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ParaMonte Fortran 2.0.0
Parallel Monte Carlo and Machine Learning Library
See the latest version documentation. |
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ParaMonte Fortran 2.0.0
Parallel Monte Carlo and Machine Learning Library
See the latest version documentation. |
This module contains classes and procedures for computing the mathematical Beta Function and its inverse. More...
Data Types | |
| interface | getBetaInc |
| Generate and return the regularized Incomplete Beta Function \(I_x(\alpha, \beta)\) as defined in the details section of pm_mathBeta. More... | |
| interface | getBetaInv |
| Generate and return the regularized Inverse Incomplete Beta Function \(I_x(\alpha, \beta)\) as defined in the details section of pm_mathBeta. More... | |
| interface | getLogBeta |
| Generate and return the natural logarithm of the Beta Function \(\mathrm{B}(\alpha, \beta)\) as defined in the details section of pm_mathBeta. More... | |
| interface | setBetaInc |
| Return the regularized Incomplete Beta Function \(I_x(\alpha, \beta)\) as defined in the details section of pm_mathBeta. More... | |
| interface | setBetaInv |
| Return the regularized Inverse Incomplete Beta Function \(I_x(\alpha, \beta)\) as defined in the details section of pm_mathBeta. More... | |
Variables | |
| character(*, SK), parameter | MODULE_NAME = "@pm_mathBeta" |
This module contains classes and procedures for computing the mathematical Beta Function and its inverse.
The Beta Function is defined by the integral,
\begin{eqnarray} \mathrm{B}(\alpha, \beta) &=& \int_{0}^{1} ~ t^{\alpha-1} (1-t)^{\beta-1}~\mathrm{d}t \\ &=& \frac{\Gamma(\alpha) \Gamma(\beta)}{\Gamma(\alpha + \beta)} ~, \end{eqnarray}
where \(\alpha\) and \(\beta\) are complex numbers such that \(\Re \alpha > 0\) and \(\Re \beta > 0\).
The incomplete beta function is a generalization of the beta function defined as,
\begin{eqnarray} \mathrm{B}(x; \alpha, \beta) = \int_{0}^{x} ~ t^{\alpha-1} (1-t)^{\beta-1}~\mathrm{d}t ~,~ x \in [0, 1] \end{eqnarray}
where for \(x = 1\) becomes the same as the equation for the (complete) beta function, that is, \(\mathrm{B}(x = 1;\alpha, \beta) = \mathrm{B}(\alpha, \beta)\).
Note that the relationship between the complete and the incomplete beta function is similar to the relationship between the complete and incomplete gamma function.
The regularized incomplete beta function (or Beta Function Ratio or regularized beta function for short) is defined in terms of the incomplete beta function and the complete beta function as,
\begin{eqnarray} I_x(\alpha, \beta) = \frac{\mathrm{B}(x; \alpha, \beta)}{\mathrm{B}(\alpha, \beta)} ~. \end{eqnarray}
Note the regularized beta function has the limiting values,
\begin{eqnarray} I_0(\alpha, \beta) &=& 0 \\ I_1(\alpha, \beta) &=& 1 ~. \end{eqnarray}
The regularized incomplete beta function is the cumulative distribution function of the beta distribution, and is related to the cumulative distribution function of a random variable \(X\) following a binomial distribution with probability of single success \(p\) and number of Bernoulli trials \(n\).
The Regularized Inverse Incomplete Beta Function \(I_y(I_x; \alpha, \beta)\) is defined such that,
\begin{equation} I_y\left(I_x(x; \alpha, \beta); \alpha, \beta\right) = x ~. \end{equation}
The Regularized Inverse Incomplete Beta Function is also the quantile function of the Beta distribution.
Final Remarks ⛓
If you believe this algorithm or its documentation can be improved, we appreciate your contribution and help to edit this page's documentation and source file on GitHub.
For details on the naming abbreviations, see this page.
For details on the naming conventions, see this page.
This software is distributed under the MIT license with additional terms outlined below.
This software is available to the public under a highly permissive license.
Help us justify its continued development and maintenance by acknowledging its benefit to society, distributing it, and contributing to it.
| character(*, SK), parameter pm_mathBeta::MODULE_NAME = "@pm_mathBeta" |
Definition at line 90 of file pm_mathBeta.F90.
1.9.3