Incomplete Beta Function Inverse

stan/math/fwd/fun.hpp

@@ -47,6 +47,7 @@
 #include <stan/math/fwd/fun/inv_erfc.hpp>
 #include <stan/math/fwd/fun/inv_Phi.hpp>
 #include <stan/math/fwd/fun/inv_cloglog.hpp>
+#include <stan/math/fwd/fun/inv_inc_beta.hpp>
 #include <stan/math/fwd/fun/inv_logit.hpp>
 #include <stan/math/fwd/fun/inv_sqrt.hpp>
 #include <stan/math/fwd/fun/inv_square.hpp>

stan/math/fwd/fun/inv_inc_beta.hpp

@@ -0,0 +1,87 @@
+#ifndef STAN_MATH_FWD_FUN_INV_INC_BETA_HPP
+#define STAN_MATH_FWD_FUN_INV_INC_BETA_HPP
+
+#include <stan/math/fwd/meta.hpp>
+#include <stan/math/fwd/core.hpp>
+#include <stan/math/prim/fun/inv_inc_beta.hpp>
+#include <stan/math/prim/fun/inc_beta.hpp>
+#include <stan/math/prim/fun/exp.hpp>
+#include <stan/math/prim/fun/log.hpp>
+#include <stan/math/prim/fun/log_diff_exp.hpp>
+#include <stan/math/prim/fun/lbeta.hpp>
+#include <stan/math/prim/fun/lgamma.hpp>
+#include <stan/math/prim/fun/digamma.hpp>
+#include <stan/math/prim/fun/F32.hpp>
+
+namespace stan {
+namespace math {
+
+/**
+ * The inverse of the normalized incomplete beta function of a, b, with
+ * probability p.
+ *
+ * Used to compute the inverse cumulative density function for the beta
+ * distribution.
+ *
+ * @param a Shape parameter a >= 0; a and b can't both be 0
+ * @param b Shape parameter b >= 0
+ * @param p Random variate. 0 <= p <= 1
+ * @throws if constraints are violated or if any argument is NaN
+ * @return The inverse of the normalized incomplete beta function.
+ */
+template <typename T1, typename T2, typename T3,
+          require_all_stan_scalar_t<T1, T2, T3>* = nullptr,
+          require_any_fvar_t<T1, T2, T3>* = nullptr>
+inline fvar<partials_return_t<T1, T2, T3>> inv_inc_beta(const T1& a,
+                                                        const T2& b,
+                                                        const T3& p) {
+  using T_return = partials_return_t<T1, T2, T3>;
+  auto a_val = value_of(a);
+  auto b_val = value_of(b);
+  auto p_val = value_of(p);
+  T_return w = inv_inc_beta(a_val, b_val, p_val);
+  T_return log_w = log(w);
+  T_return log1m_w = log1m(w);
+  auto one_m_a = 1 - a_val;
+  auto one_m_b = 1 - b_val;
+  T_return one_m_w = 1 - w;
+  auto ap1 = a_val + 1;
+  auto bp1 = b_val + 1;
+  auto lbeta_ab = lbeta(a_val, b_val);
+  auto digamma_apb = digamma(a_val + b_val);
+
+  T_return inv_d_(0);
+
+  if (is_fvar<T1>::value) {
+    auto da1 = exp(one_m_b * log1m_w + one_m_a * log_w);
+    auto da2
+        = exp(a_val * log_w + 2 * lgamma(a_val)
+              + log(F32(a_val, a_val, one_m_b, ap1, ap1, w)) - 2 * lgamma(ap1));
+    auto da3 = inc_beta(a_val, b_val, w) * exp(lbeta_ab)
+               * (log_w - digamma(a_val) + digamma_apb);
+    inv_d_ += forward_as<fvar<T_return>>(a).d_ * da1 * (da2 - da3);
+  }
+
+  if (is_fvar<T2>::value) {
+    auto db1 = (w - 1) * exp(-b_val * log1m_w + one_m_a * log_w);
+    auto db2 = 2 * lgamma(b_val)
+               + log(F32(b_val, b_val, one_m_a, bp1, bp1, one_m_w))
+               - 2 * lgamma(bp1) + b_val * log1m_w;
+
+    auto db3 = inc_beta(b_val, a_val, one_m_w) * exp(lbeta_ab)
+               * (log1m_w - digamma(b_val) + digamma_apb);
+
+    inv_d_ += forward_as<fvar<T_return>>(b).d_ * db1 * (exp(db2) - db3);
+  }
+
+  if (is_fvar<T3>::value) {
+    inv_d_ += forward_as<fvar<T_return>>(p).d_
+              * exp(one_m_b * log1m_w + one_m_a * log_w + lbeta_ab);
+  }
+
+  return fvar<T_return>(w, inv_d_);
+}
+
+}  // namespace math
+}  // namespace stan
+#endif

stan/math/prim/fun.hpp

@@ -140,6 +140,7 @@
 #include <stan/math/prim/fun/int_step.hpp>
 #include <stan/math/prim/fun/inv.hpp>
 #include <stan/math/prim/fun/inv_Phi.hpp>
+#include <stan/math/prim/fun/inv_inc_beta.hpp>
 #include <stan/math/prim/fun/inv_cloglog.hpp>
 #include <stan/math/prim/fun/inv_erfc.hpp>
 #include <stan/math/prim/fun/inv_logit.hpp>

stan/math/prim/fun/F32.hpp

@@ -49,29 +49,35 @@ namespace math {
  * @param[in] precision precision of the infinite sum. defaults to 1e-6
  * @param[in] max_steps number of steps to take. defaults to 1e5
  */
-template <typename T>
-T F32(const T& a1, const T& a2, const T& a3, const T& b1, const T& b2,
-      const T& z, double precision = 1e-6, int max_steps = 1e5) {
+template <typename Ta1, typename Ta2, typename Ta3, typename Tb1, typename Tb2,
+          typename Tz>
+return_type_t<Ta1, Ta2, Ta3, Tb1, Tb2, Tz> F32(const Ta1& a1, const Ta2& a2,
+                                               const Ta3& a3, const Tb1& b1,
+                                               const Tb2& b2, const Tz& z,
+                                               double precision = 1e-6,
+                                               int max_steps = 1e5) {
   check_3F2_converges("F32", a1, a2, a3, b1, b2, z);
 
+  using T_return = return_type_t<Ta1, Ta2, Ta3, Tb1, Tb2, Tz>;
   using std::exp;
   using std::fabs;
   using std::log;
 
-  T t_acc = 1.0;
-  T log_t = 0.0;
-  T log_z = log(z);
+  T_return t_acc = 1.0;
+  T_return log_t = 0.0;
+  Tz log_z = log(z);
   double t_sign = 1.0;
 
   for (int k = 0; k <= max_steps; ++k) {
-    T p = (a1 + k) * (a2 + k) * (a3 + k) / ((b1 + k) * (b2 + k) * (k + 1));
+    T_return p
+        = (a1 + k) * (a2 + k) * (a3 + k) / ((b1 + k) * (b2 + k) * (k + 1));
     if (p == 0.0) {
       return t_acc;
     }
 
     log_t += log(fabs(p)) + log_z;
     t_sign = p >= 0.0 ? t_sign : -t_sign;
-    T t_new = t_sign > 0.0 ? exp(log_t) : -exp(log_t);
+    T_return t_new = t_sign > 0.0 ? exp(log_t) : -exp(log_t);
     t_acc += t_new;
 
     if (fabs(t_new) <= precision) {

stan/math/prim/fun/inv_inc_beta.hpp

@@ -0,0 +1,34 @@
+#ifndef STAN_MATH_PRIM_FUN_INV_INC_BETA_HPP
+#define STAN_MATH_PRIM_FUN_INV_INC_BETA_HPP
+
+#include <stan/math/prim/meta.hpp>
+#include <stan/math/prim/err.hpp>
+#include <stan/math/prim/fun/boost_policy.hpp>
+#include <boost/math/special_functions/beta.hpp>
+
+namespace stan {
+namespace math {
+
+/**
+ * The inverse of the normalized incomplete beta function of a, b, with
+ * probability p.
+ *
+ * Used to compute the inverse cumulative density function for the beta
+ * distribution.
+ *
+ * @param a Shape parameter a >= 0; a and b can't both be 0
+ * @param b Shape parameter b >= 0
+ * @param p Random variate. 0 <= p <= 1
+ * @throws if constraints are violated or if any argument is NaN
+ * @return The inverse of the normalized incomplete beta function.
+ */
+inline double inv_inc_beta(double a, double b, double p) {
+  check_not_nan("inv_inc_beta", "a", a);
+  check_not_nan("inv_inc_beta", "b", b);
+  check_not_nan("inv_inc_beta", "p", p);
+  return boost::math::ibeta_inv(a, b, p, boost_policy_t<>());
+}
+
+}  // namespace math
+}  // namespace stan
+#endif

stan/math/rev/fun.hpp

@@ -79,6 +79,7 @@
 #include <stan/math/rev/fun/identity_free.hpp>
 #include <stan/math/rev/fun/if_else.hpp>
 #include <stan/math/rev/fun/inc_beta.hpp>
+#include <stan/math/rev/fun/inv_inc_beta.hpp>
 #include <stan/math/rev/fun/initialize_fill.hpp>
 #include <stan/math/rev/fun/initialize_variable.hpp>
 #include <stan/math/rev/fun/inv.hpp>

stan/math/rev/fun/inv_inc_beta.hpp

@@ -0,0 +1,105 @@
+#ifndef STAN_MATH_REV_FUN_INV_INC_BETA_HPP
+#define STAN_MATH_REV_FUN_INV_INC_BETA_HPP
+
+#include <stan/math/rev/meta.hpp>
+#include <stan/math/rev/core.hpp>
+#include <stan/math/prim/fun/constants.hpp>
+#include <stan/math/prim/fun/inv_inc_beta.hpp>
+#include <stan/math/prim/fun/inc_beta.hpp>
+#include <stan/math/prim/fun/exp.hpp>
+#include <stan/math/prim/fun/log.hpp>
+#include <stan/math/prim/fun/log_diff_exp.hpp>
+#include <stan/math/prim/fun/lbeta.hpp>
+#include <stan/math/prim/fun/lgamma.hpp>
+#include <stan/math/prim/fun/digamma.hpp>
+#include <stan/math/prim/fun/F32.hpp>
+#include <stan/math/prim/fun/is_any_nan.hpp>
+
+namespace stan {
+namespace math {
+
+/**
+ * The inverse of the normalized incomplete beta function of a, b, with
+ * probability p.
+ *
+ * Used to compute the inverse cumulative density function for the beta
+ * distribution.
+ *
+   \f[
+   \frac{\partial }{\partial a} =
+    (1-w)^{1-b}w^{1-a}
+      \left(
+        w^a\Gamma(a)^2 {}_3\tilde{F}_2(a,a,1-b;a+1,a+1;w)
+        - B(a,b)I_w(a,b)\left(\log(w)-\psi(a) + \psi(a+b)\right)
+      \right)/;w=I_z^{-1}(a,b)
+   \f]
+   \f[
+   \frac{\partial }{\partial b} =
+    (1-w)^{-b}w^{1-a}(w-1)
+      \left(
+        (1-w)^{b}\Gamma(b)^2 {}_3\tilde{F}_2(b,b,1-a;b+1,b+1;1-w)
+        - B_{1-w}(b,a)\left(\log(1-w)-\psi(b) + \psi(a+b)\right)
+      \right)/;w=I_z^{-1}(a,b)
+   \f]
+   \f[
+   \frac{\partial }{\partial z} = (1-w)^{1-b}w^{1-a}B(a,b)/;w=I_z^{-1}(a,b)
+   \f]
+ *
+ * @param a Shape parameter a >= 0; a and b can't both be 0
+ * @param b Shape parameter b >= 0
+ * @param p Random variate. 0 <= p <= 1
+ * @throws if constraints are violated or if any argument is NaN
+ * @return The inverse of the normalized incomplete beta function.
+ */
+template <typename T1, typename T2, typename T3,
+          require_all_stan_scalar_t<T1, T2, T3>* = nullptr,
+          require_any_var_t<T1, T2, T3>* = nullptr>
+inline var inv_inc_beta(const T1& a, const T2& b, const T3& p) {
+  double a_val = value_of(a);
+  double b_val = value_of(b);
+  double p_val = value_of(p);
+  double w = inv_inc_beta(a_val, b_val, p_val);
+  return make_callback_var(w, [a, b, p, a_val, b_val, p_val, w](auto& vi) {
+    double log_w = log(w);
+    double log1m_w = log1m(w);
+    double one_m_a = 1 - a_val;
+    double one_m_b = 1 - b_val;
+    double one_m_w = 1 - w;
+    double ap1 = a_val + 1;
+    double bp1 = b_val + 1;
+    double lbeta_ab = lbeta(a_val, b_val);
+    double digamma_apb = digamma(a_val + b_val);
+
+    if (!is_constant_all<T1>::value) {
+      double da1 = exp(one_m_b * log1m_w + one_m_a * log_w);
+      double da2 = a_val * log_w + 2 * lgamma(a_val)
+                   + log(F32(a_val, a_val, one_m_b, ap1, ap1, w))
+                   - 2 * lgamma(ap1);
+      double da3 = inc_beta(a_val, b_val, w) * exp(lbeta_ab)
+                   * (log_w - digamma(a_val) + digamma_apb);
+
+      forward_as<var>(a).adj() += vi.adj() * da1 * (exp(da2) - da3);
+    }
+
+    if (!is_constant_all<T2>::value) {
+      double db1 = (w - 1) * exp(-b_val * log1m_w + one_m_a * log_w);
+      double db2 = 2 * lgamma(b_val)
+                   + log(F32(b_val, b_val, one_m_a, bp1, bp1, one_m_w))
+                   - 2 * lgamma(bp1) + b_val * log1m_w;
+
+      double db3 = inc_beta(b_val, a_val, one_m_w) * exp(lbeta_ab)
+                   * (log1m_w - digamma(b_val) + digamma_apb);
+
+      forward_as<var>(b).adj() += vi.adj() * db1 * (exp(db2) - db3);
+    }
+
+    if (!is_constant_all<T3>::value) {
+      forward_as<var>(p).adj()
+          += vi.adj() * exp(one_m_b * log1m_w + one_m_a * log_w + lbeta_ab);
+    }
+  });
+}
+
+}  // namespace math
+}  // namespace stan
+#endif

test/unit/math/fwd/fun/inv_inc_beta_test.cpp

@@ -0,0 +1,33 @@
+#include <stan/math/fwd.hpp>
+#include <gtest/gtest.h>
+
+TEST(AgradFwdMatrixIncBetaInv, fd_scalar) {
+  using stan::math::fvar;
+  using stan::math::inv_inc_beta;
+  fvar<double> a = 6;
+  fvar<double> b = 2;
+  fvar<double> p = 0.9;
+  a.d_ = 1.0;
+  b.d_ = 1.0;
+  p.d_ = 1.0;
+
+  fvar<double> res = inv_inc_beta(a, b, p);
+
+  EXPECT_FLOAT_EQ(res.d_, 0.0117172527399 - 0.0680999818473 + 0.455387298585);
+}
+
+TEST(AgradFwdMatrixIncBetaInv, ffd_scalar) {
+  using stan::math::fvar;
+  using stan::math::inv_inc_beta;
+  fvar<fvar<double>> a = 7;
+  fvar<fvar<double>> b = 4;
+  fvar<fvar<double>> p = 0.15;
+  a.val_.d_ = 1.0;
+  b.val_.d_ = 1.0;
+  p.val_.d_ = 1.0;
+
+  fvar<fvar<double>> res = inv_inc_beta(a, b, p);
+
+  EXPECT_FLOAT_EQ(res.val_.d_,
+                  0.0428905418857 - 0.0563420377808 + 0.664919819507);
+}