Editing Dirichlet distribution (section)

===From marginal beta distributions===
A less efficient algorithm<ref>{{cite book |author1=A. Gelman |author2=J. B. Carlin |author3=H. S. Stern |author4=D. B. Rubin | year=2003 | title= Bayesian Data Analysis |url=https://archive.org/details/bayesiandataanal00gelm |url-access=limited | edition=2nd | isbn=1-58488-388-X | pages=[https://archive.org/details/bayesiandataanal00gelm/page/n607 582]|publisher=Chapman & Hall/CRC }}</ref> relies on the univariate marginal and conditional distributions being beta and proceeds as follows.  Simulate <math>x_1</math> from

<math display=block>\textrm{Beta}\left(\alpha_1, \sum_{i=2}^K \alpha_i \right)</math>

Then simulate <math>x_2, \ldots, x_{K-1}</math> in order, as follows.  For <math>j=2, \ldots, K-1</math>, simulate <math>\phi_j</math> from

<math display=block>\textrm{Beta} \left(\alpha_j, \sum_{i=j+1}^K \alpha_i \right ),</math>

and let

<math display=block>x_j= \left(1-\sum_{i=1}^{j-1} x_i \right )\phi_j.</math>

Finally, set

<math display=block>x_K=1-\sum_{i=1}^{K-1} x_i.</math>

This iterative procedure corresponds closely to the "string cutting" intuition described above.

Below is example Python code to draw the sample:

<syntaxhighlight lang="python">
params = [a1, a2, ..., ak]
xs = [random.betavariate(params[0], sum(params[1:]))]
for j in range(1, len(params) - 1):
    phi = random.betavariate(params[j], sum(params[j + 1 :]))
    xs.append((1 - sum(xs)) * phi)
xs.append(1 - sum(xs))
</syntaxhighlight>