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<title>numpy.random.dirichlet — NumPy v2.3.dev0 Manual</title>
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<section id="numpy-random-dirichlet">
<h1>numpy.random.dirichlet<a class="headerlink" href="#numpy-random-dirichlet" title="Link to this heading">#</a></h1>
<dl class="py function">
<dt class="sig sig-object py" id="numpy.random.dirichlet">
<span class="sig-prename descclassname"><span class="pre">random.</span></span><span class="sig-name descname"><span class="pre">dirichlet</span></span><span class="sig-paren">(</span><em class="sig-param"><span class="n"><span class="pre">alpha</span></span></em>, <em class="sig-param"><span class="n"><span class="pre">size</span></span><span class="o"><span class="pre">=</span></span><span class="default_value"><span class="pre">None</span></span></em><span class="sig-paren">)</span><a class="headerlink" href="#numpy.random.dirichlet" title="Link to this definition">#</a></dt>
<dd><p>Draw samples from the Dirichlet distribution.</p>
<p>Draw <a class="reference internal" href="../../generated/numpy.size.html#numpy.size" title="numpy.size"><code class="xref py py-obj docutils literal notranslate"><span class="pre">size</span></code></a> samples of dimension k from a Dirichlet distribution. A
Dirichlet-distributed random variable can be seen as a multivariate
generalization of a Beta distribution. The Dirichlet distribution
is a conjugate prior of a multinomial distribution in Bayesian
inference.</p>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>New code should use the <a class="reference internal" href="numpy.random.Generator.dirichlet.html#numpy.random.Generator.dirichlet" title="numpy.random.Generator.dirichlet"><code class="xref py py-obj docutils literal notranslate"><span class="pre">dirichlet</span></code></a>
method of a <a class="reference internal" href="../generator.html#numpy.random.Generator" title="numpy.random.Generator"><code class="xref py py-obj docutils literal notranslate"><span class="pre">Generator</span></code></a> instance instead;
please see the <a class="reference internal" href="../index.html#random-quick-start"><span class="std std-ref">Quick start</span></a>.</p>
</div>
<dl class="field-list simple">
<dt class="field-odd">Parameters<span class="colon">:</span></dt>
<dd class="field-odd"><dl class="simple">
<dt><strong>alpha</strong><span class="classifier">sequence of floats, length k</span></dt><dd><p>Parameter of the distribution (length <code class="docutils literal notranslate"><span class="pre">k</span></code> for sample of
length <code class="docutils literal notranslate"><span class="pre">k</span></code>).</p>
</dd>
<dt><strong>size</strong><span class="classifier">int or tuple of ints, optional</span></dt><dd><p>Output shape. If the given shape is, e.g., <code class="docutils literal notranslate"><span class="pre">(m,</span> <span class="pre">n)</span></code>, then
<code class="docutils literal notranslate"><span class="pre">m</span> <span class="pre">*</span> <span class="pre">n</span> <span class="pre">*</span> <span class="pre">k</span></code> samples are drawn. Default is None, in which case a
vector of length <code class="docutils literal notranslate"><span class="pre">k</span></code> is returned.</p>
</dd>
</dl>
</dd>
<dt class="field-even">Returns<span class="colon">:</span></dt>
<dd class="field-even"><dl class="simple">
<dt><strong>samples</strong><span class="classifier">ndarray,</span></dt><dd><p>The drawn samples, of shape <code class="docutils literal notranslate"><span class="pre">(size,</span> <span class="pre">k)</span></code>.</p>
</dd>
</dl>
</dd>
<dt class="field-odd">Raises<span class="colon">:</span></dt>
<dd class="field-odd"><dl class="simple">
<dt>ValueError</dt><dd><p>If any value in <code class="docutils literal notranslate"><span class="pre">alpha</span></code> is less than or equal to zero</p>
</dd>
</dl>
</dd>
</dl>
<div class="admonition seealso">
<p class="admonition-title">See also</p>
<dl class="simple">
<dt><a class="reference internal" href="numpy.random.Generator.dirichlet.html#numpy.random.Generator.dirichlet" title="numpy.random.Generator.dirichlet"><code class="xref py py-obj docutils literal notranslate"><span class="pre">random.Generator.dirichlet</span></code></a></dt><dd><p>which should be used for new code.</p>
</dd>
</dl>
</div>
<p class="rubric">Notes</p>
<p>The Dirichlet distribution is a distribution over vectors
<span class="math notranslate nohighlight">\(x\)</span> that fulfil the conditions <span class="math notranslate nohighlight">\(x_i>0\)</span> and
<span class="math notranslate nohighlight">\(\sum_{i=1}^k x_i = 1\)</span>.</p>
<p>The probability density function <span class="math notranslate nohighlight">\(p\)</span> of a
Dirichlet-distributed random vector <span class="math notranslate nohighlight">\(X\)</span> is
proportional to</p>
<div class="math notranslate nohighlight">
\[p(x) \propto \prod_{i=1}^{k}{x^{\alpha_i-1}_i},\]</div>
<p>where <span class="math notranslate nohighlight">\(\alpha\)</span> is a vector containing the positive
concentration parameters.</p>
<p>The method uses the following property for computation: let <span class="math notranslate nohighlight">\(Y\)</span>
be a random vector which has components that follow a standard gamma
distribution, then <span class="math notranslate nohighlight">\(X = \frac{1}{\sum_{i=1}^k{Y_i}} Y\)</span>
is Dirichlet-distributed</p>
<p class="rubric">References</p>
<div role="list" class="citation-list">
<div class="citation" id="r15565755312f-1" role="doc-biblioentry">
<span class="label"><span class="fn-bracket">[</span>1<span class="fn-bracket">]</span></span>
<p>David McKay, “Information Theory, Inference and Learning
Algorithms,” chapter 23,
<a class="reference external" href="https://www.inference.org.uk/mackay/itila/">https://www.inference.org.uk/mackay/itila/</a></p>
</div>
<div class="citation" id="r15565755312f-2" role="doc-biblioentry">
<span class="label"><span class="fn-bracket">[</span>2<span class="fn-bracket">]</span></span>
<p>Wikipedia, “Dirichlet distribution”,
<a class="reference external" href="https://en.wikipedia.org/wiki/Dirichlet_distribution">https://en.wikipedia.org/wiki/Dirichlet_distribution</a></p>
</div>
</div>
<p class="rubric">Examples</p>
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<p>Taking an example cited in Wikipedia, this distribution can be used if
one wanted to cut strings (each of initial length 1.0) into K pieces
with different lengths, where each piece had, on average, a designated
average length, but allowing some variation in the relative sizes of
the pieces.</p>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">s</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">dirichlet</span><span class="p">((</span><span class="mi">10</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">3</span><span class="p">),</span> <span class="mi">20</span><span class="p">)</span><span class="o">.</span><span class="n">transpose</span><span class="p">()</span>
</pre></div>
</div>
<div class="doctest highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">import</span><span class="w"> </span><span class="nn">matplotlib.pyplot</span><span class="w"> </span><span class="k">as</span><span class="w"> </span><span class="nn">plt</span>
<span class="gp">>>> </span><span class="n">plt</span><span class="o">.</span><span class="n">barh</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">20</span><span class="p">),</span> <span class="n">s</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
<span class="gp">>>> </span><span class="n">plt</span><span class="o">.</span><span class="n">barh</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">20</span><span class="p">),</span> <span class="n">s</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">left</span><span class="o">=</span><span class="n">s</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">color</span><span class="o">=</span><span class="s1">'g'</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">plt</span><span class="o">.</span><span class="n">barh</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">20</span><span class="p">),</span> <span class="n">s</span><span class="p">[</span><span class="mi">2</span><span class="p">],</span> <span class="n">left</span><span class="o">=</span><span class="n">s</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">+</span><span class="n">s</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">color</span><span class="o">=</span><span class="s1">'r'</span><span class="p">)</span>
<span class="gp">>>> </span><span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">"Lengths of Strings"</span><span class="p">)</span>
</pre></div>
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