pareto_lcdf.hpp

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#ifndef STAN_MATH_PRIM_PROB_PARETO_LCDF_HPP
#define STAN_MATH_PRIM_PROB_PARETO_LCDF_HPP
 
#include <stan/math/prim/meta.hpp>
#include <stan/math/prim/err.hpp>
#include <stan/math/prim/fun/as_column_vector_or_scalar.hpp>
#include <stan/math/prim/fun/as_array_or_scalar.hpp>
#include <stan/math/prim/fun/as_value_column_array_or_scalar.hpp>
#include <stan/math/prim/fun/constants.hpp>
#include <stan/math/prim/fun/exp.hpp>
#include <stan/math/prim/fun/log.hpp>
#include <stan/math/prim/fun/max_size.hpp>
#include <stan/math/prim/fun/promote_scalar.hpp>
#include <stan/math/prim/fun/size.hpp>
#include <stan/math/prim/fun/size_zero.hpp>
#include <stan/math/prim/fun/to_ref.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_scale, typename T_shape,
          require_all_not_nonscalar_prim_or_rev_kernel_expression_t<
              T_y, T_scale, T_shape>* = nullptr>
inline return_type_t<T_y, T_scale, T_shape> pareto_lcdf(const T_y& y,
                                                        const T_scale& y_min,
                                                        const T_shape& alpha) {
  using T_partials_return = partials_return_t<T_y, T_scale, T_shape>;
  using T_y_ref = ref_type_if_not_constant_t<T_y>;
  using T_y_min_ref = ref_type_if_not_constant_t<T_scale>;
  using T_alpha_ref = ref_type_if_not_constant_t<T_shape>;
  using std::isinf;
  static constexpr const char* function = "pareto_lcdf";
  check_consistent_sizes(function, "Random variable", y, "Scale parameter",
                         y_min, "Shape parameter", alpha);
 
  if (size_zero(y, y_min, alpha)) {
    return 0;
  }
 
  T_y_ref y_ref = y;
  T_y_min_ref y_min_ref = y_min;
  T_alpha_ref alpha_ref = alpha;
 
  decltype(auto) y_val = to_ref(as_value_column_array_or_scalar(y_ref));
  decltype(auto) y_min_val = to_ref(as_value_column_array_or_scalar(y_min_ref));
  decltype(auto) alpha_val = to_ref(as_value_column_array_or_scalar(alpha_ref));
 
  check_nonnegative(function, "Random variable", y_val);
  check_positive_finite(function, "Scale parameter", y_min_val);
  check_positive_finite(function, "Shape parameter", alpha_val);
 
  auto ops_partials = make_partials_propagator(y_ref, y_min_ref, alpha_ref);
 
  // Explicit return for extreme values
  // The gradients are technically ill-defined, but treated as zero
  if (sum(promote_scalar<int>(y_val < y_min_val))) {
    return ops_partials.build(negative_infinity());
  }
  if (sum(promote_scalar<int>(isinf(y_val)))) {
    return ops_partials.build(0.0);
  }
 
  const auto& log_quot = to_ref_if<is_any_autodiff_v<T_y, T_scale, T_shape>>(
      log(y_min_val / y_val));
  const auto& exp_prod = to_ref_if<is_any_autodiff_v<T_y, T_scale, T_shape>>(
      exp(alpha_val * log_quot));
  // TODO(Andrew) Further simplify derivatives and log1m_exp below
  T_partials_return P = sum(log1m(exp_prod));
 
  if constexpr (is_any_autodiff_v<T_y, T_scale, T_shape>) {
    const auto& common_deriv = to_ref_if<(
        is_any_autodiff_v<T_y, T_scale> && is_autodiff_v<T_shape>)>(
        exp_prod / (1 - exp_prod));
    if constexpr (is_any_autodiff_v<T_y, T_scale>) {
      const auto& y_min_inv = inv(y_min_val);
      auto common_deriv2
          = to_ref_if<(is_autodiff_v<T_y> && is_autodiff_v<T_scale>)>(
              -alpha_val * y_min_inv * common_deriv);
      if constexpr (is_autodiff_v<T_y>) {
        partials<0>(ops_partials) = -common_deriv2 * exp(log_quot);
      }
      if constexpr (is_autodiff_v<T_scale>) {
        partials<1>(ops_partials) = std::move(common_deriv2);
      }
    }
    if constexpr (is_autodiff_v<T_shape>) {
      partials<2>(ops_partials) = -common_deriv * log_quot;
    }
  }
 
  return ops_partials.build(P);
}
 
}  // namespace math
}  // namespace stan
#endif