Chris@16: // boost\math\distributions\beta.hpp
Chris@16: 
Chris@16: // Copyright John Maddock 2006.
Chris@16: // Copyright Paul A. Bristow 2006.
Chris@16: 
Chris@16: // Use, modification and distribution are subject to the
Chris@16: // Boost Software License, Version 1.0.
Chris@16: // (See accompanying file LICENSE_1_0.txt
Chris@16: // or copy at http://www.boost.org/LICENSE_1_0.txt)
Chris@16: 
Chris@16: // http://en.wikipedia.org/wiki/Beta_distribution
Chris@16: // http://www.itl.nist.gov/div898/handbook/eda/section3/eda366h.htm
Chris@16: // http://mathworld.wolfram.com/BetaDistribution.html
Chris@16: 
Chris@16: // The Beta Distribution is a continuous probability distribution.
Chris@16: // The beta distribution is used to model events which are constrained to take place
Chris@16: // within an interval defined by maxima and minima,
Chris@16: // so is used extensively in PERT and other project management systems
Chris@16: // to describe the time to completion.
Chris@16: // The cdf of the beta distribution is used as a convenient way
Chris@16: // of obtaining the sum over a set of binomial outcomes.
Chris@16: // The beta distribution is also used in Bayesian statistics.
Chris@16: 
Chris@16: #ifndef BOOST_MATH_DIST_BETA_HPP
Chris@16: #define BOOST_MATH_DIST_BETA_HPP
Chris@16: 
Chris@16: #include <boost/math/distributions/fwd.hpp>
Chris@16: #include <boost/math/special_functions/beta.hpp> // for beta.
Chris@16: #include <boost/math/distributions/complement.hpp> // complements.
Chris@16: #include <boost/math/distributions/detail/common_error_handling.hpp> // error checks
Chris@16: #include <boost/math/special_functions/fpclassify.hpp> // isnan.
Chris@16: #include <boost/math/tools/roots.hpp> // for root finding.
Chris@16: 
Chris@16: #if defined (BOOST_MSVC)
Chris@16: #  pragma warning(push)
Chris@16: #  pragma warning(disable: 4702) // unreachable code
Chris@16: // in domain_error_imp in error_handling
Chris@16: #endif
Chris@16: 
Chris@16: #include <utility>
Chris@16: 
Chris@16: namespace boost
Chris@16: {
Chris@16:   namespace math
Chris@16:   {
Chris@16:     namespace beta_detail
Chris@16:     {
Chris@16:       // Common error checking routines for beta distribution functions:
Chris@16:       template <class RealType, class Policy>
Chris@16:       inline bool check_alpha(const char* function, const RealType& alpha, RealType* result, const Policy& pol)
Chris@16:       {
Chris@16:         if(!(boost::math::isfinite)(alpha) || (alpha <= 0))
Chris@16:         {
Chris@16:           *result = policies::raise_domain_error<RealType>(
Chris@16:             function,
Chris@16:             "Alpha argument is %1%, but must be > 0 !", alpha, pol);
Chris@16:           return false;
Chris@16:         }
Chris@16:         return true;
Chris@16:       } // bool check_alpha
Chris@16: 
Chris@16:       template <class RealType, class Policy>
Chris@16:       inline bool check_beta(const char* function, const RealType& beta, RealType* result, const Policy& pol)
Chris@16:       {
Chris@16:         if(!(boost::math::isfinite)(beta) || (beta <= 0))
Chris@16:         {
Chris@16:           *result = policies::raise_domain_error<RealType>(
Chris@16:             function,
Chris@16:             "Beta argument is %1%, but must be > 0 !", beta, pol);
Chris@16:           return false;
Chris@16:         }
Chris@16:         return true;
Chris@16:       } // bool check_beta
Chris@16: 
Chris@16:       template <class RealType, class Policy>
Chris@16:       inline bool check_prob(const char* function, const RealType& p, RealType* result, const Policy& pol)
Chris@16:       {
Chris@16:         if((p < 0) || (p > 1) || !(boost::math::isfinite)(p))
Chris@16:         {
Chris@16:           *result = policies::raise_domain_error<RealType>(
Chris@16:             function,
Chris@16:             "Probability argument is %1%, but must be >= 0 and <= 1 !", p, pol);
Chris@16:           return false;
Chris@16:         }
Chris@16:         return true;
Chris@16:       } // bool check_prob
Chris@16: 
Chris@16:       template <class RealType, class Policy>
Chris@16:       inline bool check_x(const char* function, const RealType& x, RealType* result, const Policy& pol)
Chris@16:       {
Chris@16:         if(!(boost::math::isfinite)(x) || (x < 0) || (x > 1))
Chris@16:         {
Chris@16:           *result = policies::raise_domain_error<RealType>(
Chris@16:             function,
Chris@16:             "x argument is %1%, but must be >= 0 and <= 1 !", x, pol);
Chris@16:           return false;
Chris@16:         }
Chris@16:         return true;
Chris@16:       } // bool check_x
Chris@16: 
Chris@16:       template <class RealType, class Policy>
Chris@16:       inline bool check_dist(const char* function, const RealType& alpha, const RealType& beta, RealType* result, const Policy& pol)
Chris@16:       { // Check both alpha and beta.
Chris@16:         return check_alpha(function, alpha, result, pol)
Chris@16:           && check_beta(function, beta, result, pol);
Chris@16:       } // bool check_dist
Chris@16: 
Chris@16:       template <class RealType, class Policy>
Chris@16:       inline bool check_dist_and_x(const char* function, const RealType& alpha, const RealType& beta, RealType x, RealType* result, const Policy& pol)
Chris@16:       {
Chris@16:         return check_dist(function, alpha, beta, result, pol)
Chris@16:           && beta_detail::check_x(function, x, result, pol);
Chris@16:       } // bool check_dist_and_x
Chris@16: 
Chris@16:       template <class RealType, class Policy>
Chris@16:       inline bool check_dist_and_prob(const char* function, const RealType& alpha, const RealType& beta, RealType p, RealType* result, const Policy& pol)
Chris@16:       {
Chris@16:         return check_dist(function, alpha, beta, result, pol)
Chris@16:           && check_prob(function, p, result, pol);
Chris@16:       } // bool check_dist_and_prob
Chris@16: 
Chris@16:       template <class RealType, class Policy>
Chris@16:       inline bool check_mean(const char* function, const RealType& mean, RealType* result, const Policy& pol)
Chris@16:       {
Chris@16:         if(!(boost::math::isfinite)(mean) || (mean <= 0))
Chris@16:         {
Chris@16:           *result = policies::raise_domain_error<RealType>(
Chris@16:             function,
Chris@16:             "mean argument is %1%, but must be > 0 !", mean, pol);
Chris@16:           return false;
Chris@16:         }
Chris@16:         return true;
Chris@16:       } // bool check_mean
Chris@16:       template <class RealType, class Policy>
Chris@16:       inline bool check_variance(const char* function, const RealType& variance, RealType* result, const Policy& pol)
Chris@16:       {
Chris@16:         if(!(boost::math::isfinite)(variance) || (variance <= 0))
Chris@16:         {
Chris@16:           *result = policies::raise_domain_error<RealType>(
Chris@16:             function,
Chris@16:             "variance argument is %1%, but must be > 0 !", variance, pol);
Chris@16:           return false;
Chris@16:         }
Chris@16:         return true;
Chris@16:       } // bool check_variance
Chris@16:     } // namespace beta_detail
Chris@16: 
Chris@16:     // typedef beta_distribution<double> beta;
Chris@16:     // is deliberately NOT included to avoid a name clash with the beta function.
Chris@16:     // Use beta_distribution<> mybeta(...) to construct type double.
Chris@16: 
Chris@16:     template <class RealType = double, class Policy = policies::policy<> >
Chris@16:     class beta_distribution
Chris@16:     {
Chris@16:     public:
Chris@16:       typedef RealType value_type;
Chris@16:       typedef Policy policy_type;
Chris@16: 
Chris@16:       beta_distribution(RealType l_alpha = 1, RealType l_beta = 1) : m_alpha(l_alpha), m_beta(l_beta)
Chris@16:       {
Chris@16:         RealType result;
Chris@16:         beta_detail::check_dist(
Chris@16:            "boost::math::beta_distribution<%1%>::beta_distribution",
Chris@16:           m_alpha,
Chris@16:           m_beta,
Chris@16:           &result, Policy());
Chris@16:       } // beta_distribution constructor.
Chris@16:       // Accessor functions:
Chris@16:       RealType alpha() const
Chris@16:       {
Chris@16:         return m_alpha;
Chris@16:       }
Chris@16:       RealType beta() const
Chris@16:       { // .
Chris@16:         return m_beta;
Chris@16:       }
Chris@16: 
Chris@16:       // Estimation of the alpha & beta parameters.
Chris@16:       // http://en.wikipedia.org/wiki/Beta_distribution
Chris@16:       // gives formulae in section on parameter estimation.
Chris@16:       // Also NIST EDA page 3 & 4 give the same.
Chris@16:       // http://www.itl.nist.gov/div898/handbook/eda/section3/eda366h.htm
Chris@16:       // http://www.epi.ucdavis.edu/diagnostictests/betabuster.html
Chris@16: 
Chris@16:       static RealType find_alpha(
Chris@16:         RealType mean, // Expected value of mean.
Chris@16:         RealType variance) // Expected value of variance.
Chris@16:       {
Chris@16:         static const char* function = "boost::math::beta_distribution<%1%>::find_alpha";
Chris@16:         RealType result = 0; // of error checks.
Chris@16:         if(false ==
Chris@16:             (
Chris@16:               beta_detail::check_mean(function, mean, &result, Policy())
Chris@16:               && beta_detail::check_variance(function, variance, &result, Policy())
Chris@16:             )
Chris@16:           )
Chris@16:         {
Chris@16:           return result;
Chris@16:         }
Chris@16:         return mean * (( (mean * (1 - mean)) / variance)- 1);
Chris@16:       } // RealType find_alpha
Chris@16: 
Chris@16:       static RealType find_beta(
Chris@16:         RealType mean, // Expected value of mean.
Chris@16:         RealType variance) // Expected value of variance.
Chris@16:       {
Chris@16:         static const char* function = "boost::math::beta_distribution<%1%>::find_beta";
Chris@16:         RealType result = 0; // of error checks.
Chris@16:         if(false ==
Chris@16:             (
Chris@16:               beta_detail::check_mean(function, mean, &result, Policy())
Chris@16:               &&
Chris@16:               beta_detail::check_variance(function, variance, &result, Policy())
Chris@16:             )
Chris@16:           )
Chris@16:         {
Chris@16:           return result;
Chris@16:         }
Chris@16:         return (1 - mean) * (((mean * (1 - mean)) /variance)-1);
Chris@16:       } //  RealType find_beta
Chris@16: 
Chris@16:       // Estimate alpha & beta from either alpha or beta, and x and probability.
Chris@16:       // Uses for these parameter estimators are unclear.
Chris@16: 
Chris@16:       static RealType find_alpha(
Chris@16:         RealType beta, // from beta.
Chris@16:         RealType x, //  x.
Chris@16:         RealType probability) // cdf
Chris@16:       {
Chris@16:         static const char* function = "boost::math::beta_distribution<%1%>::find_alpha";
Chris@16:         RealType result = 0; // of error checks.
Chris@16:         if(false ==
Chris@16:             (
Chris@16:              beta_detail::check_prob(function, probability, &result, Policy())
Chris@16:              &&
Chris@16:              beta_detail::check_beta(function, beta, &result, Policy())
Chris@16:              &&
Chris@16:              beta_detail::check_x(function, x, &result, Policy())
Chris@16:             )
Chris@16:           )
Chris@16:         {
Chris@16:           return result;
Chris@16:         }
Chris@16:         return ibeta_inva(beta, x, probability, Policy());
Chris@16:       } // RealType find_alpha(beta, a, probability)
Chris@16: 
Chris@16:       static RealType find_beta(
Chris@16:         // ibeta_invb(T b, T x, T p); (alpha, x, cdf,)
Chris@16:         RealType alpha, // alpha.
Chris@16:         RealType x, // probability x.
Chris@16:         RealType probability) // probability cdf.
Chris@16:       {
Chris@16:         static const char* function = "boost::math::beta_distribution<%1%>::find_beta";
Chris@16:         RealType result = 0; // of error checks.
Chris@16:         if(false ==
Chris@16:             (
Chris@16:               beta_detail::check_prob(function, probability, &result, Policy())
Chris@16:               &&
Chris@16:               beta_detail::check_alpha(function, alpha, &result, Policy())
Chris@16:               &&
Chris@16:               beta_detail::check_x(function, x, &result, Policy())
Chris@16:             )
Chris@16:           )
Chris@16:         {
Chris@16:           return result;
Chris@16:         }
Chris@16:         return ibeta_invb(alpha, x, probability, Policy());
Chris@16:       } //  RealType find_beta(alpha, x, probability)
Chris@16: 
Chris@16:     private:
Chris@16:       RealType m_alpha; // Two parameters of the beta distribution.
Chris@16:       RealType m_beta;
Chris@16:     }; // template <class RealType, class Policy> class beta_distribution
Chris@16: 
Chris@16:     template <class RealType, class Policy>
Chris@16:     inline const std::pair<RealType, RealType> range(const beta_distribution<RealType, Policy>& /* dist */)
Chris@16:     { // Range of permissible values for random variable x.
Chris@16:       using boost::math::tools::max_value;
Chris@16:       return std::pair<RealType, RealType>(static_cast<RealType>(0), static_cast<RealType>(1));
Chris@16:     }
Chris@16: 
Chris@16:     template <class RealType, class Policy>
Chris@16:     inline const std::pair<RealType, RealType> support(const beta_distribution<RealType, Policy>&  /* dist */)
Chris@16:     { // Range of supported values for random variable x.
Chris@16:       // This is range where cdf rises from 0 to 1, and outside it, the pdf is zero.
Chris@16:       return std::pair<RealType, RealType>(static_cast<RealType>(0), static_cast<RealType>(1));
Chris@16:     }
Chris@16: 
Chris@16:     template <class RealType, class Policy>
Chris@16:     inline RealType mean(const beta_distribution<RealType, Policy>& dist)
Chris@16:     { // Mean of beta distribution = np.
Chris@16:       return  dist.alpha() / (dist.alpha() + dist.beta());
Chris@16:     } // mean
Chris@16: 
Chris@16:     template <class RealType, class Policy>
Chris@16:     inline RealType variance(const beta_distribution<RealType, Policy>& dist)
Chris@16:     { // Variance of beta distribution = np(1-p).
Chris@16:       RealType a = dist.alpha();
Chris@16:       RealType b = dist.beta();
Chris@16:       return  (a * b) / ((a + b ) * (a + b) * (a + b + 1));
Chris@16:     } // variance
Chris@16: 
Chris@16:     template <class RealType, class Policy>
Chris@16:     inline RealType mode(const beta_distribution<RealType, Policy>& dist)
Chris@16:     {
Chris@16:       static const char* function = "boost::math::mode(beta_distribution<%1%> const&)";
Chris@16: 
Chris@16:       RealType result;
Chris@16:       if ((dist.alpha() <= 1))
Chris@16:       {
Chris@16:         result = policies::raise_domain_error<RealType>(
Chris@16:           function,
Chris@16:           "mode undefined for alpha = %1%, must be > 1!", dist.alpha(), Policy());
Chris@16:         return result;
Chris@16:       }
Chris@16: 
Chris@16:       if ((dist.beta() <= 1))
Chris@16:       {
Chris@16:         result = policies::raise_domain_error<RealType>(
Chris@16:           function,
Chris@16:           "mode undefined for beta = %1%, must be > 1!", dist.beta(), Policy());
Chris@16:         return result;
Chris@16:       }
Chris@16:       RealType a = dist.alpha();
Chris@16:       RealType b = dist.beta();
Chris@16:       return (a-1) / (a + b - 2);
Chris@16:     } // mode
Chris@16: 
Chris@16:     //template <class RealType, class Policy>
Chris@16:     //inline RealType median(const beta_distribution<RealType, Policy>& dist)
Chris@16:     //{ // Median of beta distribution is not defined.
Chris@16:     //  return tools::domain_error<RealType>(function, "Median is not implemented, result is %1%!", std::numeric_limits<RealType>::quiet_NaN());
Chris@16:     //} // median
Chris@16: 
Chris@16:     //But WILL be provided by the derived accessor as quantile(0.5).
Chris@16: 
Chris@16:     template <class RealType, class Policy>
Chris@16:     inline RealType skewness(const beta_distribution<RealType, Policy>& dist)
Chris@16:     {
Chris@16:       BOOST_MATH_STD_USING // ADL of std functions.
Chris@16:       RealType a = dist.alpha();
Chris@16:       RealType b = dist.beta();
Chris@16:       return (2 * (b-a) * sqrt(a + b + 1)) / ((a + b + 2) * sqrt(a * b));
Chris@16:     } // skewness
Chris@16: 
Chris@16:     template <class RealType, class Policy>
Chris@16:     inline RealType kurtosis_excess(const beta_distribution<RealType, Policy>& dist)
Chris@16:     {
Chris@16:       RealType a = dist.alpha();
Chris@16:       RealType b = dist.beta();
Chris@16:       RealType a_2 = a * a;
Chris@16:       RealType n = 6 * (a_2 * a - a_2 * (2 * b - 1) + b * b * (b + 1) - 2 * a * b * (b + 2));
Chris@16:       RealType d = a * b * (a + b + 2) * (a + b + 3);
Chris@16:       return  n / d;
Chris@16:     } // kurtosis_excess
Chris@16: 
Chris@16:     template <class RealType, class Policy>
Chris@16:     inline RealType kurtosis(const beta_distribution<RealType, Policy>& dist)
Chris@16:     {
Chris@16:       return 3 + kurtosis_excess(dist);
Chris@16:     } // kurtosis
Chris@16: 
Chris@16:     template <class RealType, class Policy>
Chris@16:     inline RealType pdf(const beta_distribution<RealType, Policy>& dist, const RealType& x)
Chris@16:     { // Probability Density/Mass Function.
Chris@16:       BOOST_FPU_EXCEPTION_GUARD
Chris@16: 
Chris@16:       static const char* function = "boost::math::pdf(beta_distribution<%1%> const&, %1%)";
Chris@16: 
Chris@16:       BOOST_MATH_STD_USING // for ADL of std functions
Chris@16: 
Chris@16:       RealType a = dist.alpha();
Chris@16:       RealType b = dist.beta();
Chris@16: 
Chris@16:       // Argument checks:
Chris@16:       RealType result = 0;
Chris@16:       if(false == beta_detail::check_dist_and_x(
Chris@16:         function,
Chris@16:         a, b, x,
Chris@16:         &result, Policy()))
Chris@16:       {
Chris@16:         return result;
Chris@16:       }
Chris@16:       using boost::math::beta;
Chris@16:       return ibeta_derivative(a, b, x, Policy());
Chris@16:     } // pdf
Chris@16: 
Chris@16:     template <class RealType, class Policy>
Chris@16:     inline RealType cdf(const beta_distribution<RealType, Policy>& dist, const RealType& x)
Chris@16:     { // Cumulative Distribution Function beta.
Chris@16:       BOOST_MATH_STD_USING // for ADL of std functions
Chris@16: 
Chris@16:       static const char* function = "boost::math::cdf(beta_distribution<%1%> const&, %1%)";
Chris@16: 
Chris@16:       RealType a = dist.alpha();
Chris@16:       RealType b = dist.beta();
Chris@16: 
Chris@16:       // Argument checks:
Chris@16:       RealType result = 0;
Chris@16:       if(false == beta_detail::check_dist_and_x(
Chris@16:         function,
Chris@16:         a, b, x,
Chris@16:         &result, Policy()))
Chris@16:       {
Chris@16:         return result;
Chris@16:       }
Chris@16:       // Special cases:
Chris@16:       if (x == 0)
Chris@16:       {
Chris@16:         return 0;
Chris@16:       }
Chris@16:       else if (x == 1)
Chris@16:       {
Chris@16:         return 1;
Chris@16:       }
Chris@16:       return ibeta(a, b, x, Policy());
Chris@16:     } // beta cdf
Chris@16: 
Chris@16:     template <class RealType, class Policy>
Chris@16:     inline RealType cdf(const complemented2_type<beta_distribution<RealType, Policy>, RealType>& c)
Chris@16:     { // Complemented Cumulative Distribution Function beta.
Chris@16: 
Chris@16:       BOOST_MATH_STD_USING // for ADL of std functions
Chris@16: 
Chris@16:       static const char* function = "boost::math::cdf(beta_distribution<%1%> const&, %1%)";
Chris@16: 
Chris@16:       RealType const& x = c.param;
Chris@16:       beta_distribution<RealType, Policy> const& dist = c.dist;
Chris@16:       RealType a = dist.alpha();
Chris@16:       RealType b = dist.beta();
Chris@16: 
Chris@16:       // Argument checks:
Chris@16:       RealType result = 0;
Chris@16:       if(false == beta_detail::check_dist_and_x(
Chris@16:         function,
Chris@16:         a, b, x,
Chris@16:         &result, Policy()))
Chris@16:       {
Chris@16:         return result;
Chris@16:       }
Chris@16:       if (x == 0)
Chris@16:       {
Chris@16:         return 1;
Chris@16:       }
Chris@16:       else if (x == 1)
Chris@16:       {
Chris@16:         return 0;
Chris@16:       }
Chris@16:       // Calculate cdf beta using the incomplete beta function.
Chris@16:       // Use of ibeta here prevents cancellation errors in calculating
Chris@16:       // 1 - x if x is very small, perhaps smaller than machine epsilon.
Chris@16:       return ibetac(a, b, x, Policy());
Chris@16:     } // beta cdf
Chris@16: 
Chris@16:     template <class RealType, class Policy>
Chris@16:     inline RealType quantile(const beta_distribution<RealType, Policy>& dist, const RealType& p)
Chris@16:     { // Quantile or Percent Point beta function or
Chris@16:       // Inverse Cumulative probability distribution function CDF.
Chris@16:       // Return x (0 <= x <= 1),
Chris@16:       // for a given probability p (0 <= p <= 1).
Chris@16:       // These functions take a probability as an argument
Chris@16:       // and return a value such that the probability that a random variable x
Chris@16:       // will be less than or equal to that value
Chris@16:       // is whatever probability you supplied as an argument.
Chris@16: 
Chris@16:       static const char* function = "boost::math::quantile(beta_distribution<%1%> const&, %1%)";
Chris@16: 
Chris@16:       RealType result = 0; // of argument checks:
Chris@16:       RealType a = dist.alpha();
Chris@16:       RealType b = dist.beta();
Chris@16:       if(false == beta_detail::check_dist_and_prob(
Chris@16:         function,
Chris@16:         a, b, p,
Chris@16:         &result, Policy()))
Chris@16:       {
Chris@16:         return result;
Chris@16:       }
Chris@16:       // Special cases:
Chris@16:       if (p == 0)
Chris@16:       {
Chris@16:         return 0;
Chris@16:       }
Chris@16:       if (p == 1)
Chris@16:       {
Chris@16:         return 1;
Chris@16:       }
Chris@16:       return ibeta_inv(a, b, p, static_cast<RealType*>(0), Policy());
Chris@16:     } // quantile
Chris@16: 
Chris@16:     template <class RealType, class Policy>
Chris@16:     inline RealType quantile(const complemented2_type<beta_distribution<RealType, Policy>, RealType>& c)
Chris@16:     { // Complement Quantile or Percent Point beta function .
Chris@16:       // Return the number of expected x for a given
Chris@16:       // complement of the probability q.
Chris@16: 
Chris@16:       static const char* function = "boost::math::quantile(beta_distribution<%1%> const&, %1%)";
Chris@16: 
Chris@16:       //
Chris@16:       // Error checks:
Chris@16:       RealType q = c.param;
Chris@16:       const beta_distribution<RealType, Policy>& dist = c.dist;
Chris@16:       RealType result = 0;
Chris@16:       RealType a = dist.alpha();
Chris@16:       RealType b = dist.beta();
Chris@16:       if(false == beta_detail::check_dist_and_prob(
Chris@16:         function,
Chris@16:         a,
Chris@16:         b,
Chris@16:         q,
Chris@16:         &result, Policy()))
Chris@16:       {
Chris@16:         return result;
Chris@16:       }
Chris@16:       // Special cases:
Chris@16:       if(q == 1)
Chris@16:       {
Chris@16:         return 0;
Chris@16:       }
Chris@16:       if(q == 0)
Chris@16:       {
Chris@16:         return 1;
Chris@16:       }
Chris@16: 
Chris@16:       return ibetac_inv(a, b, q, static_cast<RealType*>(0), Policy());
Chris@16:     } // Quantile Complement
Chris@16: 
Chris@16:   } // namespace math
Chris@16: } // namespace boost
Chris@16: 
Chris@16: // This include must be at the end, *after* the accessors
Chris@16: // for this distribution have been defined, in order to
Chris@16: // keep compilers that support two-phase lookup happy.
Chris@16: #include <boost/math/distributions/detail/derived_accessors.hpp>
Chris@16: 
Chris@16: #if defined (BOOST_MSVC)
Chris@16: # pragma warning(pop)
Chris@16: #endif
Chris@16: 
Chris@16: #endif // BOOST_MATH_DIST_BETA_HPP
Chris@16: 
Chris@16: