpp>
+#include <stan/math/prim/fun/constants.hpp>
+#include <stan/math/prim/fun/digamma.hpp>
+#include <stan/math/prim/fun/exp.hpp>
+#include <stan/math/prim/fun/gamma_p.hpp>
+#include <stan/math/prim/fun/gamma_q.hpp>
+#include <stan/math/prim/fun/grad_reg_inc_gamma.hpp>
+#include <stan/math/prim/fun/lgamma.hpp>
+#include <stan/math/prim/fun/log.hpp>
+#include <stan/math/prim/fun/log1m.hpp>
+#include <stan/math/prim/fun/tgamma.hpp>
+#include <stan/math/prim/fun/value_of.hpp>
+#include <cmath>
+
+namespace stan {
+namespace math {
+
+/**
+ * Result structure containing log(Q(a,z)) and its gradient with respect to a.
+ *
+ * @tparam T return type
+ */
+template <typename T>
+struct log_gamma_q_result {
+  T log_q;      ///< log(Q(a,z)) where Q is upper regularized incomplete gamma
+  T dlog_q_da;  ///< d/da log(Q(a,z))
+};
+
+namespace internal {
+
+/**
+ * Compute log(Q(a,z)) using continued fraction expansion for upper incomplete
+ * gamma function.
+ *
+ * @tparam T_a Type of shape parameter a (double or fvar types)
+ * @tparam T_z Type of value parameter z (double or fvar types)
+ * @param a Shape parameter
+ * @param z Value at which to evaluate
+ * @param precision Convergence threshold
+ * @param max_steps Maximum number of continued fraction iterations
+ * @return log(Q(a,z)) with same type as T_a and T_z
+ */
+template <typename T_a, typename T_z>
+inline auto log_q_gamma_cf(const T_a& a, const T_z& z, double precision = 1e-16,
+                           int max_steps = 250) {
+  using stan::math::lgamma;
+  using stan::math::log;
+  using stan::math::value_of;
+  using std::fabs;
+  using T_return = return_type_t<T_a, T_z>;
+
+  const T_return a_ret = a;
+  const T_return z_ret = z;
+  const auto log_prefactor = a_ret * log(z_ret) - z_ret - lgamma(a_ret);
+
+  auto b = z_ret + 1.0 - a_ret;
+  auto C = (fabs(value_of(b)) >= EPSILON) ? b : T_return(EPSILON);
+  auto D = T_return(0.0);
+  auto f = C;
+
+  for (int i = 1; i <= max_steps; ++i) {
+    auto an = -i * (i - a_ret);
+    b += 2.0;
+
+    D = b + an * D;
+    if (fabs(value_of(D)) < EPSILON) {
+      D = T_return(EPSILON);
+    }
+    C = b + an / C;
+    if (fabs(value_of(C)) < EPSILON) {
+      C = T_return(EPSILON);
+    }
+
+    D = 1.0 / D;
+    auto delta = C * D;
+    f *= delta;
+
+    const double delta_m1 = value_of(fabs(value_of(delta) - 1.0));
+    if (delta_m1 < precision) {
+      break;
+    }
+  }
+
+  return log_prefactor - log(f);
+}
+
+}  // namespace internal
+
+/**
+ * Compute log(Q(a,z)) and its gradient with respect to a using continued
+ * fraction expansion, where Q(a,z) = Gamma(a,z) / Gamma(a) is the regularized
+ * upper incomplete gamma function.
+ *
+ * This uses a continued fraction representation for numerical stability when
+ * computing the upper incomplete gamma function in log space, along with
+ * analytical gradient computation.
+ *
+ * @tparam T_a type of the shape parameter
+ * @tparam T_z type of the value parameter
+ * @param a shape parameter (must be positive)
+ * @param z value parameter (must be non-negative)
+ * @param precision convergence threshold
+ * @param max_steps maximum iterations for continued fraction
+ * @return structure containing log(Q(a,z)) and d/da log(Q(a,z))
+ */
+template <typename T_a, typename T_z>
+inline log_gamma_q_result<return_type_t<T_a, T_z>> log_gamma_q_dgamma(
+    const T_a& a, const T_z& z, double precision = 1e-16, int max_steps = 250) {
+  using std::exp;
+  using std::log;
+  using T_return = return_type_t<T_a, T_z>;
+
+  const double a_dbl = value_of(a);
+  const double z_dbl = value_of(z);
+
+  log_gamma_q_result<T_return> result;
+
+  // For z > a + 1, use continued fraction for better numerical stability
+  if (z_dbl > a_dbl + 1.0) {
+    result.log_q = internal::log_q_gamma_cf(a_dbl, z_dbl, precision, max_steps);
+
+    // For gradient, use: d/da log(Q) = (1/Q) * dQ/da
+    // grad_reg_inc_gamma computes dQ/da
+    const double Q_val = exp(result.log_q);
+    const double dQ_da
+        = grad_reg_inc_gamma(a_dbl, z_dbl, tgamma(a_dbl), digamma(a_dbl));
+    result.dlog_q_da = dQ_da / Q_val;
+
+  } else {
+    // For z <= a + 1, use log1m(P(a,z)) for better numerical accuracy
+    const double P_val = gamma_p(a_dbl, z_dbl);
+    result.log_q = log1m(P_val);
+
+    // Gradient: d/da log(Q) = (1/Q) * dQ/da
+    // grad_reg_inc_gamma computes dQ/da
+    const double Q_val = exp(result.log_q);
+    if (Q_val > 0) {
+      const double dQ_da
+          = grad_reg_inc_gamma(a_dbl, z_dbl, tgamma(a_dbl), digamma(a_dbl));
+      result.dlog_q_da = dQ_da / Q_val;
+    } else {
+      // Fallback if Q rounds to zero - use asymptotic approximation
+      result.dlog_q_da = log(z_dbl) - digamma(a_dbl);
+    }
+  }
+
+  return result;
+}
+
+}  // namespace math
+}  // namespace stan
+
+#endif

stan/math/prim/prob/gamma_lccdf.hpp

@@ -6,28 +6,33 @@
 #include <stan/math/prim/fun/constants.hpp>
 #include <stan/math/prim/fun/digamma.hpp>
 #include <stan/math/prim/fun/exp.hpp>
-#include <stan/math/prim/fun/gamma_q.hpp>
+#include <stan/math/prim/fun/fma.hpp>
+#include <stan/math/prim/fun/gamma_p.hpp>
 #include <stan/math/prim/fun/grad_reg_inc_gamma.hpp>
+#include <stan/math/prim/fun/grad_reg_lower_inc_gamma.hpp>
+#include <stan/math/prim/fun/lgamma.hpp>
 #include <stan/math/prim/fun/log.hpp>
+#include <stan/math/prim/fun/log1m.hpp>
 #include <stan/math/prim/fun/max_size.hpp>
 #include <stan/math/prim/fun/scalar_seq_view.hpp>
 #include <stan/math/prim/fun/size.hpp>
 #include <stan/math/prim/fun/size_zero.hpp>
 #include <stan/math/prim/fun/tgamma.hpp>
 #include <stan/math/prim/fun/value_of.hpp>
+#include <stan/math/prim/fun/log_gamma_q_dgamma.hpp>
 #include <stan/math/prim/functor/partials_propagator.hpp>
 #include <cmath>
 
 namespace stan {
 namespace math {
 
 template <typename T_y, typename T_shape, typename T_inv_scale>
-inline return_type_t<T_y, T_shape, T_inv_scale> gamma_lccdf(
-    const T_y& y, const T_shape& alpha, const T_inv_scale& beta) {
+return_type_t<T_y, T_shape, T_inv_scale> gamma_lccdf(const T_y& y,
+                                                     const T_shape& alpha,
+                                                     const T_inv_scale& beta) {
   using T_partials_return = partials_return_t<T_y, T_shape, T_inv_scale>;
   using std::exp;
   using std::log;
-  using std::pow;
   using T_y_ref = ref_type_t<T_y>;
   using T_alpha_ref = ref_type_t<T_shape>;
   using T_beta_ref = ref_type_t<T_inv_scale>;
@@ -51,61 +56,159 @@ inline return_type_t<T_y, T_shape, T_inv_scale> gamma_lccdf(
   scalar_seq_view<T_y_ref> y_vec(y_ref);
   scalar_seq_view<T_alpha_ref> alpha_vec(alpha_ref);
   scalar_seq_view<T_beta_ref> beta_vec(beta_ref);
-  size_t N = max_size(y, alpha, beta);
-
-  // Explicit return for extreme values
-  // The gradients are technically ill-defined, but treated as zero
-  for (size_t i = 0; i < stan::math::size(y); i++) {
-    if (y_vec.val(i) == 0) {
-      // LCCDF(0) = log(P(Y > 0)) = log(1) = 0
-      return ops_partials.build(0.0);
-    }
-  }
+  const size_t N = max_size(y, alpha, beta);
+
+  constexpr bool need_y_beta_deriv = !is_constant_all<T_y, T_inv_scale>::value;
+  constexpr bool any_fvar = is_fvar<scalar_type_t<T_y>>::value
+                            || is_fvar<scalar_type_t<T_shape>>::value
+                            || is_fvar<scalar_type_t<T_inv_scale>>::value;
+  constexpr bool partials_fvar = is_fvar<T_partials_return>::value;
 
   for (size_t n = 0; n < N; n++) {
     // Explicit results for extreme values
     // The gradients are technically ill-defined, but treated as zero
-    if (y_vec.val(n) == INFTY) {
-      // LCCDF(∞) = log(P(Y > ∞)) = log(0) = -∞
+    const T_partials_return y_dbl = y_vec.val(n);
+    if (y_dbl == 0.0) {
+      continue;
+    }
+    if (y_dbl == INFTY) {
       return ops_partials.build(negative_infinity());
     }
 
-    const T_partials_return y_dbl = y_vec.val(n);
     const T_partials_return alpha_dbl = alpha_vec.val(n);
     const T_partials_return beta_dbl = beta_vec.val(n);
-    const T_partials_return beta_y_dbl = beta_dbl * y_dbl;
 
-    // Qn = 1 - Pn
-    const T_partials_return Qn = gamma_q(alpha_dbl, beta_y_dbl);
-    const T_partials_return log_Qn = log(Qn);
+    const T_partials_return beta_y = beta_dbl * y_dbl;
+    if (beta_y == INFTY) {
+      return ops_partials.build(negative_infinity());
+    }
+
+    bool use_cf = beta_y > alpha_dbl + 1.0;
+    T_partials_return log_Qn;
+    [[maybe_unused]] T_partials_return dlogQ_dalpha = 0.0;
+    // Extract double values for the double-only continued fraction path.
+    [[maybe_unused]] const double beta_y_dbl = value_of(value_of(beta_y));
+    [[maybe_unused]] const double alpha_dbl_val = value_of(value_of(alpha_dbl));
+
+    if (use_cf) {
+      if constexpr (!any_fvar && is_autodiff_v<T_shape>) {
+        // var-only: use analytical gradient with double inputs
+        auto log_q_result = log_gamma_q_dgamma(alpha_dbl_val, beta_y_dbl);
+        log_Qn = log_q_result.log_q;
+        dlogQ_dalpha = log_q_result.dlog_q_da;
+      } else {
+        log_Qn = internal::log_q_gamma_cf(alpha_dbl, beta_y);
+        if constexpr (is_autodiff_v<T_shape>) {
+          if constexpr (partials_fvar) {
+            auto alpha_unit = alpha_dbl;
+            alpha_unit.d_ = 1;
+            auto beta_unit = beta_y;
+            beta_unit.d_ = 0;
+            auto log_Qn_fvar = internal::log_q_gamma_cf(alpha_unit, beta_unit);
+            dlogQ_dalpha = log_Qn_fvar.d_;
+          } else {
+            const T_partials_return Qn = exp(log_Qn);
+            dlogQ_dalpha
+                = grad_reg_inc_gamma(alpha_dbl, beta_y, tgamma(alpha_dbl),
+                                     digamma(alpha_dbl))
+                  / Qn;
+          }
+        }
+      }
+    } else {
+      const T_partials_return Pn = gamma_p(alpha_dbl, beta_y);
+      log_Qn = log1m(Pn);
+
+      if (!std::isfinite(value_of(value_of(log_Qn)))) {
+        use_cf = beta_y > 0.0;
+        if (use_cf) {
+          // Fallback to continued fraction if log1m fails
+          if constexpr (!any_fvar && is_autodiff_v<T_shape>) {
+            auto log_q_result = log_gamma_q_dgamma(alpha_dbl_val, beta_y_dbl);
+            log_Qn = log_q_result.log_q;
+            dlogQ_dalpha = log_q_result.dlog_q_da;
+          } else {
+            log_Qn = internal::log_q_gamma_cf(alpha_dbl, beta_y);
+            if constexpr (is_autodiff_v<T_shape>) {
+              if constexpr (partials_fvar) {
+                auto alpha_unit = alpha_dbl;
+                alpha_unit.d_ = 1;
+                auto beta_unit = beta_y;
+                beta_unit.d_ = 0;
+                auto log_Qn_fvar
+                    = internal::log_q_gamma_cf(alpha_unit, beta_unit);
+                dlogQ_dalpha = log_Qn_fvar.d_;
+              } else {
+                const T_partials_return Qn = exp(log_Qn);
+                dlogQ_dalpha
+                    = grad_reg_inc_gamma(alpha_dbl, beta_y, tgamma(alpha_dbl),
+                                         digamma(alpha_dbl))
+                      / Qn;
+              }
+            }
+          }
+        }
+      }
 
+      if constexpr (is_autodiff_v<T_shape>) {
+        if (!use_cf) {
+          if constexpr (partials_fvar) {
+            auto alpha_unit = alpha_dbl;
+            alpha_unit.d_ = 1;
+            auto beta_unit = beta_y;
+            beta_unit.d_ = 0;
+            auto log_Qn_fvar = log1m(gamma_p(alpha_unit, beta_unit));
+            dlogQ_dalpha = log_Qn_fvar.d_;
+          } else {
+            const T_partials_return Qn = exp(log_Qn);
+            if (Qn > 0.0) {
+              dlogQ_dalpha = -grad_reg_lower_inc_gamma(alpha_dbl, beta_y) / Qn;
+            } else {
+              // Fallback to continued fraction if Q rounds to zero
+              if constexpr (!any_fvar) {
+                auto log_q_result
+                    = log_gamma_q_dgamma(alpha_dbl_val, beta_y_dbl);
+                log_Qn = log_q_result.log_q;
+                dlogQ_dalpha = log_q_result.dlog_q_da;
+              } else {
+                log_Qn = internal::log_q_gamma_cf(alpha_dbl, beta_y);
+                const T_partials_return Qn_cf = exp(log_Qn);
+                dlogQ_dalpha
+                    = grad_reg_inc_gamma(alpha_dbl, beta_y, tgamma(alpha_dbl),
+                                         digamma(alpha_dbl))
+                      / Qn_cf;
+              }
+              use_cf = true;
+            }
+          }
+        }
+      }
+    }
+    if (!std::isfinite(value_of(value_of(log_Qn)))) {
+      return ops_partials.build(negative_infinity());
+    }
     P += log_Qn;
 
-    if constexpr (is_any_autodiff_v<T_y, T_inv_scale>) {
-      const T_partials_return log_y_dbl = log(y_dbl);
-      const T_partials_return log_beta_dbl = log(beta_dbl);
+    if constexpr (need_y_beta_deriv) {
+      const T_partials_return log_y = log(y_dbl);
+      const T_partials_return log_beta = log(beta_dbl);
+      const T_partials_return lgamma_alpha = lgamma(alpha_dbl);
+      const T_partials_return alpha_minus_one = fma(alpha_dbl, log_y, -log_y);
+
       const T_partials_return log_pdf
-          = alpha_dbl * log_beta_dbl - lgamma(alpha_dbl)
-            + (alpha_dbl - 1.0) * log_y_dbl - beta_y_dbl;
-      const T_partials_return common_term = exp(log_pdf - log_Qn);
+          = alpha_dbl * log_beta - lgamma_alpha + alpha_minus_one - beta_y;
+
+      const T_partials_return hazard = exp(log_pdf - log_Qn);  // f/Q
 
       if constexpr (is_autodiff_v<T_y>) {
-        // d/dy log(1-F(y)) = -f(y)/(1-F(y))
-        partials<0>(ops_partials)[n] -= common_term;
+        partials<0>(ops_partials)[n] -= hazard;
       }
       if constexpr (is_autodiff_v<T_inv_scale>) {
-        // d/dbeta log(1-F(y)) = -y*f(y)/(beta*(1-F(y)))
-        partials<2>(ops_partials)[n] -= y_dbl / beta_dbl * common_term;
+        partials<2>(ops_partials)[n] -= (y_dbl / beta_dbl) * hazard;
       }
     }
-
     if constexpr (is_autodiff_v<T_shape>) {
-      const T_partials_return digamma_val = digamma(alpha_dbl);
-      const T_partials_return gamma_val = tgamma(alpha_dbl);
-      // d/dalpha log(1-F(y)) = grad_upper_inc_gamma / (1-F(y))
-      partials<1>(ops_partials)[n]
-          += grad_reg_inc_gamma(alpha_dbl, beta_y_dbl, gamma_val, digamma_val)
-             / Qn;
+      partials<1>(ops_partials)[n] += dlogQ_dalpha;
     }
   }
   return ops_partials.build(P);