Editing Beta distribution (section)

=====Moments of logarithmically transformed random variables=====
[[File:Logit.svg|thumbnail|right|350px|Plot of logit(''X'') =&nbsp;ln(''X''/(1&nbsp;−''X'')) (vertical axis) vs. ''X'' in the domain of 0 to 1 (horizontal axis). Logit transformations are interesting, as they usually transform various shapes (including J-shapes) into (usually skewed) bell-shaped densities over the logit variable, and they may remove the end singularities over the original variable]]

Expected values for [[Logarithm transformation|logarithmic transformations]] (useful for [[maximum likelihood]] estimates, see {{section link||Parameter estimation, Maximum likelihood}}) are discussed in this section.  The following logarithmic linear transformations are related to the geometric means ''G<sub>X</sub>'' and  ''G''<sub>(1−''X'')</sub> (see {{section link||Geometric Mean}}):

:<math>\begin{align}
\operatorname{E}[\ln(X)] &= \psi(\alpha) - \psi(\alpha + \beta)= - \operatorname{E}\left[\ln \left (\frac{1}{X} \right )\right],\\
\operatorname{E}[\ln(1-X)] &=\psi(\beta) - \psi(\alpha + \beta)= - \operatorname{E} \left[\ln \left (\frac{1}{1-X} \right )\right].
\end{align}</math>

Where the '''[[digamma function]]''' ''ψ''(''α'') is defined as the [[logarithmic derivative]] of the [[gamma function]]:<ref name=Abramowitz/>

:<math>\psi(\alpha) = \frac{d \ln\Gamma(\alpha)}{d\alpha}</math>

[[Logit]] transformations are interesting,<ref name=MacKay>{{cite book|last=MacKay|first=David|title=Information Theory, Inference and Learning Algorithms|year=2003| publisher=Cambridge University Press; First Edition |isbn=978-0521642989|bibcode=2003itil.book.....M}}</ref> as they usually transform various shapes (including J-shapes) into (usually skewed) bell-shaped densities over the logit variable, and they may remove the end singularities over the original variable:

:<math>\begin{align}
\operatorname{E}\left[\ln \left (\frac{X}{1-X} \right ) \right] &=\psi(\alpha) - \psi(\beta)= \operatorname{E}[\ln(X)] +\operatorname{E} \left[\ln \left (\frac{1}{1-X} \right) \right],\\
\operatorname{E}\left [\ln \left (\frac{1-X}{X} \right ) \right ] &=\psi(\beta) - \psi(\alpha)= - \operatorname{E} \left[\ln \left (\frac{X}{1-X} \right) \right] .
\end{align}</math>

Johnson<ref name=JohnsonLogInv>{{cite journal|last=Johnson|first=N.L.|title=Systems of frequency curves generated by methods of translation| journal=Biometrika|year=1949 |volume=36 |issue=1–2|pages=149–176|doi=10.1093/biomet/36.1-2.149|pmid=18132090|hdl=10338.dmlcz/135506|url=http://dml.cz/bitstream/handle/10338.dmlcz/135506/Kybernetika_39-2003-1_3.pdf}}</ref>  considered the distribution of the [[logit]] – transformed variable ln(''X''/1&nbsp;−&nbsp;''X''), including its moment generating function and approximations for large values of the shape parameters.  This transformation extends the finite support [0,&nbsp;1] based on the original variable ''X'' to infinite support in both directions of the real line (−∞,&nbsp;+∞). The logit of a beta variate has the [[logistic-beta distribution]].

Higher order logarithmic moments can be derived by using the representation of a beta distribution as a proportion of two gamma distributions and differentiating through the integral. They can be expressed in terms of higher order poly-gamma functions as follows:

:<math>\begin{align}
\operatorname{E} \left [\ln^2(X) \right ] &= (\psi(\alpha) - \psi(\alpha + \beta))^2+\psi_1(\alpha)-\psi_1(\alpha+\beta), \\
\operatorname{E} \left [\ln^2(1-X) \right ] &= (\psi(\beta) - \psi(\alpha + \beta))^2+\psi_1(\beta)-\psi_1(\alpha+\beta), \\
\operatorname{E} \left [\ln (X)\ln(1-X) \right ] &=(\psi(\alpha) - \psi(\alpha + \beta))(\psi(\beta) - \psi(\alpha + \beta)) -\psi_1(\alpha+\beta).
\end{align}</math>

therefore the [[variance]] of the logarithmic variables and [[covariance]] of ln(''X'') and ln(1−''X'') are:

:<math>\begin{align}
\operatorname{cov}[\ln(X), \ln(1-X)] &= \operatorname{E}\left[\ln(X)\ln(1-X)\right] - \operatorname{E}[\ln(X)]\operatorname{E}[\ln(1-X)] = -\psi_1(\alpha+\beta) \\
& \\
\operatorname{var}[\ln X] &= \operatorname{E}[\ln^2(X)] - (\operatorname{E}[\ln(X)])^2 \\
&= \psi_1(\alpha) - \psi_1(\alpha + \beta) \\
&= \psi_1(\alpha) + \operatorname{cov}[\ln(X), \ln(1-X)] \\
& \\
\operatorname{var}[\ln (1-X)] &= \operatorname{E}[\ln^2 (1-X)] - (\operatorname{E}[\ln (1-X)])^2 \\
&= \psi_1(\beta) - \psi_1(\alpha + \beta) \\
&= \psi_1(\beta) + \operatorname{cov}[\ln (X), \ln(1-X)]
\end{align}</math>

where the '''[[trigamma function]]''', denoted ''ψ''<sub>1</sub>(''α''), is the second of the [[polygamma function]]s, and is defined as the derivative of the [[digamma]] function:

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

The variances and covariance of the logarithmically transformed variables ''X'' and (1&nbsp;−&nbsp;''X'') are different, in general, because the logarithmic transformation destroys the mirror-symmetry of the original variables ''X'' and (1&nbsp;−&nbsp;''X''), as the logarithm approaches negative infinity for the variable approaching zero.

These logarithmic variances and covariance are the elements of the [[Fisher information]] matrix for the beta distribution.  They are also a measure of the curvature of the log likelihood function (see section on Maximum likelihood estimation).

The variances of the log inverse variables are identical to the variances of the log variables:

:<math>\begin{align}
\operatorname{var}\left[\ln \left (\frac{1}{X} \right ) \right] & =\operatorname{var}[\ln(X)] = \psi_1(\alpha) - \psi_1(\alpha + \beta), \\
\operatorname{var}\left[\ln \left (\frac{1}{1-X} \right ) \right] &=\operatorname{var}[\ln (1-X)] = \psi_1(\beta) - \psi_1(\alpha + \beta), \\
\operatorname{cov}\left[\ln \left (\frac{1}{X} \right), \ln \left (\frac{1}{1-X}\right ) \right] &=\operatorname{cov}[\ln(X),\ln(1-X)]= -\psi_1(\alpha + \beta).\end{align}</math>

It also follows that the variances of the [[logit]]-transformed variables are

:<math>\operatorname{var}\left[\ln \left (\frac{X}{1-X} \right )\right]=\operatorname{var}\left[\ln \left (\frac{1-X}{X} \right ) \right]=-\operatorname{cov}\left [\ln \left (\frac{X}{1-X} \right ), \ln \left (\frac{1-X}{X} \right ) \right]= \psi_1(\alpha) + \psi_1(\beta).</math>