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);
+  check_positive(function, "Shape parameter",
+                 beta_val);  // With β = +∞ this could be defined to be uniform,
+                             // but we don't support that.
+
+  if (size_zero(y, mu, alpha, beta)) {
+    return 0;
+  }
+  if (!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);
+  const auto& scaled_abs_diff_pow
+      = to_ref_if<!is_constant_all<T_scale, T_shape>::value>(
+          pow(scaled_abs_diff, beta_val));
+  const size_t N = max_size(y, mu, alpha, beta);
+
+  T_partials_return logp = -sum(scaled_abs_diff_pow);
+
+  if (include_summand<propto>::value) {
+    logp -= LOG_TWO * N;
+  }
+  if (include_summand<propto, T_scale>::value) {
+    logp -= sum(log(alpha_val)) * (N / math::size(alpha));
+  }
+  if (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 (!is_constant_all<T_y, T_loc>::value) {
+    // note: The partial derivatives for y, mu are undefined when y == mu &&
+    // beta < 1. The derivative limit as mu -> y goes:
+    //   to 0 from both sides if beta > 1 (defined as 0)
+    //   to +1/alpha from right but -1/alpha from left if beta == 1 (defined as
+    //   0, consistent with double_exponential_lpdf) to +∞ from right but -∞
+    //   from left as y -> mu if beta < 1 (undefined)
+    const auto& rep_deriv
+        = to_ref_if < !is_constant<T_y>::value
+          && !is_constant<T_loc>::value
+                 > (sign(diff) * beta_val * pow(scaled_abs_diff, beta_val - 1)
+                    * inv_alpha);
+    if (!is_constant<T_y>::value) {
+      partials<0>(ops_partials) = -rep_deriv;
+    }
+    if (!is_constant<T_loc>::value) {
+      partials<1>(ops_partials) = std::move(rep_deriv);
+    }
+  }
+  if (!is_constant<T_scale>::value) {
+    partials<2>(ops_partials)
+        = (beta_val * scaled_abs_diff_pow - 1) * inv_alpha;
+  }
+  if (!is_constant<T_shape>::value) {
+    partials<3>(ops_partials)
+        = digamma(inv_beta1p) * inv_square(beta_val)
+          - multiply_log(scaled_abs_diff_pow, scaled_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

test/prob/generalized_normal/generalized_normal_test.hpp

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+// Arguments: Doubles, Doubles, Doubles, Doubles
+#include <stan/math/prim/prob/generalized_normal_lpdf.hpp>
+#include <stan/math/prim/fun/abs.hpp>
+#include <stan/math/prim/fun/log.hpp>
+#include <stan/math/prim/fun/pow.hpp>
+#include <stan/math/prim/fun/tgamma.hpp>
+#include <stan/math/prim/fun/constants.hpp>
+
+using std::numeric_limits;
+using std::vector;
+
+class AgradDistributionGeneralizedNormal : public AgradDistributionTest {
+ public:
+  void valid_values(vector<vector<double> >& parameters,
+                    vector<double>& log_prob) {
+    vector<double> param(4);
+
+    param[0] = 0;  // y
+    param[1] = 0;  // mu
+    param[2] = 1;  // alpha
+    param[3] = 2;  // beta
+    parameters.push_back(param);
+    log_prob.push_back(
+        -0.57236494292470008707171367567652935582);  // expected log_prob
+
+    param[0] = 1;  // y
+    param[1] = 0;  // mu
+    param[2] = 1;  // alpha
+    param[3] = 2;  // beta
+    parameters.push_back(param);
+    log_prob.push_back(
+        -1.5723649429247000870717136756765293558);  // expected log_prob
+
+    param[0] = -2;  // y
+    param[1] = 0;   // mu
+    param[2] = 1;   // alpha
+    param[3] = 2;   // beta
+    parameters.push_back(param);
+    log_prob.push_back(
+        -4.5723649429247000870717136756765293558);  // expected log_prob
+
+    param[0] = -3.5;  // y
+    param[1] = 1.9;   // mu
+    param[2] = 7.2;   // alpha
+    param[3] = 2;     // beta
+    parameters.push_back(param);
+    log_prob.push_back(
+        -3.1089459689467097140959090592117462617);  // expected log_prob
+
+    param[0] = 0;  // y
+    param[1] = 0;  // mu
+    param[2] = 1;  // alpha
+    param[3] = 1;  // beta
+    parameters.push_back(param);
+    log_prob.push_back(
+        -0.69314718055994530941723212145817656808);  // expected log_prob
+
+    param[0] = 1;  // y
+    param[1] = 0;  // mu
+    param[2] = 1;  // alpha
+    param[3] = 1;  // beta
+    parameters.push_back(param);
+    log_prob.push_back(
+        -1.6931471805599453094172321214581765681);  // expected log_prob
+
+    param[0] = -2;  // y
+    param[1] = 0;   // mu
+    param[2] = 1;   // alpha
+    param[3] = 1;   // beta
+    parameters.push_back(param);
+    log_prob.push_back(
+        -2.6931471805599453094172321214581765681);  // expected log_prob
+
+    param[0] = -3.5;  // y
+    param[1] = 1.9;   // mu
+    param[2] = 7.2;   // alpha
+    param[3] = 1;     // beta
+    parameters.push_back(param);
+    log_prob.push_back(
+        -3.4172282065819549364414275049933934740);  // expected log_prob
+
+    param[0] = 0;    // y
+    param[1] = 0;    // mu
+    param[2] = 1;    // alpha
+    param[3] = 1.5;  // beta
+    parameters.push_back(param);
+    log_prob.push_back(
+        -0.59083234759930449611508182336583846717);  // expected log_prob
+
+    param[0] = 1;    // y
+    param[1] = 0;    // mu
+    param[2] = 1;    // alpha
+    param[3] = 1.5;  // beta
+    parameters.push_back(param);
+    log_prob.push_back(
+        -1.5908323475993044961150818233658384672);  // expected log_prob
+
+    param[0] = -2;   // y
+    param[1] = 0;    // mu
+    param[2] = 1;    // alpha
+    param[3] = 1.5;  // beta
+    parameters.push_back(param);
+    log_prob.push_back(
+        -3.4192594723454945937184592717852346243);  // expected log_prob
+
+    param[0] = -3.5;  // y
+    param[1] = 1.9;   // mu
+    param[2] = 7.2;   // alpha
+    param[3] = 1.5;   // beta
+    parameters.push_back(param);
+    log_prob.push_back(
+        -3.2144324264596431082120695849657575107);  // expected log_prob
+  }
+
+  void invalid_values(vector<size_t>& index, vector<double>& value) {
+    // y
+
+    // mu
+    index.push_back(1U);
+    value.push_back(numeric_limits<double>::infinity());
+
+    index.push_back(1U);
+    value.push_back(-numeric_limits<double>::infinity());
+
+    // alpha
+    index.push_back(2U);
+    value.push_back(0.0);
+
+    index.push_back(2U);
+    value.push_back(-1.0);
+
+    index.push_back(2U);
+    value.push_back(-numeric_limits<double>::infinity());
+
+    // beta
+    index.push_back(3U);
+    value.push_back(0.0);
+
+    index.push_back(3U);
+    value.push_back(-1.0);
+
+    index.push_back(3U);
+    value.push_back(-numeric_limits<double>::infinity());
+  }
+
+  template <typename T_y, typename T_loc, typename T_scale, typename T_shape,
+            typename T5, typename T6>
+  stan::return_type_t<T_y, T_loc, T_scale, T_shape> log_prob(
+      const T_y& y, const T_loc& mu, const T_scale& alpha, const T_shape& beta,
+      const T5&, const T6&) {
+    return stan::math::generalized_normal_lpdf(y, mu, alpha, beta);
+  }
+
+  template <bool propto, typename T_y, typename T_loc, typename T_scale,
+            typename T_shape, typename T5, typename T6>
+  stan::return_type_t<T_y, T_loc, T_scale, T_shape> log_prob(
+      const T_y& y, const T_loc& mu, const T_scale& alpha, const T_shape& beta,
+      const T5&, const T6&) {
+    return stan::math::generalized_normal_lpdf<propto>(y, mu, alpha, beta);
+  }
+
+  template <typename T_y, typename T_loc, typename T_scale, typename T_shape,
+            typename T5, typename T6>
+  stan::return_type_t<T_y, T_loc, T_scale, T_shape> log_prob_function(
+      const T_y& y, const T_loc& mu, const T_scale& alpha, const T_shape& beta,
+      const T5&, const T6&) {
+    using stan::math::abs;
+    using stan::math::inv;
+    using stan::math::lgamma;
+    using stan::math::log;
+    using stan::math::LOG_TWO;
+
+    return -LOG_TWO - log(alpha) - lgamma(1.0 + inv(beta))
+           - pow(abs(y - mu) / alpha, beta);
+  }
+};