File:LogDirichletDensity-alpha 0.1 to alpha 1.9.gif

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

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 0.1 to 1.9.

Maple Code

The animated plot was generated using Maple 11, with the following code:

restart; with(plots);

B := (a1, a2, a3) -> (GAMMA(1.0*a1) * GAMMA(1.0*a2) * GAMMA(1.0*a3)) / GAMMA(1.0*a1+1.0*a2+1.0*a3);

f := (x1, x2, a1, a2, a3) ->  (x1^(a1-1)) * (x2^(a2-1)) * ( (1-x1-x2)^(a3-1)) /B(a1,a2,a3) ;

animate ( plot3d,  [eval(log(f(x1, x2, a1, a2, a3)), {a1=a, a2=a, a3=a}), 
x1=0.00..1, x2=0.00..1, axes=BOXED, grid=[25,25],gridstyle=triangular,orientation=[-135, 60],
shading=zhue, contours=20, style=surfacecontour, view=-3..2 ], a=0.1..1.9, frames=100);

Dirichlet