[section:beta_dist Beta Distribution]

``#include <boost/math/distributions/beta.hpp>``

   namespace boost{ namespace math{ 
   
    template <class RealType = double, 
              class ``__Policy``   = ``__policy_class`` >
   class beta_distribution;
   
   // typedef beta_distribution<double> beta;
   // Note that this is deliberately NOT provided,
   // to avoid a clash with the function name beta.
   
   template <class RealType, class ``__Policy``>
   class beta_distribution
   {
   public:
      typedef RealType  value_type;
      typedef Policy    policy_type;
      // Constructor from two shape parameters, alpha & beta:
      BOOST_MATH_GPU_ENABLED beta_distribution(RealType a, RealType b);
      
      // Parameter accessors:
      BOOST_MATH_GPU_ENABLED RealType alpha() const;
      BOOST_MATH_GPU_ENABLED RealType beta() const;
      
      // Parameter estimators of alpha or beta from mean and variance.
      BOOST_MATH_GPU_ENABLED static RealType find_alpha(
        RealType mean, // Expected value of mean.
        RealType variance); // Expected value of variance.
      
      BOOST_MATH_GPU_ENABLED static RealType find_beta(
        RealType mean, // Expected value of mean.
        RealType variance); // Expected value of variance.
  
      // Parameter estimators from
      // either alpha or beta, and x and probability.
      
      BOOST_MATH_GPU_ENABLED static RealType find_alpha(
        RealType beta, // from beta.
        RealType x, //  x.
        RealType probability); // cdf
      
      BOOST_MATH_GPU_ENABLED static RealType find_beta(
        RealType alpha, // alpha.
        RealType x, // probability x.
        RealType probability); // probability cdf.
   };
   
   }} // namespaces
   
The class type `beta_distribution` represents a 
[@http://en.wikipedia.org/wiki/Beta_distribution beta ] 
[@http://en.wikipedia.org/wiki/Probability_distribution probability distribution function].

The [@http://mathworld.wolfram.com/BetaDistribution.htm beta distribution ]
is used as a [@http://en.wikipedia.org/wiki/Prior_distribution prior distribution]
for binomial proportions in
[@http://mathworld.wolfram.com/BayesianAnalysis.html Bayesian analysis].

See also: 
[@http://documents.wolfram.com/calculationcenter/v2/Functions/ListsMatrices/Statistics/BetaDistribution.html beta distribution]
and [@http://en.wikipedia.org/wiki/Bayesian_statistics Bayesian statistics].

How the beta distribution is used for
[@http://home.uchicago.edu/~grynav/bayes/ABSLec5.ppt 
Bayesian analysis of one parameter models]
is discussed by Jeff Grynaviski.

The [@http://en.wikipedia.org/wiki/Probability_density_function probability density function PDF]
for the [@http://en.wikipedia.org/wiki/Beta_distribution beta distribution]
defined on the interval \[0,1\] is given by:

[expression f(x;[alpha],[beta]) = x[super[alpha] - 1] (1 - x)[super[beta] -1] / B([alpha], [beta])]

where [role serif_italic B([alpha], [beta])] is the
[@http://en.wikipedia.org/wiki/Beta_function beta function],
implemented in this library as __beta.  Division by the beta function
ensures that the pdf is normalized to the range zero to unity.

The following graph illustrates examples of the pdf for various values 
of the shape parameters.  Note the ['[alpha] = [beta] = 2] (blue line) 
is dome-shaped, and might be approximated by a symmetrical triangular 
distribution.

[graph beta_pdf]

If [alpha] = [beta] = 1, then it is a[emspace]
[@http://en.wikipedia.org/wiki/Uniform_distribution_%28continuous%29 uniform distribution],
equal to unity in the entire interval x = 0 to 1.
If [alpha] and [beta] are < 1, then the pdf is U-shaped.
If [alpha] != [beta], then the shape is asymmetric
and could be approximated by a triangle
whose apex is away from the centre (where x = half).

[h4 Member Functions]

[h5 Constructor]

   BOOST_MATH_GPU_ENABLED beta_distribution(RealType alpha, RealType beta);

Constructs a beta distribution with shape parameters /alpha/ and /beta/.

Requires alpha,beta > 0,otherwise __domain_error is called.  Note that 
technically the beta distribution is defined for alpha,beta >= 0, but
it's not clear whether any program can actually make use of that latitude
or how many of the non-member functions can be usefully defined in that case.
Therefore for now, we regard it as an error if alpha or beta is zero.

For example: 
   
   beta_distribution<> mybeta(2, 5);
   
Constructs a the beta distribution with alpha=2 and beta=5 (shown in yellow
in the graph above).

[h5 Parameter Accessors]

   BOOST_MATH_GPU_ENABLED RealType alpha() const;
   
Returns the parameter /alpha/ from which this distribution was constructed.
   
   BOOST_MATH_GPU_ENABLED RealType beta() const;
   
Returns the parameter /beta/ from which this distribution was constructed.

So for example:

   beta_distribution<> mybeta(2, 5);
   assert(mybeta.alpha() == 2.);  // mybeta.alpha() returns 2
   assert(mybeta.beta() == 5.);   // mybeta.beta()  returns 5

[h4 Parameter Estimators]

Two pairs of parameter estimators are provided.

One estimates either [alpha]  or [beta] 
from presumed-known mean and variance.

The other pair estimates either [alpha] or [beta] from 
the cdf and x.

It is also possible to estimate [alpha] and  [beta]  from 
'known' mode & quantile.  For example, calculators are provided by the
[@http://www.ausvet.com.au/pprev/content.php?page=PPscript 
Pooled Prevalence Calculator] and
[@http://www.epi.ucdavis.edu/diagnostictests/betabuster.html Beta Buster]
but this is not yet implemented here.

      static RealType find_alpha(
        RealType mean, // Expected value of mean.
        RealType variance); // Expected value of variance.
        
Returns the unique value of [alpha] that corresponds to a 
beta distribution with mean /mean/ and variance /variance/.
      
      static RealType find_beta(
        RealType mean, // Expected value of mean.
        RealType variance); // Expected value of variance.
  
Returns the unique value of [beta] that corresponds to a 
beta distribution with mean /mean/ and variance /variance/.
      
      static RealType find_alpha(
        RealType beta, // from beta.
        RealType x, //  x.
        RealType probability); // probability cdf
        
Returns the value of [alpha] that gives:
`cdf(beta_distribution<RealType>(alpha, beta), x) == probability`.
      
      static RealType find_beta(
        RealType alpha, // alpha.
        RealType x, // probability x.
        RealType probability); // probability cdf.

Returns the value of [beta] that gives:
`cdf(beta_distribution<RealType>(alpha, beta), x) == probability`.
      
[h4 Non-member Accessor Functions]

All the [link math_toolkit.dist_ref.nmp usual non-member accessor functions]
that are generic to all distributions are supported: __usual_accessors.
For this distribution all non-member accessor functions are marked with `BOOST_MATH_GPU_ENABLED` and can
be run on both host and device.

The formulae for calculating these are shown in the table below, and at
[@http://mathworld.wolfram.com/BetaDistribution.html Wolfram Mathworld].

[h4 Applications]

The beta distribution can be used to model events constrained
to take place within an interval defined by a minimum and maximum value:
so it is used in project management systems. 

It is also widely used in [@http://en.wikipedia.org/wiki/Bayesian_inference Bayesian statistical inference].

[h4 Related distributions]

The beta distribution with both [alpha] and [beta] = 1 follows a
[@http://en.wikipedia.org/wiki/Uniform_distribution_%28continuous%29 uniform distribution].

The [@http://en.wikipedia.org/wiki/Triangular_distribution triangular]
is used when less precise information is available.

The [@http://en.wikipedia.org/wiki/Binomial_distribution binomial distribution]
is closely related when [alpha] and [beta]  are integers.

With integer values of [alpha]  and [beta] the distribution B(i, j) is
that of the j-th highest of a sample of i + j + 1 independent random variables
uniformly distributed between 0 and 1.
The cumulative probability from 0 to x is thus
the probability that the j-th highest value is less than x.
Or it is the probability that at least i of the random variables are less than x, 
a probability given by summing over the __binomial_distrib
with its p parameter set to x.

[h4 Accuracy]

This distribution is implemented using the 
[link math_toolkit.sf_beta.beta_function beta functions] __beta and 
[link math_toolkit.sf_beta.ibeta_function incomplete beta functions] __ibeta and __ibetac;
please refer to these functions for information on accuracy.

[h4 Implementation]

In the following table /a/ and /b/ are the parameters [alpha] and [beta],
/x/ is the random variable, /p/ is the probability and /q = 1-p/.

[table
[[Function][Implementation Notes]]
[[pdf][[role serif_italic f(x;[alpha],[beta]) = x[super[alpha] - 1] (1 - x)[super[beta] -1] / B([alpha], [beta])]
   
    Implemented using __ibeta_derivative(a, b, x).]]
[[cdf][Using the incomplete beta function __ibeta(a, b, x)]]
[[cdf complement][__ibetac(a, b, x)]]
[[quantile][Using the inverse incomplete beta function __ibeta_inv(a, b, p)]]
[[quantile from the complement][__ibetac_inv(a, b, q)]]
[[mean][`a/(a+b)`]]
[[variance][`a * b / (a+b)^2 * (a + b + 1)`]]
[[mode][`(a-1) / (a + b - 2)`]]
[[skewness][`2 (b-a) sqrt(a+b+1)/(a+b+2) * sqrt(a * b)`]]
[[kurtosis excess][ [equation beta_dist_kurtosis]  ]]
[[kurtosis][`kurtosis + 3`]]
[[parameter estimation][ ]]
[[alpha (from mean and variance)][`mean * (( (mean * (1 - mean)) / variance)- 1)`]]
[[beta (from mean and variance)][`(1 - mean) * (((mean * (1 - mean)) /variance)-1)`]]
[[The member functions `find_alpha` and `find_beta`

  from cdf and probability x
  
  and *either* `alpha` or `beta`]
      [Implemented in terms of the inverse incomplete beta functions
      
__ibeta_inva, and __ibeta_invb respectively.]]
[[`find_alpha`][`ibeta_inva(beta, x, probability)`]]
[[`find_beta`][`ibeta_invb(alpha, x, probability)`]]
] [/table]

[h4 References]

[@http://en.wikipedia.org/wiki/Beta_distribution Wikipedia Beta distribution]

[@http://www.itl.nist.gov/div898/handbook/eda/section3/eda366h.htm NIST Exploratory Data Analysis]

[@http://mathworld.wolfram.com/BetaDistribution.html Wolfram MathWorld]

[endsect] [/section:beta_dist beta]

[/ beta.qbk
  Copyright 2006 John Maddock and Paul A. Bristow.
  Distributed under the Boost Software License, Version 1.0.
  (See accompanying file LICENSE_1_0.txt or copy at
  http://www.boost.org/LICENSE_1_0.txt).
]
