# Boost C++ Libraries

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### boost/math/distributions/inverse_gamma.hpp

```// inverse_gamma.hpp

//  Copyright Paul A. Bristow 2010.
//  Use, modification and distribution are subject to the
//  Boost Software License, Version 1.0. (See accompanying file

#ifndef BOOST_STATS_INVERSE_GAMMA_HPP
#define BOOST_STATS_INVERSE_GAMMA_HPP

// Inverse Gamma Distribution is a two-parameter family
// of continuous probability distributions
// on the positive real line, which is the distribution of
// the reciprocal of a variable distributed according to the gamma distribution.

// http://en.wikipedia.org/wiki/Inverse-gamma_distribution

// http://www.itl.nist.gov/div898/handbook/eda/section3/eda366b.htm
// http://en.wikipedia.org/wiki/Gamma_distribution

#include <boost/math/distributions/fwd.hpp>
#include <boost/math/special_functions/gamma.hpp>
#include <boost/math/distributions/detail/common_error_handling.hpp>
#include <boost/math/distributions/complement.hpp>

#include <utility>
#include <cfenv>

namespace boost{ namespace math
{
namespace detail
{

template <class RealType, class Policy>
inline bool check_inverse_gamma_shape(
const char* function, // inverse_gamma
RealType shape, // shape aka alpha
RealType* result, // to update, perhaps with NaN
const Policy& pol)
{  // Sources say shape argument must be > 0
// but seems logical to allow shape zero as special case,
// returning pdf and cdf zero (but not < 0).
// (Functions like mean, variance with other limits on shape are checked
// in version including an operator & limit below).
if((shape < 0) || !(boost::math::isfinite)(shape))
{
*result = policies::raise_domain_error<RealType>(
function,
"Shape parameter is %1%, but must be >= 0 !", shape, pol);
return false;
}
return true;
} //bool check_inverse_gamma_shape

template <class RealType, class Policy>
inline bool check_inverse_gamma_x(
const char* function,
RealType const& x,
RealType* result, const Policy& pol)
{
if((x < 0) || !(boost::math::isfinite)(x))
{
*result = policies::raise_domain_error<RealType>(
function,
"Random variate is %1% but must be >= 0 !", x, pol);
return false;
}
return true;
}

template <class RealType, class Policy>
inline bool check_inverse_gamma(
const char* function, // TODO swap these over, so shape is first.
RealType scale,  // scale aka beta
RealType shape, // shape aka alpha
RealType* result, const Policy& pol)
{
return check_scale(function, scale, result, pol)
&& check_inverse_gamma_shape(function, shape, result, pol);
} // bool check_inverse_gamma

} // namespace detail

template <class RealType = double, class Policy = policies::policy<> >
class inverse_gamma_distribution
{
public:
using value_type = RealType;
using policy_type = Policy;

explicit inverse_gamma_distribution(RealType l_shape = 1, RealType l_scale = 1)
: m_shape(l_shape), m_scale(l_scale)
{
RealType result;
detail::check_inverse_gamma(
"boost::math::inverse_gamma_distribution<%1%>::inverse_gamma_distribution",
l_scale, l_shape, &result, Policy());
}

RealType shape()const
{
return m_shape;
}

RealType scale()const
{
return m_scale;
}
private:
//
// Data members:
//
RealType m_shape;     // distribution shape
RealType m_scale;     // distribution scale
};

using inverse_gamma = inverse_gamma_distribution<double>;
// typedef - but potential clash with name of inverse gamma *function*.
// but there is a typedef for the gamma distribution (gamma)

#ifdef __cpp_deduction_guides
template <class RealType>
inverse_gamma_distribution(RealType)->inverse_gamma_distribution<typename boost::math::tools::promote_args<RealType>::type>;
template <class RealType>
inverse_gamma_distribution(RealType,RealType)->inverse_gamma_distribution<typename boost::math::tools::promote_args<RealType>::type>;
#endif

// Allow random variable x to be zero, treated as a special case (unlike some definitions).

template <class RealType, class Policy>
inline std::pair<RealType, RealType> range(const inverse_gamma_distribution<RealType, Policy>& /* dist */)
{  // Range of permissible values for random variable x.
using boost::math::tools::max_value;
return std::pair<RealType, RealType>(static_cast<RealType>(0), max_value<RealType>());
}

template <class RealType, class Policy>
inline std::pair<RealType, RealType> support(const inverse_gamma_distribution<RealType, Policy>& /* dist */)
{  // Range of supported values for random variable x.
// This is range where cdf rises from 0 to 1, and outside it, the pdf is zero.
using boost::math::tools::max_value;
using boost::math::tools::min_value;
return std::pair<RealType, RealType>(static_cast<RealType>(0),  max_value<RealType>());
}

template <class RealType, class Policy>
inline RealType pdf(const inverse_gamma_distribution<RealType, Policy>& dist, const RealType& x)
{
BOOST_MATH_STD_USING  // for ADL of std functions

static const char* function = "boost::math::pdf(const inverse_gamma_distribution<%1%>&, %1%)";

RealType shape = dist.shape();
RealType scale = dist.scale();

RealType result = 0;
if(false == detail::check_inverse_gamma(function, scale, shape, &result, Policy()))
return result;
}
if(x == 0)
{ // Treat random variate zero as a special case.
return 0;
}
else if(false == detail::check_inverse_gamma_x(function, x, &result, Policy()))
return result;
}
result = scale / x;
if(result < tools::min_value<RealType>())
return 0;  // random variable is infinite or so close as to make no difference.
result = gamma_p_derivative(shape, result, Policy()) * scale;
if(0 != result)
{
if(x < 0)
{
// x * x may under or overflow, likewise our result,
// so be extra careful about the arithmetic:
RealType lim = tools::max_value<RealType>() * x;
if(lim < result)
return policies::raise_overflow_error<RealType, Policy>(function, "PDF is infinite.", Policy());
result /= x;
if(lim < result)
return policies::raise_overflow_error<RealType, Policy>(function, "PDF is infinite.", Policy());
result /= x;
}
result /= (x * x);
}

return result;
} // pdf

template <class RealType, class Policy>
inline RealType logpdf(const inverse_gamma_distribution<RealType, Policy>& dist, const RealType& x)
{
BOOST_MATH_STD_USING  // for ADL of std functions
using boost::math::lgamma;

static const char* function = "boost::math::logpdf(const inverse_gamma_distribution<%1%>&, %1%)";

RealType shape = dist.shape();
RealType scale = dist.scale();

RealType result = -std::numeric_limits<RealType>::infinity();
if(false == detail::check_inverse_gamma(function, scale, shape, &result, Policy()))
return result;
}
if(x == 0)
{ // Treat random variate zero as a special case.
return result;
}
else if(false == detail::check_inverse_gamma_x(function, x, &result, Policy()))
return result;
}
result = scale / x;
if(result < tools::min_value<RealType>())
return result;  // random variable is infinite or so close as to make no difference.

// x * x may under or overflow, likewise our result
if (!(boost::math::isfinite)(x*x))
{
return policies::raise_overflow_error<RealType, Policy>(function, "PDF is infinite.", Policy());
}

return shape * log(scale) + (-shape-1)*log(x) - lgamma(shape) - (scale/x);
} // pdf

template <class RealType, class Policy>
inline RealType cdf(const inverse_gamma_distribution<RealType, Policy>& dist, const RealType& x)
{
BOOST_MATH_STD_USING  // for ADL of std functions

static const char* function = "boost::math::cdf(const inverse_gamma_distribution<%1%>&, %1%)";

RealType shape = dist.shape();
RealType scale = dist.scale();

RealType result = 0;
if(false == detail::check_inverse_gamma(function, scale, shape, &result, Policy()))
return result;
}
if (x == 0)
{ // Treat zero as a special case.
return 0;
}
else if(false == detail::check_inverse_gamma_x(function, x, &result, Policy()))
return result;
}
result = boost::math::gamma_q(shape, scale / x, Policy());
// result = tgamma(shape, scale / x) / tgamma(shape); // naive using tgamma
return result;
} // cdf

template <class RealType, class Policy>
inline RealType quantile(const inverse_gamma_distribution<RealType, Policy>& dist, const RealType& p)
{
BOOST_MATH_STD_USING  // for ADL of std functions
using boost::math::gamma_q_inv;

static const char* function = "boost::math::quantile(const inverse_gamma_distribution<%1%>&, %1%)";

RealType shape = dist.shape();
RealType scale = dist.scale();

RealType result = 0;
if(false == detail::check_inverse_gamma(function, scale, shape, &result, Policy()))
return result;
if(false == detail::check_probability(function, p, &result, Policy()))
return result;
if(p == 1)
{
return policies::raise_overflow_error<RealType>(function, 0, Policy());
}
result = gamma_q_inv(shape, p, Policy());
if((result < 1) && (result * tools::max_value<RealType>() < scale))
return policies::raise_overflow_error<RealType, Policy>(function, "Value of random variable in inverse gamma distribution quantile is infinite.", Policy());
result = scale / result;
return result;
}

template <class RealType, class Policy>
inline RealType cdf(const complemented2_type<inverse_gamma_distribution<RealType, Policy>, RealType>& c)
{
BOOST_MATH_STD_USING  // for ADL of std functions

static const char* function = "boost::math::quantile(const gamma_distribution<%1%>&, %1%)";

RealType shape = c.dist.shape();
RealType scale = c.dist.scale();

RealType result = 0;
if(false == detail::check_inverse_gamma(function, scale, shape, &result, Policy()))
return result;
if(false == detail::check_inverse_gamma_x(function, c.param, &result, Policy()))
return result;

if(c.param == 0)
return 1; // Avoid division by zero

result = gamma_p(shape, scale/c.param, Policy());
return result;
}

template <class RealType, class Policy>
inline RealType quantile(const complemented2_type<inverse_gamma_distribution<RealType, Policy>, RealType>& c)
{
BOOST_MATH_STD_USING  // for ADL of std functions

static const char* function = "boost::math::quantile(const inverse_gamma_distribution<%1%>&, %1%)";

RealType shape = c.dist.shape();
RealType scale = c.dist.scale();
RealType q = c.param;

RealType result = 0;
if(false == detail::check_inverse_gamma(function, scale, shape, &result, Policy()))
return result;
if(false == detail::check_probability(function, q, &result, Policy()))
return result;

if(q == 0)
{
return policies::raise_overflow_error<RealType>(function, 0, Policy());
}
result = gamma_p_inv(shape, q, Policy());
if((result < 1) && (result * tools::max_value<RealType>() < scale))
return policies::raise_overflow_error<RealType, Policy>(function, "Value of random variable in inverse gamma distribution quantile is infinite.", Policy());
result = scale / result;
return result;
}

template <class RealType, class Policy>
inline RealType mean(const inverse_gamma_distribution<RealType, Policy>& dist)
{
BOOST_MATH_STD_USING  // for ADL of std functions

static const char* function = "boost::math::mean(const inverse_gamma_distribution<%1%>&)";

RealType shape = dist.shape();
RealType scale = dist.scale();

RealType result = 0;

if(false == detail::check_scale(function, scale, &result, Policy()))
{
return result;
}
if((shape <= 1) || !(boost::math::isfinite)(shape))
{
result = policies::raise_domain_error<RealType>(
function,
"Shape parameter is %1%, but for a defined mean it must be > 1", shape, Policy());
return result;
}
result = scale / (shape - 1);
return result;
} // mean

template <class RealType, class Policy>
inline RealType variance(const inverse_gamma_distribution<RealType, Policy>& dist)
{
BOOST_MATH_STD_USING  // for ADL of std functions

static const char* function = "boost::math::variance(const inverse_gamma_distribution<%1%>&)";

RealType shape = dist.shape();
RealType scale = dist.scale();

RealType result = 0;
if(false == detail::check_scale(function, scale, &result, Policy()))
{
return result;
}
if((shape <= 2) || !(boost::math::isfinite)(shape))
{
result = policies::raise_domain_error<RealType>(
function,
"Shape parameter is %1%, but for a defined variance it must be > 2", shape, Policy());
return result;
}
result = (scale * scale) / ((shape - 1) * (shape -1) * (shape -2));
return result;
}

template <class RealType, class Policy>
inline RealType mode(const inverse_gamma_distribution<RealType, Policy>& dist)
{
BOOST_MATH_STD_USING  // for ADL of std functions

static const char* function = "boost::math::mode(const inverse_gamma_distribution<%1%>&)";

RealType shape = dist.shape();
RealType scale = dist.scale();

RealType result = 0;
if(false == detail::check_inverse_gamma(function, scale, shape, &result, Policy()))
{
return result;
}
// Only defined for shape >= 0, but is checked by check_inverse_gamma.
result = scale / (shape + 1);
return result;
}

//template <class RealType, class Policy>
//inline RealType median(const gamma_distribution<RealType, Policy>& dist)
//{  // Wikipedia does not define median,
// so rely on default definition quantile(0.5) in derived accessors.
//  return result.
//}

template <class RealType, class Policy>
inline RealType skewness(const inverse_gamma_distribution<RealType, Policy>& dist)
{
BOOST_MATH_STD_USING  // for ADL of std functions

static const char* function = "boost::math::skewness(const inverse_gamma_distribution<%1%>&)";

RealType shape = dist.shape();
RealType scale = dist.scale();
RealType result = 0;

if(false == detail::check_scale(function, scale, &result, Policy()))
{
return result;
}
if((shape <= 3) || !(boost::math::isfinite)(shape))
{
result = policies::raise_domain_error<RealType>(
function,
"Shape parameter is %1%, but for a defined skewness it must be > 3", shape, Policy());
return result;
}
result = (4 * sqrt(shape - 2) ) / (shape - 3);
return result;
}

template <class RealType, class Policy>
inline RealType kurtosis_excess(const inverse_gamma_distribution<RealType, Policy>& dist)
{
BOOST_MATH_STD_USING  // for ADL of std functions

static const char* function = "boost::math::kurtosis_excess(const inverse_gamma_distribution<%1%>&)";

RealType shape = dist.shape();
RealType scale = dist.scale();

RealType result = 0;
if(false == detail::check_scale(function, scale, &result, Policy()))
{
return result;
}
if((shape <= 4) || !(boost::math::isfinite)(shape))
{
result = policies::raise_domain_error<RealType>(
function,
"Shape parameter is %1%, but for a defined kurtosis excess it must be > 4", shape, Policy());
return result;
}
result = (30 * shape - 66) / ((shape - 3) * (shape - 4));
return result;
}

template <class RealType, class Policy>
inline RealType kurtosis(const inverse_gamma_distribution<RealType, Policy>& dist)
{
static const char* function = "boost::math::kurtosis(const inverse_gamma_distribution<%1%>&)";
RealType shape = dist.shape();
RealType scale = dist.scale();

RealType result = 0;

if(false == detail::check_scale(function, scale, &result, Policy()))
{
return result;
}
if((shape <= 4) || !(boost::math::isfinite)(shape))
{
result = policies::raise_domain_error<RealType>(
function,
"Shape parameter is %1%, but for a defined kurtosis it must be > 4", shape, Policy());
return result;
}
return kurtosis_excess(dist) + 3;
}

} // namespace math
} // namespace boost

// This include must be at the end, *after* the accessors
// for this distribution have been defined, in order to
// keep compilers that support two-phase lookup happy.
#include <boost/math/distributions/detail/derived_accessors.hpp>

#endif // BOOST_STATS_INVERSE_GAMMA_HPP
```