(a) The Dirichlet distribution when $K=3$ defines a distribution over the simplex, which can be represented by the triangular surface. Points on this surface satisfy $0 \leq \theta _k \leq 1$ and ParseError: KaTeX parse error: Undefined control sequence: \sum@ at position 2: \̲s̲u̲m̲@̲ ̲@ _{k=1}^3 \the…. Generated by dirichlet_3d_triangle_plot.ipynb . (b) Plot of the Dirichlet density for ${\smallsmile }{\boldsymbol{\alpha }}=(20,20,20)$. (c) Plot of the Dirichlet density for ${\smallsmile }{\boldsymbol{\alpha }}=(3,3,20)$. (d) Plot of the Dirichlet density for ${\smallsmile }{\boldsymbol{\alpha }}=(0.1, 0.1, 0.1)$. Generated by dirichlet_3d_spiky_plot.ipynb .