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Inverse Gamma Distribution

#include <boost/math/distributions/inverse_gamma.hpp>

namespace boost{ namespace math{ 
template <class RealType = double, 
          class Policy   = policies::policy<> >
class inverse_gamma_distribution
   typedef RealType value_type;
   typedef Policy   policy_type;

   inverse_gamma_distribution(RealType shape, RealType scale = 1)

   RealType shape()const;
   RealType scale()const;

}} // namespaces

The inverse_gamma distribution is a continuous probability distribution of the reciprocal of a variable distributed according to the gamma distribution.

The inverse_gamma distribution is used in Bayesian statistics.

See inverse gamma distribution.

R inverse gamma distribution functions.

Wolfram inverse gamma distribution.

See also Gamma Distribution.

[Note] Note

In spite of potential confusion with the inverse gamma function, this distribution does provide the typedef:

typedef inverse_gamma_distribution<double> gamma;

If you want a double precision gamma distribution you can use


or you can write inverse_gamma my_ig(2, 3);

For shape parameter α and scale parameter β, it is defined by the probability density function (PDF):

  f(x;α, β) = βα * (1/x) α+1 exp(-β/x) / Γ(α)

and cumulative density function (CDF)

  F(x;α, β) = Γ(α, β/x) / Γ(α)

The following graphs illustrate how the PDF and CDF of the inverse gamma distribution varies as the parameters vary:

Member Functions
inverse_gamma_distribution(RealType shape = 1, RealType scale = 1);

Constructs an inverse gamma distribution with shape α and scale β.

Requires that the shape and scale parameters are greater than zero, otherwise calls domain_error.

RealType shape()const;

Returns the α shape parameter of this inverse gamma distribution.

RealType scale()const;

Returns the β scale parameter of this inverse gamma distribution.

Non-member Accessors

All the usual non-member accessor functions that are generic to all distributions are supported: Cumulative Distribution Function, Probability Density Function, Quantile, Hazard Function, Cumulative Hazard Function, mean, median, mode, variance, standard deviation, skewness, kurtosis, kurtosis_excess, range and support.

The domain of the random variate is [0,+∞].

[Note] Note

Unlike some definitions, this implementation supports a random variate equal to zero as a special case, returning zero for pdf and cdf.


The inverse gamma distribution is implemented in terms of the incomplete gamma functions gamma_p and gamma_q and their inverses gamma_p_inv and gamma_q_inv: refer to the accuracy data for those functions for more information. But in general, inverse_gamma results are accurate to a few epsilon, >14 decimal digits accuracy for 64-bit double.


In the following table α is the shape parameter of the distribution, α ​ is its scale parameter, x is the random variate, p is the probability and q = 1-p.


Implementation Notes


Using the relation: pdf = gamma_p_derivative(α, β/ x, β) / x * x


Using the relation: p = gamma_q(α, β / x)

cdf complement

Using the relation: q = gamma_p(α, β / x)


Using the relation: x = β ​/ gamma_q_inv(α, p)

quantile from the complement

Using the relation: x = α ​/ gamma_p_inv(α, q)


β / (α + 1)


no analytic equation is known, but is evaluated as quantile(0.5)


β / (α - 1) for α > 1, else a domain_error


(β * β) / ((α - 1) * (α - 1) * (α - 2)) for α >2, else a domain_error


4 * sqrt (α -2) / (α -3) for α >3, else a domain_error


(30 * α - 66) / ((α-3)*(α - 4)) for α >4, else a domain_error