File:LogDirichletDensity-alpha 0.3 to alpha 2.0.gif

English: We illustrate the log of the density function:

${\displaystyle \log(f(x_{1},\dots ,x_{K-1};\alpha _{1},\dots ,\alpha _{K}))=\log \left({\frac {1}{\mathrm {B} (\alpha )}}\prod _{i=1}^{K}x_{i}^{\alpha _{i}-1}\right)=+\sum _{i=1}^{K}\alpha _{i}\log(x_{i})-\sum _{i=1}^{K}\log(x_{i})-\sum _{i=1}^{K}\log(\Gamma (\alpha _{i}))+\log(\Gamma (\sum _{i=1}^{K}\alpha _{i}))}$

for ${\displaystyle K=3}$. In other words, we have two parameters ${\displaystyle x_{1},x_{2}}$ varying on the two axes, and an implicit ${\displaystyle x_{3}=1-x_{1}-x_{2}}$.

The picture illustrates the case where ${\displaystyle \alpha _{1}=\alpha _{2}=\alpha _{3}=\alpha }$ and we vary over time the parameter ${\displaystyle \alpha }$ from 0.3 to 2.0.