// 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 <istream>
#include <limits>
#include <ostream>
#include <type_traits>
#include "absl/meta/type_traits.h"
#include "absl/random/internal/fast_uniform_bits.h"
#include "absl/random/internal/fastmath.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_
