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beta_distribution.h (14704B)

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// Copyright 2017 The Abseil Authors.
//
// Licensed under the Apache License, Version 2.0 (the "License");
// you may not use this file except in compliance with the License.
// You may obtain a copy of the License at
//
//      https://www.apache.org/licenses/LICENSE-2.0
//
// Unless required by applicable law or agreed to in writing, software
// distributed under the License is distributed on an "AS IS" BASIS,
// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
// See the License for the specific language governing permissions and
// limitations under the License.

#ifndef ABSL_RANDOM_BETA_DISTRIBUTION_H_
#define ABSL_RANDOM_BETA_DISTRIBUTION_H_

#include <cassert>
#include <cmath>
#include <cstdint>
#include <istream>
#include <limits>
#include <ostream>
#include <type_traits>

#include "absl/base/attributes.h"
#include "absl/base/config.h"
#include "absl/meta/type_traits.h"
#include "absl/random/internal/fast_uniform_bits.h"
#include "absl/random/internal/generate_real.h"
#include "absl/random/internal/iostream_state_saver.h"

namespace absl {
ABSL_NAMESPACE_BEGIN

// absl::beta_distribution:
// Generate a floating-point variate conforming to a Beta distribution:
//   pdf(x) \propto x^(alpha-1) * (1-x)^(beta-1),
// where the params alpha and beta are both strictly positive real values.
//
// The support is the open interval (0, 1), but the return value might be equal
// to 0 or 1, due to numerical errors when alpha and beta are very different.
//
// Usage note: One usage is that alpha and beta are counts of number of
// successes and failures. When the total number of trials are large, consider
// approximating a beta distribution with a Gaussian distribution with the same
// mean and variance. One could use the skewness, which depends only on the
// smaller of alpha and beta when the number of trials are sufficiently large,
// to quantify how far a beta distribution is from the normal distribution.
template <typename RealType = double>
class beta_distribution {
public:
 using result_type = RealType;

 class param_type {
  public:
   using distribution_type = beta_distribution;

   explicit param_type(result_type alpha, result_type beta)
       : alpha_(alpha), beta_(beta) {
     assert(alpha >= 0);
     assert(beta >= 0);
     assert(alpha <= (std::numeric_limits<result_type>::max)());
     assert(beta <= (std::numeric_limits<result_type>::max)());
     if (alpha == 0 || beta == 0) {
       method_ = DEGENERATE_SMALL;
       x_ = (alpha >= beta) ? 1 : 0;
       return;
     }
     // a_ = min(beta, alpha), b_ = max(beta, alpha).
     if (beta < alpha) {
       inverted_ = true;
       a_ = beta;
       b_ = alpha;
     } else {
       inverted_ = false;
       a_ = alpha;
       b_ = beta;
     }
     if (a_ <= 1 && b_ >= ThresholdForLargeA()) {
       method_ = DEGENERATE_SMALL;
       x_ = inverted_ ? result_type(1) : result_type(0);
       return;
     }
     // For threshold values, see also:
     // Evaluation of Beta Generation Algorithms, Ying-Chao Hung, et. al.
     // February, 2009.
     if ((b_ < 1.0 && a_ + b_ <= 1.2) || a_ <= ThresholdForSmallA()) {
       // Choose Joehnk over Cheng when it's faster or when Cheng encounters
       // numerical issues.
       method_ = JOEHNK;
       a_ = result_type(1) / alpha_;
       b_ = result_type(1) / beta_;
       if (std::isinf(a_) || std::isinf(b_)) {
         method_ = DEGENERATE_SMALL;
         x_ = inverted_ ? result_type(1) : result_type(0);
       }
       return;
     }
     if (a_ >= ThresholdForLargeA()) {
       method_ = DEGENERATE_LARGE;
       // Note: on PPC for long double, evaluating
       // `std::numeric_limits::max() / ThresholdForLargeA` results in NaN.
       result_type r = a_ / b_;
       x_ = (inverted_ ? result_type(1) : r) / (1 + r);
       return;
     }
     x_ = a_ + b_;
     log_x_ = std::log(x_);
     if (a_ <= 1) {
       method_ = CHENG_BA;
       y_ = result_type(1) / a_;
       gamma_ = a_ + a_;
       return;
     }
     method_ = CHENG_BB;
     result_type r = (a_ - 1) / (b_ - 1);
     y_ = std::sqrt((1 + r) / (b_ * r * 2 - r + 1));
     gamma_ = a_ + result_type(1) / y_;
   }

   result_type alpha() const { return alpha_; }
   result_type beta() const { return beta_; }

   friend bool operator==(const param_type& a, const param_type& b) {
     return a.alpha_ == b.alpha_ && a.beta_ == b.beta_;
   }

   friend bool operator!=(const param_type& a, const param_type& b) {
     return !(a == b);
   }

  private:
   friend class beta_distribution;

#ifdef _MSC_VER
   // MSVC does not have constexpr implementations for std::log and std::exp
   // so they are computed at runtime.
#define ABSL_RANDOM_INTERNAL_LOG_EXP_CONSTEXPR
#else
#define ABSL_RANDOM_INTERNAL_LOG_EXP_CONSTEXPR constexpr
#endif

   // The threshold for whether std::exp(1/a) is finite.
   // Note that this value is quite large, and a smaller a_ is NOT abnormal.
   static ABSL_RANDOM_INTERNAL_LOG_EXP_CONSTEXPR result_type
   ThresholdForSmallA() {
     return result_type(1) /
            std::log((std::numeric_limits<result_type>::max)());
   }

   // The threshold for whether a * std::log(a) is finite.
   static ABSL_RANDOM_INTERNAL_LOG_EXP_CONSTEXPR result_type
   ThresholdForLargeA() {
     return std::exp(
         std::log((std::numeric_limits<result_type>::max)()) -
         std::log(std::log((std::numeric_limits<result_type>::max)())) -
         ThresholdPadding());
   }

#undef ABSL_RANDOM_INTERNAL_LOG_EXP_CONSTEXPR

   // Pad the threshold for large A for long double on PPC. This is done via a
   // template specialization below.
   static constexpr result_type ThresholdPadding() { return 0; }

   enum Method {
     JOEHNK,    // Uses algorithm Joehnk
     CHENG_BA,  // Uses algorithm BA in Cheng
     CHENG_BB,  // Uses algorithm BB in Cheng

     // Note: See also:
     //   Hung et al. Evaluation of beta generation algorithms. Communications
     //   in Statistics-Simulation and Computation 38.4 (2009): 750-770.
     // especially:
     //   Zechner, Heinz, and Ernst Stadlober. Generating beta variates via
     //   patchwork rejection. Computing 50.1 (1993): 1-18.

     DEGENERATE_SMALL,  // a_ is abnormally small.
     DEGENERATE_LARGE,  // a_ is abnormally large.
   };

   result_type alpha_;
   result_type beta_;

   result_type a_{};  // the smaller of {alpha, beta}, or 1.0/alpha_ in JOEHNK
   result_type b_{};  // the larger of {alpha, beta}, or 1.0/beta_ in JOEHNK
   result_type x_{};  // alpha + beta, or the result in degenerate cases
   result_type log_x_{};  // log(x_)
   result_type y_{};      // "beta" in Cheng
   result_type gamma_{};  // "gamma" in Cheng

   Method method_{};

   // Placing this last for optimal alignment.
   // Whether alpha_ != a_, i.e. true iff alpha_ > beta_.
   bool inverted_{};

   static_assert(std::is_floating_point<RealType>::value,
                 "Class-template absl::beta_distribution<> must be "
                 "parameterized using a floating-point type.");
 };

 beta_distribution() : beta_distribution(1) {}

 explicit beta_distribution(result_type alpha, result_type beta = 1)
     : param_(alpha, beta) {}

 explicit beta_distribution(const param_type& p) : param_(p) {}

 void reset() {}

 // Generating functions
 template <typename URBG>
 result_type operator()(URBG& g) {  // NOLINT(runtime/references)
   return (*this)(g, param_);
 }

 template <typename URBG>
 result_type operator()(URBG& g,  // NOLINT(runtime/references)
                        const param_type& p);

 param_type param() const { return param_; }
 void param(const param_type& p) { param_ = p; }

 result_type(min)() const { return 0; }
 result_type(max)() const { return 1; }

 result_type alpha() const { return param_.alpha(); }
 result_type beta() const { return param_.beta(); }

 friend bool operator==(const beta_distribution& a,
                        const beta_distribution& b) {
   return a.param_ == b.param_;
 }
 friend bool operator!=(const beta_distribution& a,
                        const beta_distribution& b) {
   return a.param_ != b.param_;
 }

private:
 template <typename URBG>
 result_type AlgorithmJoehnk(URBG& g,  // NOLINT(runtime/references)
                             const param_type& p);

 template <typename URBG>
 result_type AlgorithmCheng(URBG& g,  // NOLINT(runtime/references)
                            const param_type& p);

 template <typename URBG>
 result_type DegenerateCase(URBG& g,  // NOLINT(runtime/references)
                            const param_type& p) {
   if (p.method_ == param_type::DEGENERATE_SMALL && p.alpha_ == p.beta_) {
     // Returns 0 or 1 with equal probability.
     random_internal::FastUniformBits<uint8_t> fast_u8;
     return static_cast<result_type>((fast_u8(g) & 0x10) !=
                                     0);  // pick any single bit.
   }
   return p.x_;
 }

 param_type param_;
 random_internal::FastUniformBits<uint64_t> fast_u64_;
};

#if defined(__powerpc64__) || defined(__PPC64__) || defined(__powerpc__) || \
   defined(__ppc__) || defined(__PPC__)
// PPC needs a more stringent boundary for long double.
template <>
constexpr long double
beta_distribution<long double>::param_type::ThresholdPadding() {
 return 10;
}
#endif

template <typename RealType>
template <typename URBG>
typename beta_distribution<RealType>::result_type
beta_distribution<RealType>::AlgorithmJoehnk(
   URBG& g,  // NOLINT(runtime/references)
   const param_type& p) {
 using random_internal::GeneratePositiveTag;
 using random_internal::GenerateRealFromBits;
 using real_type =
     absl::conditional_t<std::is_same<RealType, float>::value, float, double>;

 // Based on Joehnk, M. D. Erzeugung von betaverteilten und gammaverteilten
 // Zufallszahlen. Metrika 8.1 (1964): 5-15.
 // This method is described in Knuth, Vol 2 (Third Edition), pp 134.

 result_type u, v, x, y, z;
 for (;;) {
   u = GenerateRealFromBits<real_type, GeneratePositiveTag, false>(
       fast_u64_(g));
   v = GenerateRealFromBits<real_type, GeneratePositiveTag, false>(
       fast_u64_(g));

   // Direct method. std::pow is slow for float, so rely on the optimizer to
   // remove the std::pow() path for that case.
   if (!std::is_same<float, result_type>::value) {
     x = std::pow(u, p.a_);
     y = std::pow(v, p.b_);
     z = x + y;
     if (z > 1) {
       // Reject if and only if `x + y > 1.0`
       continue;
     }
     if (z > 0) {
       // When both alpha and beta are small, x and y are both close to 0, so
       // divide by (x+y) directly may result in nan.
       return x / z;
     }
   }

   // Log transform.
   // x = log( pow(u, p.a_) ), y = log( pow(v, p.b_) )
   // since u, v <= 1.0,  x, y < 0.
   x = std::log(u) * p.a_;
   y = std::log(v) * p.b_;
   if (!std::isfinite(x) || !std::isfinite(y)) {
     continue;
   }
   // z = log( pow(u, a) + pow(v, b) )
   z = x > y ? (x + std::log(1 + std::exp(y - x)))
             : (y + std::log(1 + std::exp(x - y)));
   // Reject iff log(x+y) > 0.
   if (z > 0) {
     continue;
   }
   return std::exp(x - z);
 }
}

template <typename RealType>
template <typename URBG>
typename beta_distribution<RealType>::result_type
beta_distribution<RealType>::AlgorithmCheng(
   URBG& g,  // NOLINT(runtime/references)
   const param_type& p) {
 using random_internal::GeneratePositiveTag;
 using random_internal::GenerateRealFromBits;
 using real_type =
     absl::conditional_t<std::is_same<RealType, float>::value, float, double>;

 // Based on Cheng, Russell CH. Generating beta variates with nonintegral
 // shape parameters. Communications of the ACM 21.4 (1978): 317-322.
 // (https://dl.acm.org/citation.cfm?id=359482).
 static constexpr result_type kLogFour =
     result_type(1.3862943611198906188344642429163531361);  // log(4)
 static constexpr result_type kS =
     result_type(2.6094379124341003746007593332261876);  // 1+log(5)

 const bool use_algorithm_ba = (p.method_ == param_type::CHENG_BA);
 result_type u1, u2, v, w, z, r, s, t, bw_inv, lhs;
 for (;;) {
   u1 = GenerateRealFromBits<real_type, GeneratePositiveTag, false>(
       fast_u64_(g));
   u2 = GenerateRealFromBits<real_type, GeneratePositiveTag, false>(
       fast_u64_(g));
   v = p.y_ * std::log(u1 / (1 - u1));
   w = p.a_ * std::exp(v);
   bw_inv = result_type(1) / (p.b_ + w);
   r = p.gamma_ * v - kLogFour;
   s = p.a_ + r - w;
   z = u1 * u1 * u2;
   if (!use_algorithm_ba && s + kS >= 5 * z) {
     break;
   }
   t = std::log(z);
   if (!use_algorithm_ba && s >= t) {
     break;
   }
   lhs = p.x_ * (p.log_x_ + std::log(bw_inv)) + r;
   if (lhs >= t) {
     break;
   }
 }
 return p.inverted_ ? (1 - w * bw_inv) : w * bw_inv;
}

template <typename RealType>
template <typename URBG>
typename beta_distribution<RealType>::result_type
beta_distribution<RealType>::operator()(URBG& g,  // NOLINT(runtime/references)
                                       const param_type& p) {
 switch (p.method_) {
   case param_type::JOEHNK:
     return AlgorithmJoehnk(g, p);
   case param_type::CHENG_BA:
     ABSL_FALLTHROUGH_INTENDED;
   case param_type::CHENG_BB:
     return AlgorithmCheng(g, p);
   default:
     return DegenerateCase(g, p);
 }
}

template <typename CharT, typename Traits, typename RealType>
std::basic_ostream<CharT, Traits>& operator<<(
   std::basic_ostream<CharT, Traits>& os,  // NOLINT(runtime/references)
   const beta_distribution<RealType>& x) {
 auto saver = random_internal::make_ostream_state_saver(os);
 os.precision(random_internal::stream_precision_helper<RealType>::kPrecision);
 os << x.alpha() << os.fill() << x.beta();
 return os;
}

template <typename CharT, typename Traits, typename RealType>
std::basic_istream<CharT, Traits>& operator>>(
   std::basic_istream<CharT, Traits>& is,  // NOLINT(runtime/references)
   beta_distribution<RealType>& x) {       // NOLINT(runtime/references)
 using result_type = typename beta_distribution<RealType>::result_type;
 using param_type = typename beta_distribution<RealType>::param_type;
 result_type alpha, beta;

 auto saver = random_internal::make_istream_state_saver(is);
 alpha = random_internal::read_floating_point<result_type>(is);
 if (is.fail()) return is;
 beta = random_internal::read_floating_point<result_type>(is);
 if (!is.fail()) {
   x.param(param_type(alpha, beta));
 }
 return is;
}

ABSL_NAMESPACE_END
}  // namespace absl

#endif  // ABSL_RANDOM_BETA_DISTRIBUTION_H_