Editing Beta distribution (section)

=====Mode and concentration=====
[[Concave function|Concave]] beta distributions, which have <math>\alpha,\beta>1</math>, can be parametrized in terms of mode and "concentration". The mode, <math>\omega=\frac{\alpha-1}{\alpha+\beta-2}</math>, and concentration, <math>\kappa = \alpha + \beta</math>, can be used to define the usual shape parameters as follows:<ref name="Kruschke2015">{{cite book|last=Kruschke|first=John K.|author-link=John K. Kruschke|title=Doing Bayesian Data Analysis: A Tutorial with R, JAGS and Stan|year=2015|publisher=Academic Press / Elsevier|isbn=978-0-12-405888-0}}</ref>
:<math>\begin{align}
\alpha &= \omega (\kappa - 2) + 1\\
\beta  &= (1 - \omega)(\kappa - 2) + 1
\end{align}</math>
For the mode, <math>0<\omega<1</math>, to be well-defined, we need <math>\alpha,\beta>1</math>, or equivalently <math>\kappa>2</math>. If instead we define the concentration as <math>c=\alpha+\beta-2</math>, the condition simplifies to <math>c>0</math> and the beta density at <math>\alpha=1+c\omega</math> and <math>\beta=1+c(1-\omega)</math> can be written as:
:<math>
f(x;\omega,c) = \frac{x^{c\omega}(1-x)^{c(1-\omega)}}{\Beta\bigl(1+c\omega,1+c(1-\omega)\bigr)}
</math>
where <math>c</math> directly scales the [[sufficient statistics]], <math>\log(x)</math> and <math>\log(1-x)</math>. Note also that in the limit, <math>c\to0</math>, the distribution becomes flat.