Editing Beta distribution (section)

=====Two parameters=====
For ''X''<sub>1</sub>, ..., ''X''<sub>''N''</sub> independent random variables each having a beta distribution parametrized with shape parameters ''α'' and ''β'', the joint log likelihood function for ''N'' [[independent and identically distributed random variables|iid]] observations is:

: <math>\ln (\mathcal{L} (\alpha, \beta\mid X) )= (\alpha - 1)\sum_{i=1}^N \ln X_i + (\beta- 1)\sum_{i=1}^N  \ln (1-X_i)- N \ln \Beta(\alpha,\beta) </math>

therefore the joint log likelihood function per ''N'' [[independent and identically distributed random variables|iid]] observations is

:<math>\frac{1}{N} \ln(\mathcal{L} (\alpha, \beta\mid X)) = (\alpha - 1)\frac{1}{N}\sum_{i=1}^N  \ln X_i + (\beta- 1) \frac{1}{N}\sum_{i=1}^N  \ln (1-X_i)-\, \ln \Beta(\alpha,\beta). </math>

For the two parameter case, the Fisher information has 4 components: 2 diagonal and 2 off-diagonal. Since the Fisher information matrix is symmetric, one of these off diagonal components is independent. Therefore, the Fisher information matrix has 3 independent components (2 diagonal and 1 off diagonal).
 	
Aryal and Nadarajah<ref name=Aryal>{{cite journal|last=Aryal|first=Gokarna|author2=Saralees Nadarajah|title=Information matrix for beta distributions|journal=Serdica Mathematical Journal (Bulgarian Academy of Science)| year=2004| volume=30|pages=513–526|url=http://www.math.bas.bg/serdica/2004/2004-513-526.pdf}}</ref> calculated Fisher's information matrix for the four-parameter case, from which the two parameter case can be obtained as follows:

:<math>- \frac{\partial^2\ln \mathcal{L}(\alpha,\beta\mid X)}{N\partial \alpha^2}=  \operatorname{var}[\ln (X)]= \psi_1(\alpha) - \psi_1(\alpha + \beta) ={\mathcal{I}}_{\alpha, \alpha}= \operatorname{E}\left [- \frac{\partial^2\ln \mathcal{L}(\alpha,\beta\mid X)}{N\partial \alpha^2} \right ] = \ln \operatorname{var}_{GX} </math>
:<math>- \frac{\partial^2\ln \mathcal{L}(\alpha,\beta\mid X)}{N\,\partial \beta^2} = \operatorname{var}[\ln (1-X)] = \psi_1(\beta) - \psi_1(\alpha + \beta) ={\mathcal{I}}_{\beta, \beta}=  \operatorname{E}\left [- \frac{\partial^2\ln \mathcal{L}(\alpha,\beta\mid X)}{N\partial \beta^2} \right]= \ln \operatorname{var}_{G(1-X)} </math>
:<math>- \frac{\partial^2\ln \mathcal{L}(\alpha,\beta\mid X)}{N \, \partial \alpha \, \partial \beta} = \operatorname{cov}[\ln X,\ln(1-X)]  = -\psi_1(\alpha+\beta) ={\mathcal{I}}_{\alpha, \beta}=  \operatorname{E}\left [- \frac{\partial^2\ln \mathcal{L}(\alpha,\beta\mid X)}{N\,\partial \alpha\,\partial \beta} \right] = \ln \operatorname{cov}_{G{X,(1-X)}}</math>

Since the Fisher information matrix is symmetric

:<math> \mathcal{I}_{\alpha, \beta}= \mathcal{I}_{\beta, \alpha}= \ln \operatorname{cov}_{G{X,(1-X)}}</math>

The Fisher information components are equal to the log geometric variances and log geometric covariance. Therefore, they can be expressed as '''[[trigamma function]]s''', denoted ψ<sub>1</sub>(α),  the second of the [[polygamma function]]s, defined as the derivative of the [[digamma]] function:

:<math>\psi_1(\alpha) = \frac{d^2\ln\Gamma(\alpha)}{\partial\alpha^2}=\, \frac{\partial \psi(\alpha)}{\partial\alpha}. </math>

These derivatives are also derived in the {{section link||Two unknown parameters}} and plots of the log likelihood function are also shown in that section.  {{section link||Geometric variance and covariance}} contains plots and further discussion of the Fisher information matrix components: the log geometric variances and log geometric covariance as a function of the shape parameters α and β.  {{section link||Moments of logarithmically transformed random variables}} contains formulas for moments of logarithmically transformed random variables. Images for the Fisher information components <math>\mathcal{I}_{\alpha, \alpha}, \mathcal{I}_{\beta, \beta}</math> and <math>\mathcal{I}_{\alpha, \beta}</math> are shown in {{section link||Geometric variance}}.

The determinant of Fisher's information matrix is of interest (for example for the calculation of [[Jeffreys prior]] probability).  From the expressions for the individual components of the Fisher information matrix, it follows that the determinant of Fisher's (symmetric) information matrix for the beta distribution is:

:<math>\begin{align}
\det(\mathcal{I}(\alpha, \beta))&= \mathcal{I}_{\alpha, \alpha} \mathcal{I}_{\beta, \beta}-\mathcal{I}_{\alpha, \beta} \mathcal{I}_{\alpha, \beta} \\[4pt]
&=(\psi_1(\alpha) - \psi_1(\alpha + \beta))(\psi_1(\beta) - \psi_1(\alpha + \beta))-( -\psi_1(\alpha+\beta))( -\psi_1(\alpha+\beta))\\[4pt]
&= \psi_1(\alpha)\psi_1(\beta)-( \psi_1(\alpha)+\psi_1(\beta))\psi_1(\alpha + \beta)\\[4pt]
\lim_{\alpha\to 0} \det(\mathcal{I}(\alpha, \beta)) &=\lim_{\beta \to  0} \det(\mathcal{I}(\alpha, \beta)) = \infty\\[4pt]
\lim_{\alpha\to \infty} \det(\mathcal{I}(\alpha, \beta)) &=\lim_{\beta \to  \infty} \det(\mathcal{I}(\alpha, \beta)) = 0
\end{align}</math>

From [[Sylvester's criterion]] (checking whether the diagonal elements are all positive), it follows that the Fisher information matrix for the two parameter case is [[Positive-definite matrix|positive-definite]] (under the standard condition that the shape parameters are positive ''α''&nbsp;>&nbsp;0 and&nbsp;''β''&nbsp;>&nbsp;0).