Generalized normal distribution

stan/math/prim/prob/generalized_normal_lpdf.hpp

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+#ifndef STAN_MATH_PRIM_PROB_GENERALIZED_NORMAL_LPDF_HPP
+#define STAN_MATH_PRIM_PROB_GENERALIZED_NORMAL_LPDF_HPP
+
+#include <cmath>
+#include <stan/math/prim/err.hpp>
+#include <stan/math/prim/functor/partials_propagator.hpp>
+#include <stan/math/prim/fun/abs.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/digamma.hpp>
+#include <stan/math/prim/fun/lgamma.hpp>
+#include <stan/math/prim/fun/log.hpp>
+#include <stan/math/prim/fun/max_size.hpp>
+#include <stan/math/prim/fun/multiply_log.hpp>
+#include <stan/math/prim/fun/pow.hpp>
+#include <stan/math/prim/fun/sign.hpp>
+#include <stan/math/prim/fun/size.hpp>
+#include <stan/math/prim/fun/size_zero.hpp>
+#include <stan/math/prim/fun/square.hpp>
+#include <stan/math/prim/fun/to_ref.hpp>
+#include <stan/math/prim/fun/value_of.hpp>
+#include <stan/math/prim/meta.hpp>
+
+namespace stan {
+namespace math {
+
+/** \ingroup prob_dists
+ * The log of the generalized normal density for the specified scalar(s) given
+ * the specified location, scale and shape parameters. y, mu, alpha, or beta can
+ * each be either a scalar or a vector. Any vector inputs must be the same
+ * length.
+ *
+ * <p>The result log probability is defined to be the sum of the
+ * log probabilities for each observation/mean/scale/shape tuple.
+ *
+ * @tparam T_y type of scalar
+ * @tparam T_loc type of location parameter
+ * @tparam T_scale type of scale parameter
+ * @tparam T_shape type of shape parameter
+ * @param y (Sequence of) scalar(s)
+ * @param mu (Sequence of) location parameter(s)
+ * @param alpha (Sequence of) scale parameter(s)
+ * @param beta (Sequence of) shape parameter(s)
+ * @return The log of the product of the densities
+ * @throw std::domain_error if alpha or beta is not positive
+ */
+template <bool propto, typename T_y, typename T_loc, typename T_scale,
+          typename T_shape,
+          require_all_not_nonscalar_prim_or_rev_kernel_expression_t<
+              T_y, T_loc, T_scale, T_shape>* = nullptr>
+inline return_type_t<T_y, T_loc, T_scale, T_shape> generalized_normal_lpdf(
+    T_y&& y, T_loc&& mu, T_scale&& alpha, T_shape&& beta) {
+  using T_partials_return = partials_return_t<T_y, T_loc, T_scale, T_shape>;
+  using T_y_ref = ref_type_if_not_constant_t<T_y>;
+  using T_mu_ref = ref_type_if_not_constant_t<T_loc>;
+  using T_alpha_ref = ref_type_if_not_constant_t<T_scale>;
+  using T_beta_ref = ref_type_if_not_constant_t<T_shape>;
+  static constexpr const char* function = "generalized_normal_lpdf";
+  check_consistent_sizes(function, "Random variable", y, "Location parameter",
+                         mu, "Scale parameter", alpha, "Shape parameter", beta);
+
+  T_y_ref y_ref = std::forward<T_y>(y);
+  T_mu_ref mu_ref = std::forward<T_loc>(mu);
+  T_alpha_ref alpha_ref = std::forward<T_scale>(alpha);
+  T_beta_ref beta_ref = std::forward<T_shape>(beta);
+
+  decltype(auto) y_val = to_ref(as_value_column_array_or_scalar(y_ref));
+  decltype(auto) mu_val = to_ref(as_value_column_array_or_scalar(mu_ref));
+  decltype(auto) alpha_val = to_ref(as_value_column_array_or_scalar(alpha_ref));
+  decltype(auto) beta_val = to_ref(as_value_column_array_or_scalar(beta_ref));
+
+  check_not_nan(function, "Random variable", y_val);
+  check_finite(function, "Location parameter", mu_val);
+  check_positive(function, "Scale parameter", alpha_val);
+
+  // With β = +∞ this could be defined to be uniform, but we don't support that.
+  check_positive(function, "Shape parameter", beta_val);
+
+  if (size_zero(y, mu, alpha, beta)) {
+    return 0;
+  }
+  if constexpr (!include_summand<propto, T_y, T_loc, T_scale, T_shape>::value) {
+    return 0;
+  }
+
+  const auto& inv_beta1p
+      = to_ref_if<!is_constant<T_shape>::value>(inv(beta_val) + 1);
+  const auto& diff
+      = to_ref_if<!is_constant_all<T_y, T_loc>::value>(y_val - mu_val);
+  const auto& inv_alpha = to_ref(inv(alpha_val));
+  const auto& scaled_abs_diff
+      = to_ref_if<!is_constant_all<T_y, T_loc, T_shape>::value>(abs(diff)
+                                                                * inv_alpha);
+  // pow(0, beta) is 0 for beta > 0, but autodiff of pow at base == 0
+  // produces NaN (0 * log(0) in the tangent) and pollutes the autodiff stack.
+  // Replace 0 with 1 in the base (pow(1, p) is always clean), then zero
+  // out those results afterward.
+  const auto& sad_is_zero = eval(value_of_rec(scaled_abs_diff) == 0);
+  auto safe_abs_diff = eval(scaled_abs_diff);
+  if constexpr (is_eigen_v<std::decay_t<decltype(sad_is_zero)>>) {
+    for (Eigen::Index i = 0; i < sad_is_zero.size(); ++i)
+      if (sad_is_zero.coeff(i))
+        safe_abs_diff.coeffRef(i) += 1;
+  } else {
+    if (sad_is_zero)
+      safe_abs_diff += 1;
+  }
+  auto scaled_abs_diff_pow = eval(pow(safe_abs_diff, beta_val));
+  if constexpr (is_eigen_v<std::decay_t<decltype(sad_is_zero)>>) {
+    for (Eigen::Index i = 0; i < sad_is_zero.size(); ++i) {
+      if (sad_is_zero.coeff(i))
+        scaled_abs_diff_pow.coeffRef(i) = 0;
+    }
+  } else {
+    if (sad_is_zero)
+      scaled_abs_diff_pow = 0;
+  }
+  const size_t N = max_size(y, mu, alpha, beta);
+
+  T_partials_return logp = -sum(scaled_abs_diff_pow);
+
+  if constexpr (include_summand<propto>::value) {
+    logp -= LOG_TWO * N;
+  }
+  if constexpr (include_summand<propto, T_scale>::value) {
+    logp -= sum(log(alpha_val)) * (N / math::size(alpha));
+  }
+  if constexpr (include_summand<propto, T_shape>::value) {
+    logp -= sum(lgamma(inv_beta1p)) * (N / math::size(beta));
+  }
+
+  auto ops_partials
+      = make_partials_propagator(y_ref, mu_ref, alpha_ref, beta_ref);
+
+  if constexpr (!is_constant_all<T_y, T_loc>::value) {
+    // At y == mu, the derivative is 0 for beta >= 1, undefined for beta < 1.
+    // Use safe_abs_diff (0 replaced with 1) to avoid NaN from pow(0, beta-1).
+    auto rep_deriv = eval(sign(diff) * beta_val
+                          * pow(safe_abs_diff, beta_val - 1) * inv_alpha);
+    if constexpr (is_eigen_v<std::decay_t<decltype(sad_is_zero)>>) {
+      for (Eigen::Index i = 0; i < sad_is_zero.size(); ++i) {
+        if (sad_is_zero.coeff(i))
+          rep_deriv.coeffRef(i) = 0;
+      }
+    } else {
+      if (sad_is_zero)
+        rep_deriv = 0;
+    }
+    if constexpr (!is_constant<T_y>::value) {
+      partials<0>(ops_partials) = -rep_deriv;
+    }
+    if constexpr (!is_constant<T_loc>::value) {
+      partials<1>(ops_partials) = std::move(rep_deriv);
+    }
+  }
+  if constexpr (!is_constant<T_scale>::value) {
+    partials<2>(ops_partials)
+        = (beta_val * scaled_abs_diff_pow - 1) * inv_alpha;
+  }
+  if constexpr (!is_constant<T_shape>::value) {
+    // multiply_log(0, 0) = 0 by convention, but fvar autodiff of
+    // multiply_log at (0, 0) produces NaN.  safe_abs_diff avoids this.
+    partials<3>(ops_partials)
+        = digamma(inv_beta1p) * inv_square(beta_val)
+          - multiply_log(scaled_abs_diff_pow, safe_abs_diff);
+  }
+
+  return ops_partials.build(logp);
+}
+
+template <typename T_y, typename T_loc, typename T_scale, typename T_shape>
+inline return_type_t<T_y, T_loc, T_scale, T_shape> generalized_normal_lpdf(
+    T_y&& y, T_loc&& mu, T_scale&& alpha, T_shape&& beta) {
+  return generalized_normal_lpdf<false>(
+      std::forward<T_y>(y), std::forward<T_loc>(mu),
+      std::forward<T_scale>(alpha), std::forward<T_shape>(beta));
+}
+
+}  // namespace math
+}  // namespace stan
+#endif