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<div class="section" id="module-torch.distributions">
<span id="probability-distributions-torch-distributions"></span><h1>Probability distributions - torch.distributions<a class="headerlink" href="#module-torch.distributions" title="Permalink to this headline">¶</a></h1>
<p>The <code class="docutils literal notranslate"><span class="pre">distributions</span></code> package contains parameterizable probability distributions
and sampling functions. This allows the construction of stochastic computation
graphs and stochastic gradient estimators for optimization. This package
generally follows the design of the <a class="reference external" href="https://arxiv.org/abs/1711.10604">TensorFlow Distributions</a> package.</p>
<p>It is not possible to directly backpropagate through random samples. However,
there are two main methods for creating surrogate functions that can be
backpropagated through. These are the score function estimator/likelihood ratio
estimator/REINFORCE and the pathwise derivative estimator. REINFORCE is commonly
seen as the basis for policy gradient methods in reinforcement learning, and the
pathwise derivative estimator is commonly seen in the reparameterization trick
in variational autoencoders. Whilst the score function only requires the value
of samples <span class="math"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mclose">)</span></span></span></span>
</span>, the pathwise derivative requires the derivative
<span class="math"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>f</mi><mo mathvariant="normal" lspace="0em" rspace="0em">′</mo></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f'(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.001892em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mclose">)</span></span></span></span>
</span>. The next sections discuss these two in a reinforcement learning
example. For more details see
<a class="reference external" href="https://arxiv.org/abs/1506.05254">Gradient Estimation Using Stochastic Computation Graphs</a> .</p>
<div class="section" id="score-function">
<h2>Score function<a class="headerlink" href="#score-function" title="Permalink to this headline">¶</a></h2>
<p>When the probability density function is differentiable with respect to its
parameters, we only need <code class="xref py py-meth docutils literal notranslate"><span class="pre">sample()</span></code> and
<code class="xref py py-meth docutils literal notranslate"><span class="pre">log_prob()</span></code> to implement REINFORCE:</p>
<div class="math">
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><mi>θ</mi><mo>=</mo><mi>α</mi><mi>r</mi><mfrac><mrow><mi mathvariant="normal">∂</mi><mi>log</mi><mo></mo><mi>p</mi><mo stretchy="false">(</mo><mi>a</mi><mi mathvariant="normal">∣</mi><msup><mi>π</mi><mi>θ</mi></msup><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">∂</mi><mi>θ</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\Delta\theta = \alpha r \frac{\partial\log p(a|\pi^\theta(s))}{\partial\theta}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord">Δ</span><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.212108em;vertical-align:-0.686em;"></span><span class="mord mathdefault" style="margin-right:0.0037em;">α</span><span class="mord mathdefault" style="margin-right:0.02778em;">r</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.526108em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord mathdefault">a</span><span class="mord">∣</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">π</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.849108em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em;">θ</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathdefault">s</span><span class="mclose">)</span><span class="mclose">)</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span>
</div><p>where <span class="math"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\theta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span></span></span></span>
</span> are the parameters, <span class="math"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.0037em;">α</span></span></span></span>
</span> is the learning rate,
<span class="math"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>r</mi></mrow><annotation encoding="application/x-tex">r</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">r</span></span></span></span>
</span> is the reward and <span class="math"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>a</mi><mi mathvariant="normal">∣</mi><msup><mi>π</mi><mi>θ</mi></msup><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p(a|\pi^\theta(s))</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.099108em;vertical-align:-0.25em;"></span><span class="mord mathdefault">p</span><span class="mopen">(</span><span class="mord mathdefault">a</span><span class="mord">∣</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">π</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.849108em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em;">θ</span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathdefault">s</span><span class="mclose">)</span><span class="mclose">)</span></span></span></span>
</span> is the probability of
taking action <span class="math"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">a</span></span></span></span>
</span> in state <span class="math"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>s</mi></mrow><annotation encoding="application/x-tex">s</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">s</span></span></span></span>
</span> given policy <span class="math"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>π</mi><mi>θ</mi></msup></mrow><annotation encoding="application/x-tex">\pi^\theta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.849108em;vertical-align:0em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">π</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.849108em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.02778em;">θ</span></span></span></span></span></span></span></span></span></span></span>
</span>.</p>
<p>In practice we would sample an action from the output of a network, apply this
action in an environment, and then use <code class="docutils literal notranslate"><span class="pre">log_prob</span></code> to construct an equivalent
loss function. Note that we use a negative because optimizers use gradient
descent, whilst the rule above assumes gradient ascent. With a categorical
policy, the code for implementing REINFORCE would be as follows:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">probs</span> <span class="o">=</span> <span class="n">policy_network</span><span class="p">(</span><span class="n">state</span><span class="p">)</span>
<span class="c1"># Note that this is equivalent to what used to be called multinomial</span>
<span class="n">m</span> <span class="o">=</span> <span class="n">Categorical</span><span class="p">(</span><span class="n">probs</span><span class="p">)</span>
<span class="n">action</span> <span class="o">=</span> <span class="n">m</span><span class="o">.</span><span class="n">sample</span><span class="p">()</span>
<span class="n">next_state</span><span class="p">,</span> <span class="n">reward</span> <span class="o">=</span> <span class="n">env</span><span class="o">.</span><span class="n">step</span><span class="p">(</span><span class="n">action</span><span class="p">)</span>
<span class="n">loss</span> <span class="o">=</span> <span class="o">-</span><span class="n">m</span><span class="o">.</span><span class="n">log_prob</span><span class="p">(</span><span class="n">action</span><span class="p">)</span> <span class="o">*</span> <span class="n">reward</span>
<span class="n">loss</span><span class="o">.</span><span class="n">backward</span><span class="p">()</span>
</pre></div>
</div>
</div>
<div class="section" id="pathwise-derivative">
<h2>Pathwise derivative<a class="headerlink" href="#pathwise-derivative" title="Permalink to this headline">¶</a></h2>
<p>The other way to implement these stochastic/policy gradients would be to use the
reparameterization trick from the
<code class="xref py py-meth docutils literal notranslate"><span class="pre">rsample()</span></code> method, where the
parameterized random variable can be constructed via a parameterized
deterministic function of a parameter-free random variable. The reparameterized
sample therefore becomes differentiable. The code for implementing the pathwise
derivative would be as follows:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">params</span> <span class="o">=</span> <span class="n">policy_network</span><span class="p">(</span><span class="n">state</span><span class="p">)</span>
<span class="n">m</span> <span class="o">=</span> <span class="n">Normal</span><span class="p">(</span><span class="o">*</span><span class="n">params</span><span class="p">)</span>
<span class="c1"># Any distribution with .has_rsample == True could work based on the application</span>
<span class="n">action</span> <span class="o">=</span> <span class="n">m</span><span class="o">.</span><span class="n">rsample</span><span class="p">()</span>
<span class="n">next_state</span><span class="p">,</span> <span class="n">reward</span> <span class="o">=</span> <span class="n">env</span><span class="o">.</span><span class="n">step</span><span class="p">(</span><span class="n">action</span><span class="p">)</span> <span class="c1"># Assuming that reward is differentiable</span>
<span class="n">loss</span> <span class="o">=</span> <span class="o">-</span><span class="n">reward</span>
<span class="n">loss</span><span class="o">.</span><span class="n">backward</span><span class="p">()</span>
</pre></div>
</div>
</div>
<div class="section" id="distribution">
<h2><span class="hidden-section">Distribution</span><a class="headerlink" href="#distribution" title="Permalink to this headline">¶</a></h2>
<dl class="class">
<dt id="torch.distributions.distribution.Distribution">
<em class="property">class </em><code class="sig-prename descclassname">torch.distributions.distribution.</code><code class="sig-name descname">Distribution</code><span class="sig-paren">(</span><em class="sig-param">batch_shape=torch.Size([])</em>, <em class="sig-param">event_shape=torch.Size([])</em>, <em class="sig-param">validate_args=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/torch/distributions/distribution.html#Distribution"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#torch.distributions.distribution.Distribution" title="Permalink to this definition">¶</a></dt>
<dd><p>Bases: <a class="reference external" href="https://docs.python.org/3/library/functions.html#object" title="(in Python v3.8)"><code class="xref py py-class docutils literal notranslate"><span class="pre">object</span></code></a></p>
<p>Distribution is the abstract base class for probability distributions.</p>
<dl class="method">
<dt id="torch.distributions.distribution.Distribution.arg_constraints">
<em class="property">property </em><code class="sig-name descname">arg_constraints</code><a class="headerlink" href="#torch.distributions.distribution.Distribution.arg_constraints" title="Permalink to this definition">¶</a></dt>
<dd><p>Returns a dictionary from argument names to
<a class="reference internal" href="#torch.distributions.constraints.Constraint" title="torch.distributions.constraints.Constraint"><code class="xref py py-class docutils literal notranslate"><span class="pre">Constraint</span></code></a> objects that
should be satisfied by each argument of this distribution. Args that
are not tensors need not appear in this dict.</p>
</dd></dl>
<dl class="method">
<dt id="torch.distributions.distribution.Distribution.batch_shape">
<em class="property">property </em><code class="sig-name descname">batch_shape</code><a class="headerlink" href="#torch.distributions.distribution.Distribution.batch_shape" title="Permalink to this definition">¶</a></dt>
<dd><p>Returns the shape over which parameters are batched.</p>
</dd></dl>
<dl class="method">
<dt id="torch.distributions.distribution.Distribution.cdf">
<code class="sig-name descname">cdf</code><span class="sig-paren">(</span><em class="sig-param">value</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/torch/distributions/distribution.html#Distribution.cdf"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#torch.distributions.distribution.Distribution.cdf" title="Permalink to this definition">¶</a></dt>
<dd><p>Returns the cumulative density/mass function evaluated at
<cite>value</cite>.</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><p><strong>value</strong> (<a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a>) – </p>
</dd>
</dl>
</dd></dl>
<dl class="method">
<dt id="torch.distributions.distribution.Distribution.entropy">
<code class="sig-name descname">entropy</code><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="reference internal" href="_modules/torch/distributions/distribution.html#Distribution.entropy"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#torch.distributions.distribution.Distribution.entropy" title="Permalink to this definition">¶</a></dt>
<dd><p>Returns entropy of distribution, batched over batch_shape.</p>
<dl class="field-list simple">
<dt class="field-odd">Returns</dt>
<dd class="field-odd"><p>Tensor of shape batch_shape.</p>
</dd>
</dl>
</dd></dl>
<dl class="method">
<dt id="torch.distributions.distribution.Distribution.enumerate_support">
<code class="sig-name descname">enumerate_support</code><span class="sig-paren">(</span><em class="sig-param">expand=True</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/torch/distributions/distribution.html#Distribution.enumerate_support"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#torch.distributions.distribution.Distribution.enumerate_support" title="Permalink to this definition">¶</a></dt>
<dd><p>Returns tensor containing all values supported by a discrete
distribution. The result will enumerate over dimension 0, so the shape
of the result will be <cite>(cardinality,) + batch_shape + event_shape</cite>
(where <cite>event_shape = ()</cite> for univariate distributions).</p>
<p>Note that this enumerates over all batched tensors in lock-step
<cite>[[0, 0], [1, 1], …]</cite>. With <cite>expand=False</cite>, enumeration happens
along dim 0, but with the remaining batch dimensions being
singleton dimensions, <cite>[[0], [1], ..</cite>.</p>
<p>To iterate over the full Cartesian product use
<cite>itertools.product(m.enumerate_support())</cite>.</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><p><strong>expand</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#bool" title="(in Python v3.8)"><em>bool</em></a>) – whether to expand the support over the
batch dims to match the distribution’s <cite>batch_shape</cite>.</p>
</dd>
<dt class="field-even">Returns</dt>
<dd class="field-even"><p>Tensor iterating over dimension 0.</p>
</dd>
</dl>
</dd></dl>
<dl class="method">
<dt id="torch.distributions.distribution.Distribution.event_shape">
<em class="property">property </em><code class="sig-name descname">event_shape</code><a class="headerlink" href="#torch.distributions.distribution.Distribution.event_shape" title="Permalink to this definition">¶</a></dt>
<dd><p>Returns the shape of a single sample (without batching).</p>
</dd></dl>
<dl class="method">
<dt id="torch.distributions.distribution.Distribution.expand">
<code class="sig-name descname">expand</code><span class="sig-paren">(</span><em class="sig-param">batch_shape</em>, <em class="sig-param">_instance=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/torch/distributions/distribution.html#Distribution.expand"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#torch.distributions.distribution.Distribution.expand" title="Permalink to this definition">¶</a></dt>
<dd><p>Returns a new distribution instance (or populates an existing instance
provided by a derived class) with batch dimensions expanded to
<cite>batch_shape</cite>. This method calls <a class="reference internal" href="tensors.html#torch.Tensor.expand" title="torch.Tensor.expand"><code class="xref py py-class docutils literal notranslate"><span class="pre">expand</span></code></a> on
the distribution’s parameters. As such, this does not allocate new
memory for the expanded distribution instance. Additionally,
this does not repeat any args checking or parameter broadcasting in
<cite>__init__.py</cite>, when an instance is first created.</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><ul class="simple">
<li><p><strong>batch_shape</strong> (<em>torch.Size</em>) – the desired expanded size.</p></li>
<li><p><strong>_instance</strong> – new instance provided by subclasses that
need to override <cite>.expand</cite>.</p></li>
</ul>
</dd>
<dt class="field-even">Returns</dt>
<dd class="field-even"><p>New distribution instance with batch dimensions expanded to
<cite>batch_size</cite>.</p>
</dd>
</dl>
</dd></dl>
<dl class="method">
<dt id="torch.distributions.distribution.Distribution.icdf">
<code class="sig-name descname">icdf</code><span class="sig-paren">(</span><em class="sig-param">value</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/torch/distributions/distribution.html#Distribution.icdf"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#torch.distributions.distribution.Distribution.icdf" title="Permalink to this definition">¶</a></dt>
<dd><p>Returns the inverse cumulative density/mass function evaluated at
<cite>value</cite>.</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><p><strong>value</strong> (<a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a>) – </p>
</dd>
</dl>
</dd></dl>
<dl class="method">
<dt id="torch.distributions.distribution.Distribution.log_prob">
<code class="sig-name descname">log_prob</code><span class="sig-paren">(</span><em class="sig-param">value</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/torch/distributions/distribution.html#Distribution.log_prob"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#torch.distributions.distribution.Distribution.log_prob" title="Permalink to this definition">¶</a></dt>
<dd><p>Returns the log of the probability density/mass function evaluated at
<cite>value</cite>.</p>
<dl class="field-list simple">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><p><strong>value</strong> (<a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a>) – </p>
</dd>
</dl>
</dd></dl>
<dl class="method">
<dt id="torch.distributions.distribution.Distribution.mean">
<em class="property">property </em><code class="sig-name descname">mean</code><a class="headerlink" href="#torch.distributions.distribution.Distribution.mean" title="Permalink to this definition">¶</a></dt>
<dd><p>Returns the mean of the distribution.</p>
</dd></dl>
<dl class="method">
<dt id="torch.distributions.distribution.Distribution.perplexity">
<code class="sig-name descname">perplexity</code><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="reference internal" href="_modules/torch/distributions/distribution.html#Distribution.perplexity"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#torch.distributions.distribution.Distribution.perplexity" title="Permalink to this definition">¶</a></dt>
<dd><p>Returns perplexity of distribution, batched over batch_shape.</p>
<dl class="field-list simple">
<dt class="field-odd">Returns</dt>
<dd class="field-odd"><p>Tensor of shape batch_shape.</p>
</dd>
</dl>
</dd></dl>
<dl class="method">
<dt id="torch.distributions.distribution.Distribution.rsample">
<code class="sig-name descname">rsample</code><span class="sig-paren">(</span><em class="sig-param">sample_shape=torch.Size([])</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/torch/distributions/distribution.html#Distribution.rsample"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#torch.distributions.distribution.Distribution.rsample" title="Permalink to this definition">¶</a></dt>
<dd><p>Generates a sample_shape shaped reparameterized sample or sample_shape
shaped batch of reparameterized samples if the distribution parameters
are batched.</p>
</dd></dl>
<dl class="method">
<dt id="torch.distributions.distribution.Distribution.sample">
<code class="sig-name descname">sample</code><span class="sig-paren">(</span><em class="sig-param">sample_shape=torch.Size([])</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/torch/distributions/distribution.html#Distribution.sample"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#torch.distributions.distribution.Distribution.sample" title="Permalink to this definition">¶</a></dt>
<dd><p>Generates a sample_shape shaped sample or sample_shape shaped batch of
samples if the distribution parameters are batched.</p>
</dd></dl>
<dl class="method">
<dt id="torch.distributions.distribution.Distribution.sample_n">
<code class="sig-name descname">sample_n</code><span class="sig-paren">(</span><em class="sig-param">n</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/torch/distributions/distribution.html#Distribution.sample_n"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#torch.distributions.distribution.Distribution.sample_n" title="Permalink to this definition">¶</a></dt>
<dd><p>Generates n samples or n batches of samples if the distribution
parameters are batched.</p>
</dd></dl>
<dl class="method">
<dt id="torch.distributions.distribution.Distribution.stddev">
<em class="property">property </em><code class="sig-name descname">stddev</code><a class="headerlink" href="#torch.distributions.distribution.Distribution.stddev" title="Permalink to this definition">¶</a></dt>
<dd><p>Returns the standard deviation of the distribution.</p>
</dd></dl>
<dl class="method">
<dt id="torch.distributions.distribution.Distribution.support">
<em class="property">property </em><code class="sig-name descname">support</code><a class="headerlink" href="#torch.distributions.distribution.Distribution.support" title="Permalink to this definition">¶</a></dt>
<dd><p>Returns a <a class="reference internal" href="#torch.distributions.constraints.Constraint" title="torch.distributions.constraints.Constraint"><code class="xref py py-class docutils literal notranslate"><span class="pre">Constraint</span></code></a> object
representing this distribution’s support.</p>
</dd></dl>
<dl class="method">
<dt id="torch.distributions.distribution.Distribution.variance">
<em class="property">property </em><code class="sig-name descname">variance</code><a class="headerlink" href="#torch.distributions.distribution.Distribution.variance" title="Permalink to this definition">¶</a></dt>
<dd><p>Returns the variance of the distribution.</p>
</dd></dl>
</dd></dl>
</div>
<div class="section" id="exponentialfamily">
<h2><span class="hidden-section">ExponentialFamily</span><a class="headerlink" href="#exponentialfamily" title="Permalink to this headline">¶</a></h2>
<dl class="class">
<dt id="torch.distributions.exp_family.ExponentialFamily">
<em class="property">class </em><code class="sig-prename descclassname">torch.distributions.exp_family.</code><code class="sig-name descname">ExponentialFamily</code><span class="sig-paren">(</span><em class="sig-param">batch_shape=torch.Size([])</em>, <em class="sig-param">event_shape=torch.Size([])</em>, <em class="sig-param">validate_args=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/torch/distributions/exp_family.html#ExponentialFamily"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#torch.distributions.exp_family.ExponentialFamily" title="Permalink to this definition">¶</a></dt>
<dd><p>Bases: <a class="reference internal" href="#torch.distributions.distribution.Distribution" title="torch.distributions.distribution.Distribution"><code class="xref py py-class docutils literal notranslate"><span class="pre">torch.distributions.distribution.Distribution</span></code></a></p>
<p>ExponentialFamily is the abstract base class for probability distributions belonging to an
exponential family, whose probability mass/density function has the form is defined below</p>
<div class="math">
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>p</mi><mi>F</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">;</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>=</mo><mi>exp</mi><mo></mo><mo stretchy="false">(</mo><mo stretchy="false">⟨</mo><mi>t</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo separator="true">,</mo><mi>θ</mi><mo stretchy="false">⟩</mo><mo>−</mo><mi>F</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>+</mo><mi>k</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p_{F}(x; \theta) = \exp(\langle t(x), \theta\rangle - F(\theta) + k(x))</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathdefault">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em;">F</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">exp</span><span class="mopen">(</span><span class="mopen">⟨</span><span class="mord mathdefault">t</span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span><span class="mclose">⟩</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.03148em;">k</span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mclose">)</span><span class="mclose">)</span></span></span></span></span>
</div><p>where <span class="math"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>θ</mi></mrow><annotation encoding="application/x-tex">\theta</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span></span></span></span>
</span> denotes the natural parameters, <span class="math"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>t</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">t(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">t</span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mclose">)</span></span></span></span>
</span> denotes the sufficient statistic,
<span class="math"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(\theta)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span><span class="mclose">)</span></span></span></span>
</span> is the log normalizer function for a given family and <span class="math"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">k(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.03148em;">k</span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mclose">)</span></span></span></span>
</span> is the carrier
measure.</p>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>This class is an intermediary between the <cite>Distribution</cite> class and distributions which belong
to an exponential family mainly to check the correctness of the <cite>.entropy()</cite> and analytic KL
divergence methods. We use this class to compute the entropy and KL divergence using the AD
framework and Bregman divergences (courtesy of: Frank Nielsen and Richard Nock, Entropies and
Cross-entropies of Exponential Families).</p>
</div>
<dl class="method">
<dt id="torch.distributions.exp_family.ExponentialFamily.entropy">
<code class="sig-name descname">entropy</code><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="reference internal" href="_modules/torch/distributions/exp_family.html#ExponentialFamily.entropy"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#torch.distributions.exp_family.ExponentialFamily.entropy" title="Permalink to this definition">¶</a></dt>
<dd><p>Method to compute the entropy using Bregman divergence of the log normalizer.</p>
</dd></dl>
</dd></dl>
</div>
<div class="section" id="bernoulli">
<h2><span class="hidden-section">Bernoulli</span><a class="headerlink" href="#bernoulli" title="Permalink to this headline">¶</a></h2>
<dl class="class">
<dt id="torch.distributions.bernoulli.Bernoulli">
<em class="property">class </em><code class="sig-prename descclassname">torch.distributions.bernoulli.</code><code class="sig-name descname">Bernoulli</code><span class="sig-paren">(</span><em class="sig-param">probs=None</em>, <em class="sig-param">logits=None</em>, <em class="sig-param">validate_args=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/torch/distributions/bernoulli.html#Bernoulli"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#torch.distributions.bernoulli.Bernoulli" title="Permalink to this definition">¶</a></dt>
<dd><p>Bases: <a class="reference internal" href="#torch.distributions.exp_family.ExponentialFamily" title="torch.distributions.exp_family.ExponentialFamily"><code class="xref py py-class docutils literal notranslate"><span class="pre">torch.distributions.exp_family.ExponentialFamily</span></code></a></p>
<p>Creates a Bernoulli distribution parameterized by <a class="reference internal" href="#torch.distributions.bernoulli.Bernoulli.probs" title="torch.distributions.bernoulli.Bernoulli.probs"><code class="xref py py-attr docutils literal notranslate"><span class="pre">probs</span></code></a>
or <a class="reference internal" href="#torch.distributions.bernoulli.Bernoulli.logits" title="torch.distributions.bernoulli.Bernoulli.logits"><code class="xref py py-attr docutils literal notranslate"><span class="pre">logits</span></code></a> (but not both).</p>
<p>Samples are binary (0 or 1). They take the value <cite>1</cite> with probability <cite>p</cite>
and <cite>0</cite> with probability <cite>1 - p</cite>.</p>
<p>Example:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">m</span> <span class="o">=</span> <span class="n">Bernoulli</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([</span><span class="mf">0.3</span><span class="p">]))</span>
<span class="gp">>>> </span><span class="n">m</span><span class="o">.</span><span class="n">sample</span><span class="p">()</span> <span class="c1"># 30% chance 1; 70% chance 0</span>
<span class="go">tensor([ 0.])</span>
</pre></div>
</div>
<dl class="field-list simple">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><ul class="simple">
<li><p><strong>probs</strong> (<em>Number</em><em>, </em><a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a>) – the probability of sampling <cite>1</cite></p></li>
<li><p><strong>logits</strong> (<em>Number</em><em>, </em><a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a>) – the log-odds of sampling <cite>1</cite></p></li>
</ul>
</dd>
</dl>
<dl class="attribute">
<dt id="torch.distributions.bernoulli.Bernoulli.arg_constraints">
<code class="sig-name descname">arg_constraints</code><em class="property"> = {'logits': Real(), 'probs': Interval(lower_bound=0.0, upper_bound=1.0)}</em><a class="headerlink" href="#torch.distributions.bernoulli.Bernoulli.arg_constraints" title="Permalink to this definition">¶</a></dt>
<dd></dd></dl>
<dl class="method">
<dt id="torch.distributions.bernoulli.Bernoulli.entropy">
<code class="sig-name descname">entropy</code><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="reference internal" href="_modules/torch/distributions/bernoulli.html#Bernoulli.entropy"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#torch.distributions.bernoulli.Bernoulli.entropy" title="Permalink to this definition">¶</a></dt>
<dd></dd></dl>
<dl class="method">
<dt id="torch.distributions.bernoulli.Bernoulli.enumerate_support">
<code class="sig-name descname">enumerate_support</code><span class="sig-paren">(</span><em class="sig-param">expand=True</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/torch/distributions/bernoulli.html#Bernoulli.enumerate_support"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#torch.distributions.bernoulli.Bernoulli.enumerate_support" title="Permalink to this definition">¶</a></dt>
<dd></dd></dl>
<dl class="method">
<dt id="torch.distributions.bernoulli.Bernoulli.expand">
<code class="sig-name descname">expand</code><span class="sig-paren">(</span><em class="sig-param">batch_shape</em>, <em class="sig-param">_instance=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/torch/distributions/bernoulli.html#Bernoulli.expand"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#torch.distributions.bernoulli.Bernoulli.expand" title="Permalink to this definition">¶</a></dt>
<dd></dd></dl>
<dl class="attribute">
<dt id="torch.distributions.bernoulli.Bernoulli.has_enumerate_support">
<code class="sig-name descname">has_enumerate_support</code><em class="property"> = True</em><a class="headerlink" href="#torch.distributions.bernoulli.Bernoulli.has_enumerate_support" title="Permalink to this definition">¶</a></dt>
<dd></dd></dl>
<dl class="method">
<dt id="torch.distributions.bernoulli.Bernoulli.log_prob">
<code class="sig-name descname">log_prob</code><span class="sig-paren">(</span><em class="sig-param">value</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/torch/distributions/bernoulli.html#Bernoulli.log_prob"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#torch.distributions.bernoulli.Bernoulli.log_prob" title="Permalink to this definition">¶</a></dt>
<dd></dd></dl>
<dl class="attribute">
<dt id="torch.distributions.bernoulli.Bernoulli.logits">
<code class="sig-name descname">logits</code><a class="reference internal" href="_modules/torch/distributions/bernoulli.html#Bernoulli.logits"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#torch.distributions.bernoulli.Bernoulli.logits" title="Permalink to this definition">¶</a></dt>
<dd></dd></dl>
<dl class="method">
<dt id="torch.distributions.bernoulli.Bernoulli.mean">
<em class="property">property </em><code class="sig-name descname">mean</code><a class="headerlink" href="#torch.distributions.bernoulli.Bernoulli.mean" title="Permalink to this definition">¶</a></dt>
<dd></dd></dl>
<dl class="method">
<dt id="torch.distributions.bernoulli.Bernoulli.param_shape">
<em class="property">property </em><code class="sig-name descname">param_shape</code><a class="headerlink" href="#torch.distributions.bernoulli.Bernoulli.param_shape" title="Permalink to this definition">¶</a></dt>
<dd></dd></dl>
<dl class="attribute">
<dt id="torch.distributions.bernoulli.Bernoulli.probs">
<code class="sig-name descname">probs</code><a class="reference internal" href="_modules/torch/distributions/bernoulli.html#Bernoulli.probs"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#torch.distributions.bernoulli.Bernoulli.probs" title="Permalink to this definition">¶</a></dt>
<dd></dd></dl>
<dl class="method">
<dt id="torch.distributions.bernoulli.Bernoulli.sample">
<code class="sig-name descname">sample</code><span class="sig-paren">(</span><em class="sig-param">sample_shape=torch.Size([])</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/torch/distributions/bernoulli.html#Bernoulli.sample"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#torch.distributions.bernoulli.Bernoulli.sample" title="Permalink to this definition">¶</a></dt>
<dd></dd></dl>
<dl class="attribute">
<dt id="torch.distributions.bernoulli.Bernoulli.support">
<code class="sig-name descname">support</code><em class="property"> = Boolean()</em><a class="headerlink" href="#torch.distributions.bernoulli.Bernoulli.support" title="Permalink to this definition">¶</a></dt>
<dd></dd></dl>
<dl class="method">
<dt id="torch.distributions.bernoulli.Bernoulli.variance">
<em class="property">property </em><code class="sig-name descname">variance</code><a class="headerlink" href="#torch.distributions.bernoulli.Bernoulli.variance" title="Permalink to this definition">¶</a></dt>
<dd></dd></dl>
</dd></dl>
</div>
<div class="section" id="beta">
<h2><span class="hidden-section">Beta</span><a class="headerlink" href="#beta" title="Permalink to this headline">¶</a></h2>
<dl class="class">
<dt id="torch.distributions.beta.Beta">
<em class="property">class </em><code class="sig-prename descclassname">torch.distributions.beta.</code><code class="sig-name descname">Beta</code><span class="sig-paren">(</span><em class="sig-param">concentration1</em>, <em class="sig-param">concentration0</em>, <em class="sig-param">validate_args=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/torch/distributions/beta.html#Beta"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#torch.distributions.beta.Beta" title="Permalink to this definition">¶</a></dt>
<dd><p>Bases: <a class="reference internal" href="#torch.distributions.exp_family.ExponentialFamily" title="torch.distributions.exp_family.ExponentialFamily"><code class="xref py py-class docutils literal notranslate"><span class="pre">torch.distributions.exp_family.ExponentialFamily</span></code></a></p>
<p>Beta distribution parameterized by <a class="reference internal" href="#torch.distributions.beta.Beta.concentration1" title="torch.distributions.beta.Beta.concentration1"><code class="xref py py-attr docutils literal notranslate"><span class="pre">concentration1</span></code></a> and <a class="reference internal" href="#torch.distributions.beta.Beta.concentration0" title="torch.distributions.beta.Beta.concentration0"><code class="xref py py-attr docutils literal notranslate"><span class="pre">concentration0</span></code></a>.</p>
<p>Example:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">m</span> <span class="o">=</span> <span class="n">Beta</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([</span><span class="mf">0.5</span><span class="p">]),</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([</span><span class="mf">0.5</span><span class="p">]))</span>
<span class="gp">>>> </span><span class="n">m</span><span class="o">.</span><span class="n">sample</span><span class="p">()</span> <span class="c1"># Beta distributed with concentration concentration1 and concentration0</span>
<span class="go">tensor([ 0.1046])</span>
</pre></div>
</div>
<dl class="field-list simple">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><ul class="simple">
<li><p><strong>concentration1</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#float" title="(in Python v3.8)"><em>float</em></a><em> or </em><a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a>) – 1st concentration parameter of the distribution
(often referred to as alpha)</p></li>
<li><p><strong>concentration0</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#float" title="(in Python v3.8)"><em>float</em></a><em> or </em><a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a>) – 2nd concentration parameter of the distribution
(often referred to as beta)</p></li>
</ul>
</dd>
</dl>
<dl class="attribute">
<dt id="torch.distributions.beta.Beta.arg_constraints">
<code class="sig-name descname">arg_constraints</code><em class="property"> = {'concentration0': GreaterThan(lower_bound=0.0), 'concentration1': GreaterThan(lower_bound=0.0)}</em><a class="headerlink" href="#torch.distributions.beta.Beta.arg_constraints" title="Permalink to this definition">¶</a></dt>
<dd></dd></dl>
<dl class="method">
<dt id="torch.distributions.beta.Beta.concentration0">
<em class="property">property </em><code class="sig-name descname">concentration0</code><a class="headerlink" href="#torch.distributions.beta.Beta.concentration0" title="Permalink to this definition">¶</a></dt>
<dd></dd></dl>
<dl class="method">
<dt id="torch.distributions.beta.Beta.concentration1">
<em class="property">property </em><code class="sig-name descname">concentration1</code><a class="headerlink" href="#torch.distributions.beta.Beta.concentration1" title="Permalink to this definition">¶</a></dt>
<dd></dd></dl>
<dl class="method">
<dt id="torch.distributions.beta.Beta.entropy">
<code class="sig-name descname">entropy</code><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="reference internal" href="_modules/torch/distributions/beta.html#Beta.entropy"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#torch.distributions.beta.Beta.entropy" title="Permalink to this definition">¶</a></dt>
<dd></dd></dl>
<dl class="method">
<dt id="torch.distributions.beta.Beta.expand">
<code class="sig-name descname">expand</code><span class="sig-paren">(</span><em class="sig-param">batch_shape</em>, <em class="sig-param">_instance=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/torch/distributions/beta.html#Beta.expand"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#torch.distributions.beta.Beta.expand" title="Permalink to this definition">¶</a></dt>
<dd></dd></dl>
<dl class="attribute">
<dt id="torch.distributions.beta.Beta.has_rsample">
<code class="sig-name descname">has_rsample</code><em class="property"> = True</em><a class="headerlink" href="#torch.distributions.beta.Beta.has_rsample" title="Permalink to this definition">¶</a></dt>
<dd></dd></dl>
<dl class="method">
<dt id="torch.distributions.beta.Beta.log_prob">
<code class="sig-name descname">log_prob</code><span class="sig-paren">(</span><em class="sig-param">value</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/torch/distributions/beta.html#Beta.log_prob"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#torch.distributions.beta.Beta.log_prob" title="Permalink to this definition">¶</a></dt>
<dd></dd></dl>
<dl class="method">
<dt id="torch.distributions.beta.Beta.mean">
<em class="property">property </em><code class="sig-name descname">mean</code><a class="headerlink" href="#torch.distributions.beta.Beta.mean" title="Permalink to this definition">¶</a></dt>
<dd></dd></dl>
<dl class="method">
<dt id="torch.distributions.beta.Beta.rsample">
<code class="sig-name descname">rsample</code><span class="sig-paren">(</span><em class="sig-param">sample_shape=()</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/torch/distributions/beta.html#Beta.rsample"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#torch.distributions.beta.Beta.rsample" title="Permalink to this definition">¶</a></dt>
<dd></dd></dl>
<dl class="attribute">
<dt id="torch.distributions.beta.Beta.support">
<code class="sig-name descname">support</code><em class="property"> = Interval(lower_bound=0.0, upper_bound=1.0)</em><a class="headerlink" href="#torch.distributions.beta.Beta.support" title="Permalink to this definition">¶</a></dt>
<dd></dd></dl>
<dl class="method">
<dt id="torch.distributions.beta.Beta.variance">
<em class="property">property </em><code class="sig-name descname">variance</code><a class="headerlink" href="#torch.distributions.beta.Beta.variance" title="Permalink to this definition">¶</a></dt>
<dd></dd></dl>
</dd></dl>
</div>
<div class="section" id="binomial">
<h2><span class="hidden-section">Binomial</span><a class="headerlink" href="#binomial" title="Permalink to this headline">¶</a></h2>
<dl class="class">
<dt id="torch.distributions.binomial.Binomial">
<em class="property">class </em><code class="sig-prename descclassname">torch.distributions.binomial.</code><code class="sig-name descname">Binomial</code><span class="sig-paren">(</span><em class="sig-param">total_count=1</em>, <em class="sig-param">probs=None</em>, <em class="sig-param">logits=None</em>, <em class="sig-param">validate_args=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/torch/distributions/binomial.html#Binomial"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#torch.distributions.binomial.Binomial" title="Permalink to this definition">¶</a></dt>
<dd><p>Bases: <a class="reference internal" href="#torch.distributions.distribution.Distribution" title="torch.distributions.distribution.Distribution"><code class="xref py py-class docutils literal notranslate"><span class="pre">torch.distributions.distribution.Distribution</span></code></a></p>
<p>Creates a Binomial distribution parameterized by <code class="xref py py-attr docutils literal notranslate"><span class="pre">total_count</span></code> and
either <a class="reference internal" href="#torch.distributions.binomial.Binomial.probs" title="torch.distributions.binomial.Binomial.probs"><code class="xref py py-attr docutils literal notranslate"><span class="pre">probs</span></code></a> or <a class="reference internal" href="#torch.distributions.binomial.Binomial.logits" title="torch.distributions.binomial.Binomial.logits"><code class="xref py py-attr docutils literal notranslate"><span class="pre">logits</span></code></a> (but not both). <code class="xref py py-attr docutils literal notranslate"><span class="pre">total_count</span></code> must be
broadcastable with <a class="reference internal" href="#torch.distributions.binomial.Binomial.probs" title="torch.distributions.binomial.Binomial.probs"><code class="xref py py-attr docutils literal notranslate"><span class="pre">probs</span></code></a>/<a class="reference internal" href="#torch.distributions.binomial.Binomial.logits" title="torch.distributions.binomial.Binomial.logits"><code class="xref py py-attr docutils literal notranslate"><span class="pre">logits</span></code></a>.</p>
<p>Example:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">m</span> <span class="o">=</span> <span class="n">Binomial</span><span class="p">(</span><span class="mi">100</span><span class="p">,</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([</span><span class="mi">0</span> <span class="p">,</span> <span class="o">.</span><span class="mi">2</span><span class="p">,</span> <span class="o">.</span><span class="mi">8</span><span class="p">,</span> <span class="mi">1</span><span class="p">]))</span>
<span class="gp">>>> </span><span class="n">x</span> <span class="o">=</span> <span class="n">m</span><span class="o">.</span><span class="n">sample</span><span class="p">()</span>
<span class="go">tensor([ 0., 22., 71., 100.])</span>
<span class="gp">>>> </span><span class="n">m</span> <span class="o">=</span> <span class="n">Binomial</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([[</span><span class="mf">5.</span><span class="p">],</span> <span class="p">[</span><span class="mf">10.</span><span class="p">]]),</span> <span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([</span><span class="mf">0.5</span><span class="p">,</span> <span class="mf">0.8</span><span class="p">]))</span>
<span class="gp">>>> </span><span class="n">x</span> <span class="o">=</span> <span class="n">m</span><span class="o">.</span><span class="n">sample</span><span class="p">()</span>
<span class="go">tensor([[ 4., 5.],</span>
<span class="go"> [ 7., 6.]])</span>
</pre></div>
</div>
<dl class="field-list simple">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><ul class="simple">
<li><p><strong>total_count</strong> (<a class="reference external" href="https://docs.python.org/3/library/functions.html#int" title="(in Python v3.8)"><em>int</em></a><em> or </em><a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a>) – number of Bernoulli trials</p></li>
<li><p><strong>probs</strong> (<a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a>) – Event probabilities</p></li>
<li><p><strong>logits</strong> (<a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a>) – Event log-odds</p></li>
</ul>
</dd>
</dl>
<dl class="attribute">
<dt id="torch.distributions.binomial.Binomial.arg_constraints">
<code class="sig-name descname">arg_constraints</code><em class="property"> = {'logits': Real(), 'probs': Interval(lower_bound=0.0, upper_bound=1.0), 'total_count': IntegerGreaterThan(lower_bound=0)}</em><a class="headerlink" href="#torch.distributions.binomial.Binomial.arg_constraints" title="Permalink to this definition">¶</a></dt>
<dd></dd></dl>
<dl class="method">
<dt id="torch.distributions.binomial.Binomial.enumerate_support">
<code class="sig-name descname">enumerate_support</code><span class="sig-paren">(</span><em class="sig-param">expand=True</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/torch/distributions/binomial.html#Binomial.enumerate_support"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#torch.distributions.binomial.Binomial.enumerate_support" title="Permalink to this definition">¶</a></dt>
<dd></dd></dl>
<dl class="method">
<dt id="torch.distributions.binomial.Binomial.expand">
<code class="sig-name descname">expand</code><span class="sig-paren">(</span><em class="sig-param">batch_shape</em>, <em class="sig-param">_instance=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/torch/distributions/binomial.html#Binomial.expand"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#torch.distributions.binomial.Binomial.expand" title="Permalink to this definition">¶</a></dt>
<dd></dd></dl>
<dl class="attribute">
<dt id="torch.distributions.binomial.Binomial.has_enumerate_support">
<code class="sig-name descname">has_enumerate_support</code><em class="property"> = True</em><a class="headerlink" href="#torch.distributions.binomial.Binomial.has_enumerate_support" title="Permalink to this definition">¶</a></dt>
<dd></dd></dl>
<dl class="method">
<dt id="torch.distributions.binomial.Binomial.log_prob">
<code class="sig-name descname">log_prob</code><span class="sig-paren">(</span><em class="sig-param">value</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/torch/distributions/binomial.html#Binomial.log_prob"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#torch.distributions.binomial.Binomial.log_prob" title="Permalink to this definition">¶</a></dt>
<dd></dd></dl>
<dl class="attribute">
<dt id="torch.distributions.binomial.Binomial.logits">
<code class="sig-name descname">logits</code><a class="reference internal" href="_modules/torch/distributions/binomial.html#Binomial.logits"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#torch.distributions.binomial.Binomial.logits" title="Permalink to this definition">¶</a></dt>
<dd></dd></dl>
<dl class="method">
<dt id="torch.distributions.binomial.Binomial.mean">
<em class="property">property </em><code class="sig-name descname">mean</code><a class="headerlink" href="#torch.distributions.binomial.Binomial.mean" title="Permalink to this definition">¶</a></dt>
<dd></dd></dl>
<dl class="method">
<dt id="torch.distributions.binomial.Binomial.param_shape">
<em class="property">property </em><code class="sig-name descname">param_shape</code><a class="headerlink" href="#torch.distributions.binomial.Binomial.param_shape" title="Permalink to this definition">¶</a></dt>
<dd></dd></dl>
<dl class="attribute">
<dt id="torch.distributions.binomial.Binomial.probs">
<code class="sig-name descname">probs</code><a class="reference internal" href="_modules/torch/distributions/binomial.html#Binomial.probs"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#torch.distributions.binomial.Binomial.probs" title="Permalink to this definition">¶</a></dt>
<dd></dd></dl>
<dl class="method">
<dt id="torch.distributions.binomial.Binomial.sample">
<code class="sig-name descname">sample</code><span class="sig-paren">(</span><em class="sig-param">sample_shape=torch.Size([])</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/torch/distributions/binomial.html#Binomial.sample"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#torch.distributions.binomial.Binomial.sample" title="Permalink to this definition">¶</a></dt>
<dd></dd></dl>
<dl class="method">
<dt id="torch.distributions.binomial.Binomial.support">
<em class="property">property </em><code class="sig-name descname">support</code><a class="headerlink" href="#torch.distributions.binomial.Binomial.support" title="Permalink to this definition">¶</a></dt>
<dd></dd></dl>
<dl class="method">
<dt id="torch.distributions.binomial.Binomial.variance">
<em class="property">property </em><code class="sig-name descname">variance</code><a class="headerlink" href="#torch.distributions.binomial.Binomial.variance" title="Permalink to this definition">¶</a></dt>
<dd></dd></dl>
</dd></dl>
</div>
<div class="section" id="categorical">
<h2><span class="hidden-section">Categorical</span><a class="headerlink" href="#categorical" title="Permalink to this headline">¶</a></h2>
<dl class="class">
<dt id="torch.distributions.categorical.Categorical">
<em class="property">class </em><code class="sig-prename descclassname">torch.distributions.categorical.</code><code class="sig-name descname">Categorical</code><span class="sig-paren">(</span><em class="sig-param">probs=None</em>, <em class="sig-param">logits=None</em>, <em class="sig-param">validate_args=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/torch/distributions/categorical.html#Categorical"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#torch.distributions.categorical.Categorical" title="Permalink to this definition">¶</a></dt>
<dd><p>Bases: <a class="reference internal" href="#torch.distributions.distribution.Distribution" title="torch.distributions.distribution.Distribution"><code class="xref py py-class docutils literal notranslate"><span class="pre">torch.distributions.distribution.Distribution</span></code></a></p>
<p>Creates a categorical distribution parameterized by either <a class="reference internal" href="#torch.distributions.categorical.Categorical.probs" title="torch.distributions.categorical.Categorical.probs"><code class="xref py py-attr docutils literal notranslate"><span class="pre">probs</span></code></a> or
<a class="reference internal" href="#torch.distributions.categorical.Categorical.logits" title="torch.distributions.categorical.Categorical.logits"><code class="xref py py-attr docutils literal notranslate"><span class="pre">logits</span></code></a> (but not both).</p>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>It is equivalent to the distribution that <a class="reference internal" href="generated/torch.multinomial.html#torch.multinomial" title="torch.multinomial"><code class="xref py py-func docutils literal notranslate"><span class="pre">torch.multinomial()</span></code></a>
samples from.</p>
</div>
<p>Samples are integers from <span class="math"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">{</mo><mn>0</mn><mo separator="true">,</mo><mo>…</mo><mo separator="true">,</mo><mi>K</mi><mo>−</mo><mn>1</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{0, \ldots, K-1\}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{</span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.07153em;">K</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">}</span></span></span></span>
</span> where <cite>K</cite> is <code class="docutils literal notranslate"><span class="pre">probs.size(-1)</span></code>.</p>
<p>If <a class="reference internal" href="#torch.distributions.categorical.Categorical.probs" title="torch.distributions.categorical.Categorical.probs"><code class="xref py py-attr docutils literal notranslate"><span class="pre">probs</span></code></a> is 1D with length-<cite>K</cite>, each element is the relative
probability of sampling the class at that index.</p>
<p>If <a class="reference internal" href="#torch.distributions.categorical.Categorical.probs" title="torch.distributions.categorical.Categorical.probs"><code class="xref py py-attr docutils literal notranslate"><span class="pre">probs</span></code></a> is 2D, it is treated as a batch of relative probability
vectors.</p>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p><a class="reference internal" href="#torch.distributions.categorical.Categorical.probs" title="torch.distributions.categorical.Categorical.probs"><code class="xref py py-attr docutils literal notranslate"><span class="pre">probs</span></code></a> must be non-negative, finite and have a non-zero sum,
and it will be normalized to sum to 1.</p>
</div>
<p>See also: <a class="reference internal" href="generated/torch.multinomial.html#torch.multinomial" title="torch.multinomial"><code class="xref py py-func docutils literal notranslate"><span class="pre">torch.multinomial()</span></code></a></p>
<p>Example:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="n">m</span> <span class="o">=</span> <span class="n">Categorical</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">tensor</span><span class="p">([</span> <span class="mf">0.25</span><span class="p">,</span> <span class="mf">0.25</span><span class="p">,</span> <span class="mf">0.25</span><span class="p">,</span> <span class="mf">0.25</span> <span class="p">]))</span>
<span class="gp">>>> </span><span class="n">m</span><span class="o">.</span><span class="n">sample</span><span class="p">()</span> <span class="c1"># equal probability of 0, 1, 2, 3</span>
<span class="go">tensor(3)</span>
</pre></div>
</div>
<dl class="field-list simple">
<dt class="field-odd">Parameters</dt>
<dd class="field-odd"><ul class="simple">
<li><p><strong>probs</strong> (<a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a>) – event probabilities</p></li>
<li><p><strong>logits</strong> (<a class="reference internal" href="tensors.html#torch.Tensor" title="torch.Tensor"><em>Tensor</em></a>) – event log-odds</p></li>
</ul>
</dd>
</dl>
<dl class="attribute">
<dt id="torch.distributions.categorical.Categorical.arg_constraints">
<code class="sig-name descname">arg_constraints</code><em class="property"> = {'logits': Real(), 'probs': Simplex()}</em><a class="headerlink" href="#torch.distributions.categorical.Categorical.arg_constraints" title="Permalink to this definition">¶</a></dt>
<dd></dd></dl>
<dl class="method">
<dt id="torch.distributions.categorical.Categorical.entropy">
<code class="sig-name descname">entropy</code><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="reference internal" href="_modules/torch/distributions/categorical.html#Categorical.entropy"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#torch.distributions.categorical.Categorical.entropy" title="Permalink to this definition">¶</a></dt>
<dd></dd></dl>
<dl class="method">
<dt id="torch.distributions.categorical.Categorical.enumerate_support">
<code class="sig-name descname">enumerate_support</code><span class="sig-paren">(</span><em class="sig-param">expand=True</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/torch/distributions/categorical.html#Categorical.enumerate_support"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#torch.distributions.categorical.Categorical.enumerate_support" title="Permalink to this definition">¶</a></dt>
<dd></dd></dl>
<dl class="method">
<dt id="torch.distributions.categorical.Categorical.expand">
<code class="sig-name descname">expand</code><span class="sig-paren">(</span><em class="sig-param">batch_shape</em>, <em class="sig-param">_instance=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/torch/distributions/categorical.html#Categorical.expand"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#torch.distributions.categorical.Categorical.expand" title="Permalink to this definition">¶</a></dt>
<dd></dd></dl>
<dl class="attribute">
<dt id="torch.distributions.categorical.Categorical.has_enumerate_support">
<code class="sig-name descname">has_enumerate_support</code><em class="property"> = True</em><a class="headerlink" href="#torch.distributions.categorical.Categorical.has_enumerate_support" title="Permalink to this definition">¶</a></dt>
<dd></dd></dl>
<dl class="method">
<dt id="torch.distributions.categorical.Categorical.log_prob">
<code class="sig-name descname">log_prob</code><span class="sig-paren">(</span><em class="sig-param">value</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/torch/distributions/categorical.html#Categorical.log_prob"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#torch.distributions.categorical.Categorical.log_prob" title="Permalink to this definition">¶</a></dt>
<dd></dd></dl>
<dl class="attribute">
<dt id="torch.distributions.categorical.Categorical.logits">
<code class="sig-name descname">logits</code><a class="reference internal" href="_modules/torch/distributions/categorical.html#Categorical.logits"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#torch.distributions.categorical.Categorical.logits" title="Permalink to this definition">¶</a></dt>
<dd></dd></dl>
<dl class="method">
<dt id="torch.distributions.categorical.Categorical.mean">
<em class="property">property </em><code class="sig-name descname">mean</code><a class="headerlink" href="#torch.distributions.categorical.Categorical.mean" title="Permalink to this definition">¶</a></dt>
<dd></dd></dl>
<dl class="method">
<dt id="torch.distributions.categorical.Categorical.param_shape">
<em class="property">property </em><code class="sig-name descname">param_shape</code><a class="headerlink" href="#torch.distributions.categorical.Categorical.param_shape" title="Permalink to this definition">¶</a></dt>
<dd></dd></dl>
<dl class="attribute">
<dt id="torch.distributions.categorical.Categorical.probs">
<code class="sig-name descname">probs</code><a class="reference internal" href="_modules/torch/distributions/categorical.html#Categorical.probs"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#torch.distributions.categorical.Categorical.probs" title="Permalink to this definition">¶</a></dt>
<dd></dd></dl>
<dl class="method">
<dt id="torch.distributions.categorical.Categorical.sample">
<code class="sig-name descname">sample</code><span class="sig-paren">(</span><em class="sig-param">sample_shape=torch.Size([])</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/torch/distributions/categorical.html#Categorical.sample"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#torch.distributions.categorical.Categorical.sample" title="Permalink to this definition">¶</a></dt>