student_t_lccdf.hpp

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#ifndef STAN_MATH_PRIM_PROB_STUDENT_T_LCCDF_HPP
#define STAN_MATH_PRIM_PROB_STUDENT_T_LCCDF_HPP
 
#include <stan/math/prim/meta.hpp>
#include <stan/math/prim/err.hpp>
#include <stan/math/prim/fun/beta.hpp>
#include <stan/math/prim/fun/constants.hpp>
#include <stan/math/prim/fun/digamma.hpp>
#include <stan/math/prim/fun/grad_reg_inc_beta.hpp>
#include <stan/math/prim/fun/inc_beta.hpp>
#include <stan/math/prim/fun/log.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/value_of.hpp>
#include <stan/math/prim/functor/partials_propagator.hpp>
#include <cmath>
 
namespace stan {
namespace math {
 
template <typename T_y, typename T_dof, typename T_loc, typename T_scale>
return_type_t<T_y, T_dof, T_loc, T_scale> student_t_lccdf(
    const T_y& y, const T_dof& nu, const T_loc& mu, const T_scale& sigma) {
  using T_partials_return = partials_return_t<T_y, T_dof, T_loc, T_scale>;
  using T_y_ref = ref_type_t<T_y>;
  using T_nu_ref = ref_type_t<T_dof>;
  using T_mu_ref = ref_type_t<T_loc>;
  using T_sigma_ref = ref_type_t<T_scale>;
  using std::exp;
  using std::log;
  using std::pow;
  static constexpr const char* function = "student_t_lccdf";
  T_y_ref y_ref = y;
  T_nu_ref nu_ref = nu;
  T_mu_ref mu_ref = mu;
  T_sigma_ref sigma_ref = sigma;
  check_not_nan(function, "Random variable", y_ref);
  check_positive_finite(function, "Degrees of freedom parameter", nu_ref);
  check_finite(function, "Location parameter", mu_ref);
  check_positive_finite(function, "Scale parameter", sigma_ref);
 
  if (size_zero(y, nu, mu, sigma)) {
    return 0;
  }
 
  T_partials_return P(0.0);
  auto ops_partials
      = make_partials_propagator(y_ref, nu_ref, mu_ref, sigma_ref);
  scalar_seq_view<T_y_ref> y_vec(y_ref);
  scalar_seq_view<T_nu_ref> nu_vec(nu_ref);
  scalar_seq_view<T_mu_ref> mu_vec(mu_ref);
  scalar_seq_view<T_sigma_ref> sigma_vec(sigma_ref);
  size_t N = max_size(y, nu, mu, sigma);
 
  // 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) == NEGATIVE_INFTY) {
      return ops_partials.build(0.0);
    }
  }
 
  T_partials_return digammaHalf = 0;
 
  VectorBuilder<!is_constant_all<T_dof>::value, T_partials_return, T_dof>
      digamma_vec(math::size(nu));
  VectorBuilder<!is_constant_all<T_dof>::value, T_partials_return, T_dof>
      digammaNu_vec(math::size(nu));
  VectorBuilder<!is_constant_all<T_dof>::value, T_partials_return, T_dof>
      digammaNuPlusHalf_vec(math::size(nu));
 
  if (!is_constant_all<T_dof>::value) {
    digammaHalf = digamma(0.5);
 
    for (size_t i = 0; i < stan::math::size(nu); i++) {
      const T_partials_return nu_dbl = nu_vec.val(i);
 
      digammaNu_vec[i] = digamma(0.5 * nu_dbl);
      digammaNuPlusHalf_vec[i] = digamma(0.5 + 0.5 * nu_dbl);
    }
  }
 
  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) {
      return ops_partials.build(negative_infinity());
    }
 
    const T_partials_return sigma_inv = 1.0 / sigma_vec.val(n);
    const T_partials_return t = (y_vec.val(n) - mu_vec.val(n)) * sigma_inv;
    const T_partials_return nu_dbl = nu_vec.val(n);
    const T_partials_return q = nu_dbl / (t * t);
    const T_partials_return r = 1.0 / (1.0 + q);
    const T_partials_return J = 2 * r * r * q / t;
    const T_partials_return betaNuHalf = beta(0.5, 0.5 * nu_dbl);
    T_partials_return zJacobian = t > 0 ? -0.5 : 0.5;
 
    if (q < 2) {
      T_partials_return z
          = inc_beta(0.5 * nu_dbl, (T_partials_return)0.5, 1.0 - r);
      const T_partials_return Pn = t > 0 ? 0.5 * z : 1.0 - 0.5 * z;
      const T_partials_return d_ibeta
          = pow(r, -0.5) * pow(1.0 - r, 0.5 * nu_dbl - 1) / betaNuHalf;
 
      P += log(Pn);
 
      if (!is_constant_all<T_y>::value) {
        partials<0>(ops_partials)[n]
            += zJacobian * d_ibeta * J * sigma_inv / Pn;
      }
 
      if (!is_constant_all<T_dof>::value) {
        T_partials_return g1 = 0;
        T_partials_return g2 = 0;
 
        grad_reg_inc_beta(g1, g2, 0.5 * nu_dbl, (T_partials_return)0.5, 1.0 - r,
                          digammaNu_vec[n], digammaHalf,
                          digammaNuPlusHalf_vec[n], betaNuHalf);
 
        partials<1>(ops_partials)[n]
            -= zJacobian * (d_ibeta * (r / t) * (r / t) + 0.5 * g1) / Pn;
      }
 
      if (!is_constant_all<T_loc>::value) {
        partials<2>(ops_partials)[n]
            -= zJacobian * d_ibeta * J * sigma_inv / Pn;
      }
      if (!is_constant_all<T_scale>::value) {
        partials<3>(ops_partials)[n]
            -= zJacobian * d_ibeta * J * sigma_inv * t / Pn;
      }
 
    } else {
      T_partials_return z
          = 1.0 - inc_beta((T_partials_return)0.5, 0.5 * nu_dbl, r);
      zJacobian *= -1;
 
      const T_partials_return Pn = t > 0 ? 0.5 * z : 1.0 - 0.5 * z;
 
      T_partials_return d_ibeta
          = pow(1.0 - r, 0.5 * nu_dbl - 1) * pow(r, -0.5) / betaNuHalf;
 
      P += log(Pn);
 
      if (!is_constant_all<T_y>::value) {
        partials<0>(ops_partials)[n]
            -= zJacobian * d_ibeta * J * sigma_inv / Pn;
      }
 
      if (!is_constant_all<T_dof>::value) {
        T_partials_return g1 = 0;
        T_partials_return g2 = 0;
 
        grad_reg_inc_beta(g1, g2, (T_partials_return)0.5, 0.5 * nu_dbl, r,
                          digammaHalf, digammaNu_vec[n],
                          digammaNuPlusHalf_vec[n], betaNuHalf);
 
        partials<1>(ops_partials)[n]
            -= zJacobian * (-d_ibeta * (r / t) * (r / t) + 0.5 * g2) / Pn;
      }
 
      if (!is_constant_all<T_loc>::value) {
        partials<2>(ops_partials)[n]
            += zJacobian * d_ibeta * J * sigma_inv / Pn;
      }
      if (!is_constant_all<T_scale>::value) {
        partials<3>(ops_partials)[n]
            += zJacobian * d_ibeta * J * sigma_inv * t / Pn;
      }
    }
  }
  return ops_partials.build(P);
}
 
}  // namespace math
}  // namespace stan
#endif