Editing Dirichlet distribution (section)

===Conjugate prior of the Dirichlet distribution===
Because the Dirichlet distribution is an [[exponential family|exponential family distribution]] it has a conjugate prior.
The conjugate prior is of the form:<ref name=Lefkimmiatis2009>{{cite journal |first1=Stamatios |last1=Lefkimmiatis |first2=Petros |last2=Maragos |first3=George |last3=Papandreou |year=2009 |title=Bayesian Inference on Multiscale Models for Poisson Intensity Estimation: Applications to Photon-Limited Image Denoising |journal=IEEE Transactions on Image Processing |volume=18 |issue=8 |pages=1724–1741 |doi=10.1109/TIP.2009.2022008 |pmid=19414285 |bibcode=2009ITIP...18.1724L |s2cid=859561 }}</ref>

<math display=block>\operatorname{CD}(\boldsymbol\alpha \mid \boldsymbol{v},\eta) \propto \left(\frac{1}{\operatorname{B}(\boldsymbol\alpha)}\right)^\eta \exp\left(-\sum_k v_k \alpha_k\right).</math>

Here <math>\boldsymbol{v}</math> is a {{mvar|K}}-dimensional real vector and <math>\eta</math> is a scalar parameter.  The domain of <math>(\boldsymbol{v},\eta)</math> is restricted to the set of parameters for which the above unnormalized density function can be normalized. The (necessary and sufficient) condition is:<ref name=Andreoli2018>{{cite arXiv |last=Andreoli |first=Jean-Marc |year=2018 |eprint=1811.05266 |title=A conjugate prior for the Dirichlet distribution |class=cs.LG }}</ref>

<math display=block>
\forall k\;\;v_k>0\;\;\;\;\text{ and } \;\;\;\;\eta>-1 \;\;\;\;\text{ and } \;\;\;\;(\eta\leq0\;\;\;\;\text{ or }\;\;\;\;\sum_k \exp-\frac{v_k} \eta < 1)
</math>

The conjugation property can be expressed as

: if [''prior'': <math>\boldsymbol{\alpha}\sim\operatorname{CD}(\cdot \mid \boldsymbol{v},\eta)</math>] and [''observation'': <math>\boldsymbol{x}\mid\boldsymbol{\alpha}\sim\operatorname{Dirichlet}(\cdot \mid \boldsymbol{\alpha})</math>] then [''posterior'': <math>\boldsymbol{\alpha}\mid\boldsymbol{x}\sim\operatorname{CD}(\cdot \mid \boldsymbol{v}-\log \boldsymbol{x}, \eta+1)</math>].

In the published literature there is no practical algorithm to efficiently generate samples from <math>\operatorname{CD}(\boldsymbol{\alpha} \mid \boldsymbol{v},\eta)</math>.