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<li class="toctree-l1"><a class="reference internal" href="../../../notes/amp_examples.html">CUDA Automatic Mixed Precision examples</a></li>
<li class="toctree-l1"><a class="reference internal" href="../../../notes/autograd.html">Autograd mechanics</a></li>
<li class="toctree-l1"><a class="reference internal" href="../../../notes/broadcasting.html">Broadcasting semantics</a></li>
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<h1>Source code for torch.distributions.kl</h1><div class="highlight"><pre>
<span></span><span class="kn">import</span> <span class="nn">math</span>
<span class="kn">import</span> <span class="nn">warnings</span>
<span class="kn">from</span> <span class="nn">functools</span> <span class="kn">import</span> <span class="n">total_ordering</span>
<span class="kn">from</span> <span class="nn">typing</span> <span class="kn">import</span> <span class="n">Type</span><span class="p">,</span> <span class="n">Dict</span><span class="p">,</span> <span class="n">Callable</span><span class="p">,</span> <span class="n">Tuple</span>
<span class="kn">import</span> <span class="nn">torch</span>
<span class="kn">from</span> <span class="nn">torch</span> <span class="kn">import</span> <span class="n">inf</span>
<span class="kn">from</span> <span class="nn">.bernoulli</span> <span class="kn">import</span> <span class="n">Bernoulli</span>
<span class="kn">from</span> <span class="nn">.beta</span> <span class="kn">import</span> <span class="n">Beta</span>
<span class="kn">from</span> <span class="nn">.binomial</span> <span class="kn">import</span> <span class="n">Binomial</span>
<span class="kn">from</span> <span class="nn">.categorical</span> <span class="kn">import</span> <span class="n">Categorical</span>
<span class="kn">from</span> <span class="nn">.cauchy</span> <span class="kn">import</span> <span class="n">Cauchy</span>
<span class="kn">from</span> <span class="nn">.continuous_bernoulli</span> <span class="kn">import</span> <span class="n">ContinuousBernoulli</span>
<span class="kn">from</span> <span class="nn">.dirichlet</span> <span class="kn">import</span> <span class="n">Dirichlet</span>
<span class="kn">from</span> <span class="nn">.distribution</span> <span class="kn">import</span> <span class="n">Distribution</span>
<span class="kn">from</span> <span class="nn">.exponential</span> <span class="kn">import</span> <span class="n">Exponential</span>
<span class="kn">from</span> <span class="nn">.exp_family</span> <span class="kn">import</span> <span class="n">ExponentialFamily</span>
<span class="kn">from</span> <span class="nn">.gamma</span> <span class="kn">import</span> <span class="n">Gamma</span>
<span class="kn">from</span> <span class="nn">.geometric</span> <span class="kn">import</span> <span class="n">Geometric</span>
<span class="kn">from</span> <span class="nn">.gumbel</span> <span class="kn">import</span> <span class="n">Gumbel</span>
<span class="kn">from</span> <span class="nn">.half_normal</span> <span class="kn">import</span> <span class="n">HalfNormal</span>
<span class="kn">from</span> <span class="nn">.independent</span> <span class="kn">import</span> <span class="n">Independent</span>
<span class="kn">from</span> <span class="nn">.laplace</span> <span class="kn">import</span> <span class="n">Laplace</span>
<span class="kn">from</span> <span class="nn">.lowrank_multivariate_normal</span> <span class="kn">import</span> <span class="p">(</span><span class="n">LowRankMultivariateNormal</span><span class="p">,</span> <span class="n">_batch_lowrank_logdet</span><span class="p">,</span>
<span class="n">_batch_lowrank_mahalanobis</span><span class="p">)</span>
<span class="kn">from</span> <span class="nn">.multivariate_normal</span> <span class="kn">import</span> <span class="p">(</span><span class="n">MultivariateNormal</span><span class="p">,</span> <span class="n">_batch_mahalanobis</span><span class="p">)</span>
<span class="kn">from</span> <span class="nn">.normal</span> <span class="kn">import</span> <span class="n">Normal</span>
<span class="kn">from</span> <span class="nn">.one_hot_categorical</span> <span class="kn">import</span> <span class="n">OneHotCategorical</span>
<span class="kn">from</span> <span class="nn">.pareto</span> <span class="kn">import</span> <span class="n">Pareto</span>
<span class="kn">from</span> <span class="nn">.poisson</span> <span class="kn">import</span> <span class="n">Poisson</span>
<span class="kn">from</span> <span class="nn">.transformed_distribution</span> <span class="kn">import</span> <span class="n">TransformedDistribution</span>
<span class="kn">from</span> <span class="nn">.uniform</span> <span class="kn">import</span> <span class="n">Uniform</span>
<span class="kn">from</span> <span class="nn">.utils</span> <span class="kn">import</span> <span class="n">_sum_rightmost</span><span class="p">,</span> <span class="n">euler_constant</span> <span class="k">as</span> <span class="n">_euler_gamma</span>
<span class="n">_KL_REGISTRY</span> <span class="o">=</span> <span class="p">{}</span> <span class="c1"># Source of truth mapping a few general (type, type) pairs to functions.</span>
<span class="n">_KL_MEMOIZE</span><span class="p">:</span> <span class="n">Dict</span><span class="p">[</span><span class="n">Tuple</span><span class="p">[</span><span class="n">Type</span><span class="p">,</span> <span class="n">Type</span><span class="p">],</span> <span class="n">Callable</span><span class="p">]</span> <span class="o">=</span> <span class="p">{}</span> <span class="c1"># Memoized version mapping many specific (type, type) pairs to functions.</span>
<span class="n">__all__</span> <span class="o">=</span> <span class="p">[</span><span class="s2">"register_kl"</span><span class="p">,</span> <span class="s2">"kl_divergence"</span><span class="p">]</span>
<div class="viewcode-block" id="register_kl"><a class="viewcode-back" href="../../../distributions.html#torch.distributions.kl.register_kl">[docs]</a><span class="k">def</span> <span class="nf">register_kl</span><span class="p">(</span><span class="n">type_p</span><span class="p">,</span> <span class="n">type_q</span><span class="p">):</span>
<span class="sd">"""</span>
<span class="sd"> Decorator to register a pairwise function with :meth:`kl_divergence`.</span>
<span class="sd"> Usage::</span>
<span class="sd"> @register_kl(Normal, Normal)</span>
<span class="sd"> def kl_normal_normal(p, q):</span>
<span class="sd"> # insert implementation here</span>
<span class="sd"> Lookup returns the most specific (type,type) match ordered by subclass. If</span>
<span class="sd"> the match is ambiguous, a `RuntimeWarning` is raised. For example to</span>
<span class="sd"> resolve the ambiguous situation::</span>
<span class="sd"> @register_kl(BaseP, DerivedQ)</span>
<span class="sd"> def kl_version1(p, q): ...</span>
<span class="sd"> @register_kl(DerivedP, BaseQ)</span>
<span class="sd"> def kl_version2(p, q): ...</span>
<span class="sd"> you should register a third most-specific implementation, e.g.::</span>
<span class="sd"> register_kl(DerivedP, DerivedQ)(kl_version1) # Break the tie.</span>
<span class="sd"> Args:</span>
<span class="sd"> type_p (type): A subclass of :class:`~torch.distributions.Distribution`.</span>
<span class="sd"> type_q (type): A subclass of :class:`~torch.distributions.Distribution`.</span>
<span class="sd"> """</span>
<span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">type_p</span><span class="p">,</span> <span class="nb">type</span><span class="p">)</span> <span class="ow">and</span> <span class="nb">issubclass</span><span class="p">(</span><span class="n">type_p</span><span class="p">,</span> <span class="n">Distribution</span><span class="p">):</span>
<span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s1">'Expected type_p to be a Distribution subclass but got </span><span class="si">{}</span><span class="s1">'</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">type_p</span><span class="p">))</span>
<span class="k">if</span> <span class="ow">not</span> <span class="nb">isinstance</span><span class="p">(</span><span class="n">type_q</span><span class="p">,</span> <span class="nb">type</span><span class="p">)</span> <span class="ow">and</span> <span class="nb">issubclass</span><span class="p">(</span><span class="n">type_q</span><span class="p">,</span> <span class="n">Distribution</span><span class="p">):</span>
<span class="k">raise</span> <span class="ne">TypeError</span><span class="p">(</span><span class="s1">'Expected type_q to be a Distribution subclass but got </span><span class="si">{}</span><span class="s1">'</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">type_q</span><span class="p">))</span>
<span class="k">def</span> <span class="nf">decorator</span><span class="p">(</span><span class="n">fun</span><span class="p">):</span>
<span class="n">_KL_REGISTRY</span><span class="p">[</span><span class="n">type_p</span><span class="p">,</span> <span class="n">type_q</span><span class="p">]</span> <span class="o">=</span> <span class="n">fun</span>
<span class="n">_KL_MEMOIZE</span><span class="o">.</span><span class="n">clear</span><span class="p">()</span> <span class="c1"># reset since lookup order may have changed</span>
<span class="k">return</span> <span class="n">fun</span>
<span class="k">return</span> <span class="n">decorator</span></div>
<span class="nd">@total_ordering</span>
<span class="k">class</span> <span class="nc">_Match</span><span class="p">:</span>
<span class="vm">__slots__</span> <span class="o">=</span> <span class="p">[</span><span class="s1">'types'</span><span class="p">]</span>
<span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="n">types</span><span class="p">):</span>
<span class="bp">self</span><span class="o">.</span><span class="n">types</span> <span class="o">=</span> <span class="n">types</span>
<span class="k">def</span> <span class="fm">__eq__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
<span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">types</span> <span class="o">==</span> <span class="n">other</span><span class="o">.</span><span class="n">types</span>
<span class="k">def</span> <span class="fm">__le__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">other</span><span class="p">):</span>
<span class="k">for</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">types</span><span class="p">,</span> <span class="n">other</span><span class="o">.</span><span class="n">types</span><span class="p">):</span>
<span class="k">if</span> <span class="ow">not</span> <span class="nb">issubclass</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">y</span><span class="p">):</span>
<span class="k">return</span> <span class="kc">False</span>
<span class="k">if</span> <span class="n">x</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">y</span><span class="p">:</span>
<span class="k">break</span>
<span class="k">return</span> <span class="kc">True</span>
<span class="k">def</span> <span class="nf">_dispatch_kl</span><span class="p">(</span><span class="n">type_p</span><span class="p">,</span> <span class="n">type_q</span><span class="p">):</span>
<span class="sd">"""</span>
<span class="sd"> Find the most specific approximate match, assuming single inheritance.</span>
<span class="sd"> """</span>
<span class="n">matches</span> <span class="o">=</span> <span class="p">[(</span><span class="n">super_p</span><span class="p">,</span> <span class="n">super_q</span><span class="p">)</span> <span class="k">for</span> <span class="n">super_p</span><span class="p">,</span> <span class="n">super_q</span> <span class="ow">in</span> <span class="n">_KL_REGISTRY</span>
<span class="k">if</span> <span class="nb">issubclass</span><span class="p">(</span><span class="n">type_p</span><span class="p">,</span> <span class="n">super_p</span><span class="p">)</span> <span class="ow">and</span> <span class="nb">issubclass</span><span class="p">(</span><span class="n">type_q</span><span class="p">,</span> <span class="n">super_q</span><span class="p">)]</span>
<span class="k">if</span> <span class="ow">not</span> <span class="n">matches</span><span class="p">:</span>
<span class="k">return</span> <span class="bp">NotImplemented</span>
<span class="c1"># Check that the left- and right- lexicographic orders agree.</span>
<span class="c1"># mypy isn't smart enough to know that _Match implements __lt__</span>
<span class="c1"># see: https://github.com/python/typing/issues/760#issuecomment-710670503</span>
<span class="n">left_p</span><span class="p">,</span> <span class="n">left_q</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">_Match</span><span class="p">(</span><span class="o">*</span><span class="n">m</span><span class="p">)</span> <span class="k">for</span> <span class="n">m</span> <span class="ow">in</span> <span class="n">matches</span><span class="p">)</span><span class="o">.</span><span class="n">types</span> <span class="c1"># type: ignore[type-var]</span>
<span class="n">right_q</span><span class="p">,</span> <span class="n">right_p</span> <span class="o">=</span> <span class="nb">min</span><span class="p">(</span><span class="n">_Match</span><span class="p">(</span><span class="o">*</span><span class="nb">reversed</span><span class="p">(</span><span class="n">m</span><span class="p">))</span> <span class="k">for</span> <span class="n">m</span> <span class="ow">in</span> <span class="n">matches</span><span class="p">)</span><span class="o">.</span><span class="n">types</span> <span class="c1"># type: ignore[type-var]</span>
<span class="n">left_fun</span> <span class="o">=</span> <span class="n">_KL_REGISTRY</span><span class="p">[</span><span class="n">left_p</span><span class="p">,</span> <span class="n">left_q</span><span class="p">]</span>
<span class="n">right_fun</span> <span class="o">=</span> <span class="n">_KL_REGISTRY</span><span class="p">[</span><span class="n">right_p</span><span class="p">,</span> <span class="n">right_q</span><span class="p">]</span>
<span class="k">if</span> <span class="n">left_fun</span> <span class="ow">is</span> <span class="ow">not</span> <span class="n">right_fun</span><span class="p">:</span>
<span class="n">warnings</span><span class="o">.</span><span class="n">warn</span><span class="p">(</span><span class="s1">'Ambiguous kl_divergence(</span><span class="si">{}</span><span class="s1">, </span><span class="si">{}</span><span class="s1">). Please register_kl(</span><span class="si">{}</span><span class="s1">, </span><span class="si">{}</span><span class="s1">)'</span><span class="o">.</span><span class="n">format</span><span class="p">(</span>
<span class="n">type_p</span><span class="o">.</span><span class="vm">__name__</span><span class="p">,</span> <span class="n">type_q</span><span class="o">.</span><span class="vm">__name__</span><span class="p">,</span> <span class="n">left_p</span><span class="o">.</span><span class="vm">__name__</span><span class="p">,</span> <span class="n">right_q</span><span class="o">.</span><span class="vm">__name__</span><span class="p">),</span>
<span class="ne">RuntimeWarning</span><span class="p">)</span>
<span class="k">return</span> <span class="n">left_fun</span>
<span class="k">def</span> <span class="nf">_infinite_like</span><span class="p">(</span><span class="n">tensor</span><span class="p">):</span>
<span class="sd">"""</span>
<span class="sd"> Helper function for obtaining infinite KL Divergence throughout</span>
<span class="sd"> """</span>
<span class="k">return</span> <span class="n">torch</span><span class="o">.</span><span class="n">full_like</span><span class="p">(</span><span class="n">tensor</span><span class="p">,</span> <span class="n">inf</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">_x_log_x</span><span class="p">(</span><span class="n">tensor</span><span class="p">):</span>
<span class="sd">"""</span>
<span class="sd"> Utility function for calculating x log x</span>
<span class="sd"> """</span>
<span class="k">return</span> <span class="n">tensor</span> <span class="o">*</span> <span class="n">tensor</span><span class="o">.</span><span class="n">log</span><span class="p">()</span>
<span class="k">def</span> <span class="nf">_batch_trace_XXT</span><span class="p">(</span><span class="n">bmat</span><span class="p">):</span>
<span class="sd">"""</span>
<span class="sd"> Utility function for calculating the trace of XX^{T} with X having arbitrary trailing batch dimensions</span>
<span class="sd"> """</span>
<span class="n">n</span> <span class="o">=</span> <span class="n">bmat</span><span class="o">.</span><span class="n">size</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
<span class="n">m</span> <span class="o">=</span> <span class="n">bmat</span><span class="o">.</span><span class="n">size</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">)</span>
<span class="n">flat_trace</span> <span class="o">=</span> <span class="n">bmat</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="n">m</span> <span class="o">*</span> <span class="n">n</span><span class="p">)</span><span class="o">.</span><span class="n">pow</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
<span class="k">return</span> <span class="n">flat_trace</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="n">bmat</span><span class="o">.</span><span class="n">shape</span><span class="p">[:</span><span class="o">-</span><span class="mi">2</span><span class="p">])</span>
<div class="viewcode-block" id="kl_divergence"><a class="viewcode-back" href="../../../distributions.html#torch.distributions.kl.kl_divergence">[docs]</a><span class="k">def</span> <span class="nf">kl_divergence</span><span class="p">(</span><span class="n">p</span><span class="p">:</span> <span class="n">Distribution</span><span class="p">,</span> <span class="n">q</span><span class="p">:</span> <span class="n">Distribution</span><span class="p">)</span> <span class="o">-></span> <span class="n">torch</span><span class="o">.</span><span class="n">Tensor</span><span class="p">:</span>
<span class="sa">r</span><span class="sd">"""</span>
<span class="sd"> Compute Kullback-Leibler divergence :math:`KL(p \| q)` between two distributions.</span>
<span class="sd"> .. math::</span>
<span class="sd"> KL(p \| q) = \int p(x) \log\frac {p(x)} {q(x)} \,dx</span>
<span class="sd"> Args:</span>
<span class="sd"> p (Distribution): A :class:`~torch.distributions.Distribution` object.</span>
<span class="sd"> q (Distribution): A :class:`~torch.distributions.Distribution` object.</span>
<span class="sd"> Returns:</span>
<span class="sd"> Tensor: A batch of KL divergences of shape `batch_shape`.</span>
<span class="sd"> Raises:</span>
<span class="sd"> NotImplementedError: If the distribution types have not been registered via</span>
<span class="sd"> :meth:`register_kl`.</span>
<span class="sd"> """</span>
<span class="k">try</span><span class="p">:</span>
<span class="n">fun</span> <span class="o">=</span> <span class="n">_KL_MEMOIZE</span><span class="p">[</span><span class="nb">type</span><span class="p">(</span><span class="n">p</span><span class="p">),</span> <span class="nb">type</span><span class="p">(</span><span class="n">q</span><span class="p">)]</span>
<span class="k">except</span> <span class="ne">KeyError</span><span class="p">:</span>
<span class="n">fun</span> <span class="o">=</span> <span class="n">_dispatch_kl</span><span class="p">(</span><span class="nb">type</span><span class="p">(</span><span class="n">p</span><span class="p">),</span> <span class="nb">type</span><span class="p">(</span><span class="n">q</span><span class="p">))</span>
<span class="n">_KL_MEMOIZE</span><span class="p">[</span><span class="nb">type</span><span class="p">(</span><span class="n">p</span><span class="p">),</span> <span class="nb">type</span><span class="p">(</span><span class="n">q</span><span class="p">)]</span> <span class="o">=</span> <span class="n">fun</span>
<span class="k">if</span> <span class="n">fun</span> <span class="ow">is</span> <span class="bp">NotImplemented</span><span class="p">:</span>
<span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s2">"No KL(p || q) is implemented for p type </span><span class="si">{}</span><span class="s2"> and q type </span><span class="si">{}</span><span class="s2">"</span>
<span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="vm">__class__</span><span class="o">.</span><span class="vm">__name__</span><span class="p">,</span> <span class="n">q</span><span class="o">.</span><span class="vm">__class__</span><span class="o">.</span><span class="vm">__name__</span><span class="p">))</span>
<span class="k">return</span> <span class="n">fun</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">)</span></div>
<span class="c1">################################################################################</span>
<span class="c1"># KL Divergence Implementations</span>
<span class="c1">################################################################################</span>
<span class="c1"># Same distributions</span>
<span class="nd">@register_kl</span><span class="p">(</span><span class="n">Bernoulli</span><span class="p">,</span> <span class="n">Bernoulli</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">_kl_bernoulli_bernoulli</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">):</span>
<span class="n">t1</span> <span class="o">=</span> <span class="n">p</span><span class="o">.</span><span class="n">probs</span> <span class="o">*</span> <span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">nn</span><span class="o">.</span><span class="n">functional</span><span class="o">.</span><span class="n">softplus</span><span class="p">(</span><span class="o">-</span><span class="n">q</span><span class="o">.</span><span class="n">logits</span><span class="p">)</span> <span class="o">-</span> <span class="n">torch</span><span class="o">.</span><span class="n">nn</span><span class="o">.</span><span class="n">functional</span><span class="o">.</span><span class="n">softplus</span><span class="p">(</span><span class="o">-</span><span class="n">p</span><span class="o">.</span><span class="n">logits</span><span class="p">))</span>
<span class="n">t1</span><span class="p">[</span><span class="n">q</span><span class="o">.</span><span class="n">probs</span> <span class="o">==</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="n">inf</span>
<span class="n">t1</span><span class="p">[</span><span class="n">p</span><span class="o">.</span><span class="n">probs</span> <span class="o">==</span> <span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="mi">0</span>
<span class="n">t2</span> <span class="o">=</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">p</span><span class="o">.</span><span class="n">probs</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">nn</span><span class="o">.</span><span class="n">functional</span><span class="o">.</span><span class="n">softplus</span><span class="p">(</span><span class="n">q</span><span class="o">.</span><span class="n">logits</span><span class="p">)</span> <span class="o">-</span> <span class="n">torch</span><span class="o">.</span><span class="n">nn</span><span class="o">.</span><span class="n">functional</span><span class="o">.</span><span class="n">softplus</span><span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">logits</span><span class="p">))</span>
<span class="n">t2</span><span class="p">[</span><span class="n">q</span><span class="o">.</span><span class="n">probs</span> <span class="o">==</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="n">inf</span>
<span class="n">t2</span><span class="p">[</span><span class="n">p</span><span class="o">.</span><span class="n">probs</span> <span class="o">==</span> <span class="mi">1</span><span class="p">]</span> <span class="o">=</span> <span class="mi">0</span>
<span class="k">return</span> <span class="n">t1</span> <span class="o">+</span> <span class="n">t2</span>
<span class="nd">@register_kl</span><span class="p">(</span><span class="n">Beta</span><span class="p">,</span> <span class="n">Beta</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">_kl_beta_beta</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">):</span>
<span class="n">sum_params_p</span> <span class="o">=</span> <span class="n">p</span><span class="o">.</span><span class="n">concentration1</span> <span class="o">+</span> <span class="n">p</span><span class="o">.</span><span class="n">concentration0</span>
<span class="n">sum_params_q</span> <span class="o">=</span> <span class="n">q</span><span class="o">.</span><span class="n">concentration1</span> <span class="o">+</span> <span class="n">q</span><span class="o">.</span><span class="n">concentration0</span>
<span class="n">t1</span> <span class="o">=</span> <span class="n">q</span><span class="o">.</span><span class="n">concentration1</span><span class="o">.</span><span class="n">lgamma</span><span class="p">()</span> <span class="o">+</span> <span class="n">q</span><span class="o">.</span><span class="n">concentration0</span><span class="o">.</span><span class="n">lgamma</span><span class="p">()</span> <span class="o">+</span> <span class="p">(</span><span class="n">sum_params_p</span><span class="p">)</span><span class="o">.</span><span class="n">lgamma</span><span class="p">()</span>
<span class="n">t2</span> <span class="o">=</span> <span class="n">p</span><span class="o">.</span><span class="n">concentration1</span><span class="o">.</span><span class="n">lgamma</span><span class="p">()</span> <span class="o">+</span> <span class="n">p</span><span class="o">.</span><span class="n">concentration0</span><span class="o">.</span><span class="n">lgamma</span><span class="p">()</span> <span class="o">+</span> <span class="p">(</span><span class="n">sum_params_q</span><span class="p">)</span><span class="o">.</span><span class="n">lgamma</span><span class="p">()</span>
<span class="n">t3</span> <span class="o">=</span> <span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">concentration1</span> <span class="o">-</span> <span class="n">q</span><span class="o">.</span><span class="n">concentration1</span><span class="p">)</span> <span class="o">*</span> <span class="n">torch</span><span class="o">.</span><span class="n">digamma</span><span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">concentration1</span><span class="p">)</span>
<span class="n">t4</span> <span class="o">=</span> <span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">concentration0</span> <span class="o">-</span> <span class="n">q</span><span class="o">.</span><span class="n">concentration0</span><span class="p">)</span> <span class="o">*</span> <span class="n">torch</span><span class="o">.</span><span class="n">digamma</span><span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">concentration0</span><span class="p">)</span>
<span class="n">t5</span> <span class="o">=</span> <span class="p">(</span><span class="n">sum_params_q</span> <span class="o">-</span> <span class="n">sum_params_p</span><span class="p">)</span> <span class="o">*</span> <span class="n">torch</span><span class="o">.</span><span class="n">digamma</span><span class="p">(</span><span class="n">sum_params_p</span><span class="p">)</span>
<span class="k">return</span> <span class="n">t1</span> <span class="o">-</span> <span class="n">t2</span> <span class="o">+</span> <span class="n">t3</span> <span class="o">+</span> <span class="n">t4</span> <span class="o">+</span> <span class="n">t5</span>
<span class="nd">@register_kl</span><span class="p">(</span><span class="n">Binomial</span><span class="p">,</span> <span class="n">Binomial</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">_kl_binomial_binomial</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">):</span>
<span class="c1"># from https://math.stackexchange.com/questions/2214993/</span>
<span class="c1"># kullback-leibler-divergence-for-binomial-distributions-p-and-q</span>
<span class="k">if</span> <span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">total_count</span> <span class="o"><</span> <span class="n">q</span><span class="o">.</span><span class="n">total_count</span><span class="p">)</span><span class="o">.</span><span class="n">any</span><span class="p">():</span>
<span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s1">'KL between Binomials where q.total_count > p.total_count is not implemented'</span><span class="p">)</span>
<span class="n">kl</span> <span class="o">=</span> <span class="n">p</span><span class="o">.</span><span class="n">total_count</span> <span class="o">*</span> <span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">probs</span> <span class="o">*</span> <span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">logits</span> <span class="o">-</span> <span class="n">q</span><span class="o">.</span><span class="n">logits</span><span class="p">)</span> <span class="o">+</span> <span class="p">(</span><span class="o">-</span><span class="n">p</span><span class="o">.</span><span class="n">probs</span><span class="p">)</span><span class="o">.</span><span class="n">log1p</span><span class="p">()</span> <span class="o">-</span> <span class="p">(</span><span class="o">-</span><span class="n">q</span><span class="o">.</span><span class="n">probs</span><span class="p">)</span><span class="o">.</span><span class="n">log1p</span><span class="p">())</span>
<span class="n">inf_idxs</span> <span class="o">=</span> <span class="n">p</span><span class="o">.</span><span class="n">total_count</span> <span class="o">></span> <span class="n">q</span><span class="o">.</span><span class="n">total_count</span>
<span class="n">kl</span><span class="p">[</span><span class="n">inf_idxs</span><span class="p">]</span> <span class="o">=</span> <span class="n">_infinite_like</span><span class="p">(</span><span class="n">kl</span><span class="p">[</span><span class="n">inf_idxs</span><span class="p">])</span>
<span class="k">return</span> <span class="n">kl</span>
<span class="nd">@register_kl</span><span class="p">(</span><span class="n">Categorical</span><span class="p">,</span> <span class="n">Categorical</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">_kl_categorical_categorical</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">):</span>
<span class="n">t</span> <span class="o">=</span> <span class="n">p</span><span class="o">.</span><span class="n">probs</span> <span class="o">*</span> <span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">logits</span> <span class="o">-</span> <span class="n">q</span><span class="o">.</span><span class="n">logits</span><span class="p">)</span>
<span class="n">t</span><span class="p">[(</span><span class="n">q</span><span class="o">.</span><span class="n">probs</span> <span class="o">==</span> <span class="mi">0</span><span class="p">)</span><span class="o">.</span><span class="n">expand_as</span><span class="p">(</span><span class="n">t</span><span class="p">)]</span> <span class="o">=</span> <span class="n">inf</span>
<span class="n">t</span><span class="p">[(</span><span class="n">p</span><span class="o">.</span><span class="n">probs</span> <span class="o">==</span> <span class="mi">0</span><span class="p">)</span><span class="o">.</span><span class="n">expand_as</span><span class="p">(</span><span class="n">t</span><span class="p">)]</span> <span class="o">=</span> <span class="mi">0</span>
<span class="k">return</span> <span class="n">t</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
<span class="nd">@register_kl</span><span class="p">(</span><span class="n">ContinuousBernoulli</span><span class="p">,</span> <span class="n">ContinuousBernoulli</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">_kl_continuous_bernoulli_continuous_bernoulli</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">):</span>
<span class="n">t1</span> <span class="o">=</span> <span class="n">p</span><span class="o">.</span><span class="n">mean</span> <span class="o">*</span> <span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">logits</span> <span class="o">-</span> <span class="n">q</span><span class="o">.</span><span class="n">logits</span><span class="p">)</span>
<span class="n">t2</span> <span class="o">=</span> <span class="n">p</span><span class="o">.</span><span class="n">_cont_bern_log_norm</span><span class="p">()</span> <span class="o">+</span> <span class="n">torch</span><span class="o">.</span><span class="n">log1p</span><span class="p">(</span><span class="o">-</span><span class="n">p</span><span class="o">.</span><span class="n">probs</span><span class="p">)</span>
<span class="n">t3</span> <span class="o">=</span> <span class="o">-</span> <span class="n">q</span><span class="o">.</span><span class="n">_cont_bern_log_norm</span><span class="p">()</span> <span class="o">-</span> <span class="n">torch</span><span class="o">.</span><span class="n">log1p</span><span class="p">(</span><span class="o">-</span><span class="n">q</span><span class="o">.</span><span class="n">probs</span><span class="p">)</span>
<span class="k">return</span> <span class="n">t1</span> <span class="o">+</span> <span class="n">t2</span> <span class="o">+</span> <span class="n">t3</span>
<span class="nd">@register_kl</span><span class="p">(</span><span class="n">Dirichlet</span><span class="p">,</span> <span class="n">Dirichlet</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">_kl_dirichlet_dirichlet</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">):</span>
<span class="c1"># From http://bariskurt.com/kullback-leibler-divergence-between-two-dirichlet-and-beta-distributions/</span>
<span class="n">sum_p_concentration</span> <span class="o">=</span> <span class="n">p</span><span class="o">.</span><span class="n">concentration</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
<span class="n">sum_q_concentration</span> <span class="o">=</span> <span class="n">q</span><span class="o">.</span><span class="n">concentration</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
<span class="n">t1</span> <span class="o">=</span> <span class="n">sum_p_concentration</span><span class="o">.</span><span class="n">lgamma</span><span class="p">()</span> <span class="o">-</span> <span class="n">sum_q_concentration</span><span class="o">.</span><span class="n">lgamma</span><span class="p">()</span>
<span class="n">t2</span> <span class="o">=</span> <span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">concentration</span><span class="o">.</span><span class="n">lgamma</span><span class="p">()</span> <span class="o">-</span> <span class="n">q</span><span class="o">.</span><span class="n">concentration</span><span class="o">.</span><span class="n">lgamma</span><span class="p">())</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
<span class="n">t3</span> <span class="o">=</span> <span class="n">p</span><span class="o">.</span><span class="n">concentration</span> <span class="o">-</span> <span class="n">q</span><span class="o">.</span><span class="n">concentration</span>
<span class="n">t4</span> <span class="o">=</span> <span class="n">p</span><span class="o">.</span><span class="n">concentration</span><span class="o">.</span><span class="n">digamma</span><span class="p">()</span> <span class="o">-</span> <span class="n">sum_p_concentration</span><span class="o">.</span><span class="n">digamma</span><span class="p">()</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
<span class="k">return</span> <span class="n">t1</span> <span class="o">-</span> <span class="n">t2</span> <span class="o">+</span> <span class="p">(</span><span class="n">t3</span> <span class="o">*</span> <span class="n">t4</span><span class="p">)</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
<span class="nd">@register_kl</span><span class="p">(</span><span class="n">Exponential</span><span class="p">,</span> <span class="n">Exponential</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">_kl_exponential_exponential</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">):</span>
<span class="n">rate_ratio</span> <span class="o">=</span> <span class="n">q</span><span class="o">.</span><span class="n">rate</span> <span class="o">/</span> <span class="n">p</span><span class="o">.</span><span class="n">rate</span>
<span class="n">t1</span> <span class="o">=</span> <span class="o">-</span><span class="n">rate_ratio</span><span class="o">.</span><span class="n">log</span><span class="p">()</span>
<span class="k">return</span> <span class="n">t1</span> <span class="o">+</span> <span class="n">rate_ratio</span> <span class="o">-</span> <span class="mi">1</span>
<span class="nd">@register_kl</span><span class="p">(</span><span class="n">ExponentialFamily</span><span class="p">,</span> <span class="n">ExponentialFamily</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">_kl_expfamily_expfamily</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">):</span>
<span class="k">if</span> <span class="ow">not</span> <span class="nb">type</span><span class="p">(</span><span class="n">p</span><span class="p">)</span> <span class="o">==</span> <span class="nb">type</span><span class="p">(</span><span class="n">q</span><span class="p">):</span>
<span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">(</span><span class="s2">"The cross KL-divergence between different exponential families cannot </span><span class="se">\</span>
<span class="s2"> be computed using Bregman divergences"</span><span class="p">)</span>
<span class="n">p_nparams</span> <span class="o">=</span> <span class="p">[</span><span class="n">np</span><span class="o">.</span><span class="n">detach</span><span class="p">()</span><span class="o">.</span><span class="n">requires_grad_</span><span class="p">()</span> <span class="k">for</span> <span class="n">np</span> <span class="ow">in</span> <span class="n">p</span><span class="o">.</span><span class="n">_natural_params</span><span class="p">]</span>
<span class="n">q_nparams</span> <span class="o">=</span> <span class="n">q</span><span class="o">.</span><span class="n">_natural_params</span>
<span class="n">lg_normal</span> <span class="o">=</span> <span class="n">p</span><span class="o">.</span><span class="n">_log_normalizer</span><span class="p">(</span><span class="o">*</span><span class="n">p_nparams</span><span class="p">)</span>
<span class="n">gradients</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">autograd</span><span class="o">.</span><span class="n">grad</span><span class="p">(</span><span class="n">lg_normal</span><span class="o">.</span><span class="n">sum</span><span class="p">(),</span> <span class="n">p_nparams</span><span class="p">,</span> <span class="n">create_graph</span><span class="o">=</span><span class="kc">True</span><span class="p">)</span>
<span class="n">result</span> <span class="o">=</span> <span class="n">q</span><span class="o">.</span><span class="n">_log_normalizer</span><span class="p">(</span><span class="o">*</span><span class="n">q_nparams</span><span class="p">)</span> <span class="o">-</span> <span class="n">lg_normal</span>
<span class="k">for</span> <span class="n">pnp</span><span class="p">,</span> <span class="n">qnp</span><span class="p">,</span> <span class="n">g</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="n">p_nparams</span><span class="p">,</span> <span class="n">q_nparams</span><span class="p">,</span> <span class="n">gradients</span><span class="p">):</span>
<span class="n">term</span> <span class="o">=</span> <span class="p">(</span><span class="n">qnp</span> <span class="o">-</span> <span class="n">pnp</span><span class="p">)</span> <span class="o">*</span> <span class="n">g</span>
<span class="n">result</span> <span class="o">-=</span> <span class="n">_sum_rightmost</span><span class="p">(</span><span class="n">term</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">q</span><span class="o">.</span><span class="n">event_shape</span><span class="p">))</span>
<span class="k">return</span> <span class="n">result</span>
<span class="nd">@register_kl</span><span class="p">(</span><span class="n">Gamma</span><span class="p">,</span> <span class="n">Gamma</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">_kl_gamma_gamma</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">):</span>
<span class="n">t1</span> <span class="o">=</span> <span class="n">q</span><span class="o">.</span><span class="n">concentration</span> <span class="o">*</span> <span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">rate</span> <span class="o">/</span> <span class="n">q</span><span class="o">.</span><span class="n">rate</span><span class="p">)</span><span class="o">.</span><span class="n">log</span><span class="p">()</span>
<span class="n">t2</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">lgamma</span><span class="p">(</span><span class="n">q</span><span class="o">.</span><span class="n">concentration</span><span class="p">)</span> <span class="o">-</span> <span class="n">torch</span><span class="o">.</span><span class="n">lgamma</span><span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">concentration</span><span class="p">)</span>
<span class="n">t3</span> <span class="o">=</span> <span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">concentration</span> <span class="o">-</span> <span class="n">q</span><span class="o">.</span><span class="n">concentration</span><span class="p">)</span> <span class="o">*</span> <span class="n">torch</span><span class="o">.</span><span class="n">digamma</span><span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">concentration</span><span class="p">)</span>
<span class="n">t4</span> <span class="o">=</span> <span class="p">(</span><span class="n">q</span><span class="o">.</span><span class="n">rate</span> <span class="o">-</span> <span class="n">p</span><span class="o">.</span><span class="n">rate</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">concentration</span> <span class="o">/</span> <span class="n">p</span><span class="o">.</span><span class="n">rate</span><span class="p">)</span>
<span class="k">return</span> <span class="n">t1</span> <span class="o">+</span> <span class="n">t2</span> <span class="o">+</span> <span class="n">t3</span> <span class="o">+</span> <span class="n">t4</span>
<span class="nd">@register_kl</span><span class="p">(</span><span class="n">Gumbel</span><span class="p">,</span> <span class="n">Gumbel</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">_kl_gumbel_gumbel</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">):</span>
<span class="n">ct1</span> <span class="o">=</span> <span class="n">p</span><span class="o">.</span><span class="n">scale</span> <span class="o">/</span> <span class="n">q</span><span class="o">.</span><span class="n">scale</span>
<span class="n">ct2</span> <span class="o">=</span> <span class="n">q</span><span class="o">.</span><span class="n">loc</span> <span class="o">/</span> <span class="n">q</span><span class="o">.</span><span class="n">scale</span>
<span class="n">ct3</span> <span class="o">=</span> <span class="n">p</span><span class="o">.</span><span class="n">loc</span> <span class="o">/</span> <span class="n">q</span><span class="o">.</span><span class="n">scale</span>
<span class="n">t1</span> <span class="o">=</span> <span class="o">-</span><span class="n">ct1</span><span class="o">.</span><span class="n">log</span><span class="p">()</span> <span class="o">-</span> <span class="n">ct2</span> <span class="o">+</span> <span class="n">ct3</span>
<span class="n">t2</span> <span class="o">=</span> <span class="n">ct1</span> <span class="o">*</span> <span class="n">_euler_gamma</span>
<span class="n">t3</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="n">ct2</span> <span class="o">+</span> <span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">ct1</span><span class="p">)</span><span class="o">.</span><span class="n">lgamma</span><span class="p">()</span> <span class="o">-</span> <span class="n">ct3</span><span class="p">)</span>
<span class="k">return</span> <span class="n">t1</span> <span class="o">+</span> <span class="n">t2</span> <span class="o">+</span> <span class="n">t3</span> <span class="o">-</span> <span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">_euler_gamma</span><span class="p">)</span>
<span class="nd">@register_kl</span><span class="p">(</span><span class="n">Geometric</span><span class="p">,</span> <span class="n">Geometric</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">_kl_geometric_geometric</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">):</span>
<span class="k">return</span> <span class="o">-</span><span class="n">p</span><span class="o">.</span><span class="n">entropy</span><span class="p">()</span> <span class="o">-</span> <span class="n">torch</span><span class="o">.</span><span class="n">log1p</span><span class="p">(</span><span class="o">-</span><span class="n">q</span><span class="o">.</span><span class="n">probs</span><span class="p">)</span> <span class="o">/</span> <span class="n">p</span><span class="o">.</span><span class="n">probs</span> <span class="o">-</span> <span class="n">q</span><span class="o">.</span><span class="n">logits</span>
<span class="nd">@register_kl</span><span class="p">(</span><span class="n">HalfNormal</span><span class="p">,</span> <span class="n">HalfNormal</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">_kl_halfnormal_halfnormal</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">):</span>
<span class="k">return</span> <span class="n">_kl_normal_normal</span><span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">base_dist</span><span class="p">,</span> <span class="n">q</span><span class="o">.</span><span class="n">base_dist</span><span class="p">)</span>
<span class="nd">@register_kl</span><span class="p">(</span><span class="n">Laplace</span><span class="p">,</span> <span class="n">Laplace</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">_kl_laplace_laplace</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">):</span>
<span class="c1"># From http://www.mast.queensu.ca/~communications/Papers/gil-msc11.pdf</span>
<span class="n">scale_ratio</span> <span class="o">=</span> <span class="n">p</span><span class="o">.</span><span class="n">scale</span> <span class="o">/</span> <span class="n">q</span><span class="o">.</span><span class="n">scale</span>
<span class="n">loc_abs_diff</span> <span class="o">=</span> <span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">loc</span> <span class="o">-</span> <span class="n">q</span><span class="o">.</span><span class="n">loc</span><span class="p">)</span><span class="o">.</span><span class="n">abs</span><span class="p">()</span>
<span class="n">t1</span> <span class="o">=</span> <span class="o">-</span><span class="n">scale_ratio</span><span class="o">.</span><span class="n">log</span><span class="p">()</span>
<span class="n">t2</span> <span class="o">=</span> <span class="n">loc_abs_diff</span> <span class="o">/</span> <span class="n">q</span><span class="o">.</span><span class="n">scale</span>
<span class="n">t3</span> <span class="o">=</span> <span class="n">scale_ratio</span> <span class="o">*</span> <span class="n">torch</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">loc_abs_diff</span> <span class="o">/</span> <span class="n">p</span><span class="o">.</span><span class="n">scale</span><span class="p">)</span>
<span class="k">return</span> <span class="n">t1</span> <span class="o">+</span> <span class="n">t2</span> <span class="o">+</span> <span class="n">t3</span> <span class="o">-</span> <span class="mi">1</span>
<span class="nd">@register_kl</span><span class="p">(</span><span class="n">LowRankMultivariateNormal</span><span class="p">,</span> <span class="n">LowRankMultivariateNormal</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">_kl_lowrankmultivariatenormal_lowrankmultivariatenormal</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">):</span>
<span class="k">if</span> <span class="n">p</span><span class="o">.</span><span class="n">event_shape</span> <span class="o">!=</span> <span class="n">q</span><span class="o">.</span><span class="n">event_shape</span><span class="p">:</span>
<span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s2">"KL-divergence between two Low Rank Multivariate Normals with</span><span class="se">\</span>
<span class="s2"> different event shapes cannot be computed"</span><span class="p">)</span>
<span class="n">term1</span> <span class="o">=</span> <span class="p">(</span><span class="n">_batch_lowrank_logdet</span><span class="p">(</span><span class="n">q</span><span class="o">.</span><span class="n">_unbroadcasted_cov_factor</span><span class="p">,</span> <span class="n">q</span><span class="o">.</span><span class="n">_unbroadcasted_cov_diag</span><span class="p">,</span>
<span class="n">q</span><span class="o">.</span><span class="n">_capacitance_tril</span><span class="p">)</span> <span class="o">-</span>
<span class="n">_batch_lowrank_logdet</span><span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">_unbroadcasted_cov_factor</span><span class="p">,</span> <span class="n">p</span><span class="o">.</span><span class="n">_unbroadcasted_cov_diag</span><span class="p">,</span>
<span class="n">p</span><span class="o">.</span><span class="n">_capacitance_tril</span><span class="p">))</span>
<span class="n">term3</span> <span class="o">=</span> <span class="n">_batch_lowrank_mahalanobis</span><span class="p">(</span><span class="n">q</span><span class="o">.</span><span class="n">_unbroadcasted_cov_factor</span><span class="p">,</span> <span class="n">q</span><span class="o">.</span><span class="n">_unbroadcasted_cov_diag</span><span class="p">,</span>
<span class="n">q</span><span class="o">.</span><span class="n">loc</span> <span class="o">-</span> <span class="n">p</span><span class="o">.</span><span class="n">loc</span><span class="p">,</span>
<span class="n">q</span><span class="o">.</span><span class="n">_capacitance_tril</span><span class="p">)</span>
<span class="c1"># Expands term2 according to</span>
<span class="c1"># inv(qcov) @ pcov = [inv(qD) - inv(qD) @ qW @ inv(qC) @ qW.T @ inv(qD)] @ (pW @ pW.T + pD)</span>
<span class="c1"># = [inv(qD) - A.T @ A] @ (pD + pW @ pW.T)</span>
<span class="n">qWt_qDinv</span> <span class="o">=</span> <span class="p">(</span><span class="n">q</span><span class="o">.</span><span class="n">_unbroadcasted_cov_factor</span><span class="o">.</span><span class="n">mT</span> <span class="o">/</span>
<span class="n">q</span><span class="o">.</span><span class="n">_unbroadcasted_cov_diag</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">))</span>
<span class="n">A</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">solve_triangular</span><span class="p">(</span><span class="n">q</span><span class="o">.</span><span class="n">_capacitance_tril</span><span class="p">,</span> <span class="n">qWt_qDinv</span><span class="p">,</span> <span class="n">upper</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
<span class="n">term21</span> <span class="o">=</span> <span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">_unbroadcasted_cov_diag</span> <span class="o">/</span> <span class="n">q</span><span class="o">.</span><span class="n">_unbroadcasted_cov_diag</span><span class="p">)</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
<span class="n">term22</span> <span class="o">=</span> <span class="n">_batch_trace_XXT</span><span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">_unbroadcasted_cov_factor</span> <span class="o">*</span>
<span class="n">q</span><span class="o">.</span><span class="n">_unbroadcasted_cov_diag</span><span class="o">.</span><span class="n">rsqrt</span><span class="p">()</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">))</span>
<span class="n">term23</span> <span class="o">=</span> <span class="n">_batch_trace_XXT</span><span class="p">(</span><span class="n">A</span> <span class="o">*</span> <span class="n">p</span><span class="o">.</span><span class="n">_unbroadcasted_cov_diag</span><span class="o">.</span><span class="n">sqrt</span><span class="p">()</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">))</span>
<span class="n">term24</span> <span class="o">=</span> <span class="n">_batch_trace_XXT</span><span class="p">(</span><span class="n">A</span><span class="o">.</span><span class="n">matmul</span><span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">_unbroadcasted_cov_factor</span><span class="p">))</span>
<span class="n">term2</span> <span class="o">=</span> <span class="n">term21</span> <span class="o">+</span> <span class="n">term22</span> <span class="o">-</span> <span class="n">term23</span> <span class="o">-</span> <span class="n">term24</span>
<span class="k">return</span> <span class="mf">0.5</span> <span class="o">*</span> <span class="p">(</span><span class="n">term1</span> <span class="o">+</span> <span class="n">term2</span> <span class="o">+</span> <span class="n">term3</span> <span class="o">-</span> <span class="n">p</span><span class="o">.</span><span class="n">event_shape</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
<span class="nd">@register_kl</span><span class="p">(</span><span class="n">MultivariateNormal</span><span class="p">,</span> <span class="n">LowRankMultivariateNormal</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">_kl_multivariatenormal_lowrankmultivariatenormal</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">):</span>
<span class="k">if</span> <span class="n">p</span><span class="o">.</span><span class="n">event_shape</span> <span class="o">!=</span> <span class="n">q</span><span class="o">.</span><span class="n">event_shape</span><span class="p">:</span>
<span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s2">"KL-divergence between two (Low Rank) Multivariate Normals with</span><span class="se">\</span>
<span class="s2"> different event shapes cannot be computed"</span><span class="p">)</span>
<span class="n">term1</span> <span class="o">=</span> <span class="p">(</span><span class="n">_batch_lowrank_logdet</span><span class="p">(</span><span class="n">q</span><span class="o">.</span><span class="n">_unbroadcasted_cov_factor</span><span class="p">,</span> <span class="n">q</span><span class="o">.</span><span class="n">_unbroadcasted_cov_diag</span><span class="p">,</span>
<span class="n">q</span><span class="o">.</span><span class="n">_capacitance_tril</span><span class="p">)</span> <span class="o">-</span>
<span class="mi">2</span> <span class="o">*</span> <span class="n">p</span><span class="o">.</span><span class="n">_unbroadcasted_scale_tril</span><span class="o">.</span><span class="n">diagonal</span><span class="p">(</span><span class="n">dim1</span><span class="o">=-</span><span class="mi">2</span><span class="p">,</span> <span class="n">dim2</span><span class="o">=-</span><span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">log</span><span class="p">()</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">))</span>
<span class="n">term3</span> <span class="o">=</span> <span class="n">_batch_lowrank_mahalanobis</span><span class="p">(</span><span class="n">q</span><span class="o">.</span><span class="n">_unbroadcasted_cov_factor</span><span class="p">,</span> <span class="n">q</span><span class="o">.</span><span class="n">_unbroadcasted_cov_diag</span><span class="p">,</span>
<span class="n">q</span><span class="o">.</span><span class="n">loc</span> <span class="o">-</span> <span class="n">p</span><span class="o">.</span><span class="n">loc</span><span class="p">,</span>
<span class="n">q</span><span class="o">.</span><span class="n">_capacitance_tril</span><span class="p">)</span>
<span class="c1"># Expands term2 according to</span>
<span class="c1"># inv(qcov) @ pcov = [inv(qD) - inv(qD) @ qW @ inv(qC) @ qW.T @ inv(qD)] @ p_tril @ p_tril.T</span>
<span class="c1"># = [inv(qD) - A.T @ A] @ p_tril @ p_tril.T</span>
<span class="n">qWt_qDinv</span> <span class="o">=</span> <span class="p">(</span><span class="n">q</span><span class="o">.</span><span class="n">_unbroadcasted_cov_factor</span><span class="o">.</span><span class="n">mT</span> <span class="o">/</span>
<span class="n">q</span><span class="o">.</span><span class="n">_unbroadcasted_cov_diag</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="o">-</span><span class="mi">2</span><span class="p">))</span>
<span class="n">A</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">solve_triangular</span><span class="p">(</span><span class="n">q</span><span class="o">.</span><span class="n">_capacitance_tril</span><span class="p">,</span> <span class="n">qWt_qDinv</span><span class="p">,</span> <span class="n">upper</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
<span class="n">term21</span> <span class="o">=</span> <span class="n">_batch_trace_XXT</span><span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">_unbroadcasted_scale_tril</span> <span class="o">*</span>
<span class="n">q</span><span class="o">.</span><span class="n">_unbroadcasted_cov_diag</span><span class="o">.</span><span class="n">rsqrt</span><span class="p">()</span><span class="o">.</span><span class="n">unsqueeze</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">))</span>
<span class="n">term22</span> <span class="o">=</span> <span class="n">_batch_trace_XXT</span><span class="p">(</span><span class="n">A</span><span class="o">.</span><span class="n">matmul</span><span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">_unbroadcasted_scale_tril</span><span class="p">))</span>
<span class="n">term2</span> <span class="o">=</span> <span class="n">term21</span> <span class="o">-</span> <span class="n">term22</span>
<span class="k">return</span> <span class="mf">0.5</span> <span class="o">*</span> <span class="p">(</span><span class="n">term1</span> <span class="o">+</span> <span class="n">term2</span> <span class="o">+</span> <span class="n">term3</span> <span class="o">-</span> <span class="n">p</span><span class="o">.</span><span class="n">event_shape</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
<span class="nd">@register_kl</span><span class="p">(</span><span class="n">LowRankMultivariateNormal</span><span class="p">,</span> <span class="n">MultivariateNormal</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">_kl_lowrankmultivariatenormal_multivariatenormal</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">):</span>
<span class="k">if</span> <span class="n">p</span><span class="o">.</span><span class="n">event_shape</span> <span class="o">!=</span> <span class="n">q</span><span class="o">.</span><span class="n">event_shape</span><span class="p">:</span>
<span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s2">"KL-divergence between two (Low Rank) Multivariate Normals with</span><span class="se">\</span>
<span class="s2"> different event shapes cannot be computed"</span><span class="p">)</span>
<span class="n">term1</span> <span class="o">=</span> <span class="p">(</span><span class="mi">2</span> <span class="o">*</span> <span class="n">q</span><span class="o">.</span><span class="n">_unbroadcasted_scale_tril</span><span class="o">.</span><span class="n">diagonal</span><span class="p">(</span><span class="n">dim1</span><span class="o">=-</span><span class="mi">2</span><span class="p">,</span> <span class="n">dim2</span><span class="o">=-</span><span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">log</span><span class="p">()</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span> <span class="o">-</span>
<span class="n">_batch_lowrank_logdet</span><span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">_unbroadcasted_cov_factor</span><span class="p">,</span> <span class="n">p</span><span class="o">.</span><span class="n">_unbroadcasted_cov_diag</span><span class="p">,</span>
<span class="n">p</span><span class="o">.</span><span class="n">_capacitance_tril</span><span class="p">))</span>
<span class="n">term3</span> <span class="o">=</span> <span class="n">_batch_mahalanobis</span><span class="p">(</span><span class="n">q</span><span class="o">.</span><span class="n">_unbroadcasted_scale_tril</span><span class="p">,</span> <span class="p">(</span><span class="n">q</span><span class="o">.</span><span class="n">loc</span> <span class="o">-</span> <span class="n">p</span><span class="o">.</span><span class="n">loc</span><span class="p">))</span>
<span class="c1"># Expands term2 according to</span>
<span class="c1"># inv(qcov) @ pcov = inv(q_tril @ q_tril.T) @ (pW @ pW.T + pD)</span>
<span class="n">combined_batch_shape</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">_C</span><span class="o">.</span><span class="n">_infer_size</span><span class="p">(</span><span class="n">q</span><span class="o">.</span><span class="n">_unbroadcasted_scale_tril</span><span class="o">.</span><span class="n">shape</span><span class="p">[:</span><span class="o">-</span><span class="mi">2</span><span class="p">],</span>
<span class="n">p</span><span class="o">.</span><span class="n">_unbroadcasted_cov_factor</span><span class="o">.</span><span class="n">shape</span><span class="p">[:</span><span class="o">-</span><span class="mi">2</span><span class="p">])</span>
<span class="n">n</span> <span class="o">=</span> <span class="n">p</span><span class="o">.</span><span class="n">event_shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
<span class="n">q_scale_tril</span> <span class="o">=</span> <span class="n">q</span><span class="o">.</span><span class="n">_unbroadcasted_scale_tril</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">combined_batch_shape</span> <span class="o">+</span> <span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">n</span><span class="p">))</span>
<span class="n">p_cov_factor</span> <span class="o">=</span> <span class="n">p</span><span class="o">.</span><span class="n">_unbroadcasted_cov_factor</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">combined_batch_shape</span> <span class="o">+</span>
<span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">p</span><span class="o">.</span><span class="n">cov_factor</span><span class="o">.</span><span class="n">size</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)))</span>
<span class="n">p_cov_diag</span> <span class="o">=</span> <span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">diag_embed</span><span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">_unbroadcasted_cov_diag</span><span class="o">.</span><span class="n">sqrt</span><span class="p">())</span>
<span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">combined_batch_shape</span> <span class="o">+</span> <span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">n</span><span class="p">)))</span>
<span class="n">term21</span> <span class="o">=</span> <span class="n">_batch_trace_XXT</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">solve_triangular</span><span class="p">(</span><span class="n">q_scale_tril</span><span class="p">,</span> <span class="n">p_cov_factor</span><span class="p">,</span> <span class="n">upper</span><span class="o">=</span><span class="kc">False</span><span class="p">))</span>
<span class="n">term22</span> <span class="o">=</span> <span class="n">_batch_trace_XXT</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">solve_triangular</span><span class="p">(</span><span class="n">q_scale_tril</span><span class="p">,</span> <span class="n">p_cov_diag</span><span class="p">,</span> <span class="n">upper</span><span class="o">=</span><span class="kc">False</span><span class="p">))</span>
<span class="n">term2</span> <span class="o">=</span> <span class="n">term21</span> <span class="o">+</span> <span class="n">term22</span>
<span class="k">return</span> <span class="mf">0.5</span> <span class="o">*</span> <span class="p">(</span><span class="n">term1</span> <span class="o">+</span> <span class="n">term2</span> <span class="o">+</span> <span class="n">term3</span> <span class="o">-</span> <span class="n">p</span><span class="o">.</span><span class="n">event_shape</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
<span class="nd">@register_kl</span><span class="p">(</span><span class="n">MultivariateNormal</span><span class="p">,</span> <span class="n">MultivariateNormal</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">_kl_multivariatenormal_multivariatenormal</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">):</span>
<span class="c1"># From https://en.wikipedia.org/wiki/Multivariate_normal_distribution#Kullback%E2%80%93Leibler_divergence</span>
<span class="k">if</span> <span class="n">p</span><span class="o">.</span><span class="n">event_shape</span> <span class="o">!=</span> <span class="n">q</span><span class="o">.</span><span class="n">event_shape</span><span class="p">:</span>
<span class="k">raise</span> <span class="ne">ValueError</span><span class="p">(</span><span class="s2">"KL-divergence between two Multivariate Normals with</span><span class="se">\</span>
<span class="s2"> different event shapes cannot be computed"</span><span class="p">)</span>
<span class="n">half_term1</span> <span class="o">=</span> <span class="p">(</span><span class="n">q</span><span class="o">.</span><span class="n">_unbroadcasted_scale_tril</span><span class="o">.</span><span class="n">diagonal</span><span class="p">(</span><span class="n">dim1</span><span class="o">=-</span><span class="mi">2</span><span class="p">,</span> <span class="n">dim2</span><span class="o">=-</span><span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">log</span><span class="p">()</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span> <span class="o">-</span>
<span class="n">p</span><span class="o">.</span><span class="n">_unbroadcasted_scale_tril</span><span class="o">.</span><span class="n">diagonal</span><span class="p">(</span><span class="n">dim1</span><span class="o">=-</span><span class="mi">2</span><span class="p">,</span> <span class="n">dim2</span><span class="o">=-</span><span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">log</span><span class="p">()</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">))</span>
<span class="n">combined_batch_shape</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">_C</span><span class="o">.</span><span class="n">_infer_size</span><span class="p">(</span><span class="n">q</span><span class="o">.</span><span class="n">_unbroadcasted_scale_tril</span><span class="o">.</span><span class="n">shape</span><span class="p">[:</span><span class="o">-</span><span class="mi">2</span><span class="p">],</span>
<span class="n">p</span><span class="o">.</span><span class="n">_unbroadcasted_scale_tril</span><span class="o">.</span><span class="n">shape</span><span class="p">[:</span><span class="o">-</span><span class="mi">2</span><span class="p">])</span>
<span class="n">n</span> <span class="o">=</span> <span class="n">p</span><span class="o">.</span><span class="n">event_shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
<span class="n">q_scale_tril</span> <span class="o">=</span> <span class="n">q</span><span class="o">.</span><span class="n">_unbroadcasted_scale_tril</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">combined_batch_shape</span> <span class="o">+</span> <span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">n</span><span class="p">))</span>
<span class="n">p_scale_tril</span> <span class="o">=</span> <span class="n">p</span><span class="o">.</span><span class="n">_unbroadcasted_scale_tril</span><span class="o">.</span><span class="n">expand</span><span class="p">(</span><span class="n">combined_batch_shape</span> <span class="o">+</span> <span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">n</span><span class="p">))</span>
<span class="n">term2</span> <span class="o">=</span> <span class="n">_batch_trace_XXT</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">solve_triangular</span><span class="p">(</span><span class="n">q_scale_tril</span><span class="p">,</span> <span class="n">p_scale_tril</span><span class="p">,</span> <span class="n">upper</span><span class="o">=</span><span class="kc">False</span><span class="p">))</span>
<span class="n">term3</span> <span class="o">=</span> <span class="n">_batch_mahalanobis</span><span class="p">(</span><span class="n">q</span><span class="o">.</span><span class="n">_unbroadcasted_scale_tril</span><span class="p">,</span> <span class="p">(</span><span class="n">q</span><span class="o">.</span><span class="n">loc</span> <span class="o">-</span> <span class="n">p</span><span class="o">.</span><span class="n">loc</span><span class="p">))</span>
<span class="k">return</span> <span class="n">half_term1</span> <span class="o">+</span> <span class="mf">0.5</span> <span class="o">*</span> <span class="p">(</span><span class="n">term2</span> <span class="o">+</span> <span class="n">term3</span> <span class="o">-</span> <span class="n">n</span><span class="p">)</span>
<span class="nd">@register_kl</span><span class="p">(</span><span class="n">Normal</span><span class="p">,</span> <span class="n">Normal</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">_kl_normal_normal</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">):</span>
<span class="n">var_ratio</span> <span class="o">=</span> <span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">scale</span> <span class="o">/</span> <span class="n">q</span><span class="o">.</span><span class="n">scale</span><span class="p">)</span><span class="o">.</span><span class="n">pow</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
<span class="n">t1</span> <span class="o">=</span> <span class="p">((</span><span class="n">p</span><span class="o">.</span><span class="n">loc</span> <span class="o">-</span> <span class="n">q</span><span class="o">.</span><span class="n">loc</span><span class="p">)</span> <span class="o">/</span> <span class="n">q</span><span class="o">.</span><span class="n">scale</span><span class="p">)</span><span class="o">.</span><span class="n">pow</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
<span class="k">return</span> <span class="mf">0.5</span> <span class="o">*</span> <span class="p">(</span><span class="n">var_ratio</span> <span class="o">+</span> <span class="n">t1</span> <span class="o">-</span> <span class="mi">1</span> <span class="o">-</span> <span class="n">var_ratio</span><span class="o">.</span><span class="n">log</span><span class="p">())</span>
<span class="nd">@register_kl</span><span class="p">(</span><span class="n">OneHotCategorical</span><span class="p">,</span> <span class="n">OneHotCategorical</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">_kl_onehotcategorical_onehotcategorical</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">):</span>
<span class="k">return</span> <span class="n">_kl_categorical_categorical</span><span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">_categorical</span><span class="p">,</span> <span class="n">q</span><span class="o">.</span><span class="n">_categorical</span><span class="p">)</span>
<span class="nd">@register_kl</span><span class="p">(</span><span class="n">Pareto</span><span class="p">,</span> <span class="n">Pareto</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">_kl_pareto_pareto</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">):</span>
<span class="c1"># From http://www.mast.queensu.ca/~communications/Papers/gil-msc11.pdf</span>
<span class="n">scale_ratio</span> <span class="o">=</span> <span class="n">p</span><span class="o">.</span><span class="n">scale</span> <span class="o">/</span> <span class="n">q</span><span class="o">.</span><span class="n">scale</span>
<span class="n">alpha_ratio</span> <span class="o">=</span> <span class="n">q</span><span class="o">.</span><span class="n">alpha</span> <span class="o">/</span> <span class="n">p</span><span class="o">.</span><span class="n">alpha</span>
<span class="n">t1</span> <span class="o">=</span> <span class="n">q</span><span class="o">.</span><span class="n">alpha</span> <span class="o">*</span> <span class="n">scale_ratio</span><span class="o">.</span><span class="n">log</span><span class="p">()</span>
<span class="n">t2</span> <span class="o">=</span> <span class="o">-</span><span class="n">alpha_ratio</span><span class="o">.</span><span class="n">log</span><span class="p">()</span>
<span class="n">result</span> <span class="o">=</span> <span class="n">t1</span> <span class="o">+</span> <span class="n">t2</span> <span class="o">+</span> <span class="n">alpha_ratio</span> <span class="o">-</span> <span class="mi">1</span>
<span class="n">result</span><span class="p">[</span><span class="n">p</span><span class="o">.</span><span class="n">support</span><span class="o">.</span><span class="n">lower_bound</span> <span class="o"><</span> <span class="n">q</span><span class="o">.</span><span class="n">support</span><span class="o">.</span><span class="n">lower_bound</span><span class="p">]</span> <span class="o">=</span> <span class="n">inf</span>
<span class="k">return</span> <span class="n">result</span>
<span class="nd">@register_kl</span><span class="p">(</span><span class="n">Poisson</span><span class="p">,</span> <span class="n">Poisson</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">_kl_poisson_poisson</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">):</span>
<span class="k">return</span> <span class="n">p</span><span class="o">.</span><span class="n">rate</span> <span class="o">*</span> <span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">rate</span><span class="o">.</span><span class="n">log</span><span class="p">()</span> <span class="o">-</span> <span class="n">q</span><span class="o">.</span><span class="n">rate</span><span class="o">.</span><span class="n">log</span><span class="p">())</span> <span class="o">-</span> <span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">rate</span> <span class="o">-</span> <span class="n">q</span><span class="o">.</span><span class="n">rate</span><span class="p">)</span>
<span class="nd">@register_kl</span><span class="p">(</span><span class="n">TransformedDistribution</span><span class="p">,</span> <span class="n">TransformedDistribution</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">_kl_transformed_transformed</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">):</span>
<span class="k">if</span> <span class="n">p</span><span class="o">.</span><span class="n">transforms</span> <span class="o">!=</span> <span class="n">q</span><span class="o">.</span><span class="n">transforms</span><span class="p">:</span>
<span class="k">raise</span> <span class="ne">NotImplementedError</span>
<span class="k">if</span> <span class="n">p</span><span class="o">.</span><span class="n">event_shape</span> <span class="o">!=</span> <span class="n">q</span><span class="o">.</span><span class="n">event_shape</span><span class="p">:</span>
<span class="k">raise</span> <span class="ne">NotImplementedError</span>
<span class="k">return</span> <span class="n">kl_divergence</span><span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">base_dist</span><span class="p">,</span> <span class="n">q</span><span class="o">.</span><span class="n">base_dist</span><span class="p">)</span>
<span class="nd">@register_kl</span><span class="p">(</span><span class="n">Uniform</span><span class="p">,</span> <span class="n">Uniform</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">_kl_uniform_uniform</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">):</span>
<span class="n">result</span> <span class="o">=</span> <span class="p">((</span><span class="n">q</span><span class="o">.</span><span class="n">high</span> <span class="o">-</span> <span class="n">q</span><span class="o">.</span><span class="n">low</span><span class="p">)</span> <span class="o">/</span> <span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">high</span> <span class="o">-</span> <span class="n">p</span><span class="o">.</span><span class="n">low</span><span class="p">))</span><span class="o">.</span><span class="n">log</span><span class="p">()</span>
<span class="n">result</span><span class="p">[(</span><span class="n">q</span><span class="o">.</span><span class="n">low</span> <span class="o">></span> <span class="n">p</span><span class="o">.</span><span class="n">low</span><span class="p">)</span> <span class="o">|</span> <span class="p">(</span><span class="n">q</span><span class="o">.</span><span class="n">high</span> <span class="o"><</span> <span class="n">p</span><span class="o">.</span><span class="n">high</span><span class="p">)]</span> <span class="o">=</span> <span class="n">inf</span>
<span class="k">return</span> <span class="n">result</span>
<span class="c1"># Different distributions</span>
<span class="nd">@register_kl</span><span class="p">(</span><span class="n">Bernoulli</span><span class="p">,</span> <span class="n">Poisson</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">_kl_bernoulli_poisson</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">):</span>
<span class="k">return</span> <span class="o">-</span><span class="n">p</span><span class="o">.</span><span class="n">entropy</span><span class="p">()</span> <span class="o">-</span> <span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">probs</span> <span class="o">*</span> <span class="n">q</span><span class="o">.</span><span class="n">rate</span><span class="o">.</span><span class="n">log</span><span class="p">()</span> <span class="o">-</span> <span class="n">q</span><span class="o">.</span><span class="n">rate</span><span class="p">)</span>
<span class="nd">@register_kl</span><span class="p">(</span><span class="n">Beta</span><span class="p">,</span> <span class="n">ContinuousBernoulli</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">_kl_beta_continuous_bernoulli</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">):</span>
<span class="k">return</span> <span class="o">-</span><span class="n">p</span><span class="o">.</span><span class="n">entropy</span><span class="p">()</span> <span class="o">-</span> <span class="n">p</span><span class="o">.</span><span class="n">mean</span> <span class="o">*</span> <span class="n">q</span><span class="o">.</span><span class="n">logits</span> <span class="o">-</span> <span class="n">torch</span><span class="o">.</span><span class="n">log1p</span><span class="p">(</span><span class="o">-</span><span class="n">q</span><span class="o">.</span><span class="n">probs</span><span class="p">)</span> <span class="o">-</span> <span class="n">q</span><span class="o">.</span><span class="n">_cont_bern_log_norm</span><span class="p">()</span>
<span class="nd">@register_kl</span><span class="p">(</span><span class="n">Beta</span><span class="p">,</span> <span class="n">Pareto</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">_kl_beta_infinity</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">):</span>
<span class="k">return</span> <span class="n">_infinite_like</span><span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">concentration1</span><span class="p">)</span>
<span class="nd">@register_kl</span><span class="p">(</span><span class="n">Beta</span><span class="p">,</span> <span class="n">Exponential</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">_kl_beta_exponential</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">):</span>
<span class="k">return</span> <span class="o">-</span><span class="n">p</span><span class="o">.</span><span class="n">entropy</span><span class="p">()</span> <span class="o">-</span> <span class="n">q</span><span class="o">.</span><span class="n">rate</span><span class="o">.</span><span class="n">log</span><span class="p">()</span> <span class="o">+</span> <span class="n">q</span><span class="o">.</span><span class="n">rate</span> <span class="o">*</span> <span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">concentration1</span> <span class="o">/</span> <span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">concentration1</span> <span class="o">+</span> <span class="n">p</span><span class="o">.</span><span class="n">concentration0</span><span class="p">))</span>
<span class="nd">@register_kl</span><span class="p">(</span><span class="n">Beta</span><span class="p">,</span> <span class="n">Gamma</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">_kl_beta_gamma</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">):</span>
<span class="n">t1</span> <span class="o">=</span> <span class="o">-</span><span class="n">p</span><span class="o">.</span><span class="n">entropy</span><span class="p">()</span>
<span class="n">t2</span> <span class="o">=</span> <span class="n">q</span><span class="o">.</span><span class="n">concentration</span><span class="o">.</span><span class="n">lgamma</span><span class="p">()</span> <span class="o">-</span> <span class="n">q</span><span class="o">.</span><span class="n">concentration</span> <span class="o">*</span> <span class="n">q</span><span class="o">.</span><span class="n">rate</span><span class="o">.</span><span class="n">log</span><span class="p">()</span>
<span class="n">t3</span> <span class="o">=</span> <span class="p">(</span><span class="n">q</span><span class="o">.</span><span class="n">concentration</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">concentration1</span><span class="o">.</span><span class="n">digamma</span><span class="p">()</span> <span class="o">-</span> <span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">concentration1</span> <span class="o">+</span> <span class="n">p</span><span class="o">.</span><span class="n">concentration0</span><span class="p">)</span><span class="o">.</span><span class="n">digamma</span><span class="p">())</span>
<span class="n">t4</span> <span class="o">=</span> <span class="n">q</span><span class="o">.</span><span class="n">rate</span> <span class="o">*</span> <span class="n">p</span><span class="o">.</span><span class="n">concentration1</span> <span class="o">/</span> <span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">concentration1</span> <span class="o">+</span> <span class="n">p</span><span class="o">.</span><span class="n">concentration0</span><span class="p">)</span>
<span class="k">return</span> <span class="n">t1</span> <span class="o">+</span> <span class="n">t2</span> <span class="o">-</span> <span class="n">t3</span> <span class="o">+</span> <span class="n">t4</span>
<span class="c1"># TODO: Add Beta-Laplace KL Divergence</span>
<span class="nd">@register_kl</span><span class="p">(</span><span class="n">Beta</span><span class="p">,</span> <span class="n">Normal</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">_kl_beta_normal</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">):</span>
<span class="n">E_beta</span> <span class="o">=</span> <span class="n">p</span><span class="o">.</span><span class="n">concentration1</span> <span class="o">/</span> <span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">concentration1</span> <span class="o">+</span> <span class="n">p</span><span class="o">.</span><span class="n">concentration0</span><span class="p">)</span>
<span class="n">var_normal</span> <span class="o">=</span> <span class="n">q</span><span class="o">.</span><span class="n">scale</span><span class="o">.</span><span class="n">pow</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span>
<span class="n">t1</span> <span class="o">=</span> <span class="o">-</span><span class="n">p</span><span class="o">.</span><span class="n">entropy</span><span class="p">()</span>
<span class="n">t2</span> <span class="o">=</span> <span class="mf">0.5</span> <span class="o">*</span> <span class="p">(</span><span class="n">var_normal</span> <span class="o">*</span> <span class="mi">2</span> <span class="o">*</span> <span class="n">math</span><span class="o">.</span><span class="n">pi</span><span class="p">)</span><span class="o">.</span><span class="n">log</span><span class="p">()</span>
<span class="n">t3</span> <span class="o">=</span> <span class="p">(</span><span class="n">E_beta</span> <span class="o">*</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">E_beta</span><span class="p">)</span> <span class="o">/</span> <span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">concentration1</span> <span class="o">+</span> <span class="n">p</span><span class="o">.</span><span class="n">concentration0</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">+</span> <span class="n">E_beta</span><span class="o">.</span><span class="n">pow</span><span class="p">(</span><span class="mi">2</span><span class="p">))</span> <span class="o">*</span> <span class="mf">0.5</span>
<span class="n">t4</span> <span class="o">=</span> <span class="n">q</span><span class="o">.</span><span class="n">loc</span> <span class="o">*</span> <span class="n">E_beta</span>
<span class="n">t5</span> <span class="o">=</span> <span class="n">q</span><span class="o">.</span><span class="n">loc</span><span class="o">.</span><span class="n">pow</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span> <span class="o">*</span> <span class="mf">0.5</span>
<span class="k">return</span> <span class="n">t1</span> <span class="o">+</span> <span class="n">t2</span> <span class="o">+</span> <span class="p">(</span><span class="n">t3</span> <span class="o">-</span> <span class="n">t4</span> <span class="o">+</span> <span class="n">t5</span><span class="p">)</span> <span class="o">/</span> <span class="n">var_normal</span>
<span class="nd">@register_kl</span><span class="p">(</span><span class="n">Beta</span><span class="p">,</span> <span class="n">Uniform</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">_kl_beta_uniform</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">):</span>
<span class="n">result</span> <span class="o">=</span> <span class="o">-</span><span class="n">p</span><span class="o">.</span><span class="n">entropy</span><span class="p">()</span> <span class="o">+</span> <span class="p">(</span><span class="n">q</span><span class="o">.</span><span class="n">high</span> <span class="o">-</span> <span class="n">q</span><span class="o">.</span><span class="n">low</span><span class="p">)</span><span class="o">.</span><span class="n">log</span><span class="p">()</span>
<span class="n">result</span><span class="p">[(</span><span class="n">q</span><span class="o">.</span><span class="n">low</span> <span class="o">></span> <span class="n">p</span><span class="o">.</span><span class="n">support</span><span class="o">.</span><span class="n">lower_bound</span><span class="p">)</span> <span class="o">|</span> <span class="p">(</span><span class="n">q</span><span class="o">.</span><span class="n">high</span> <span class="o"><</span> <span class="n">p</span><span class="o">.</span><span class="n">support</span><span class="o">.</span><span class="n">upper_bound</span><span class="p">)]</span> <span class="o">=</span> <span class="n">inf</span>
<span class="k">return</span> <span class="n">result</span>
<span class="c1"># Note that the KL between a ContinuousBernoulli and Beta has no closed form</span>
<span class="nd">@register_kl</span><span class="p">(</span><span class="n">ContinuousBernoulli</span><span class="p">,</span> <span class="n">Pareto</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">_kl_continuous_bernoulli_infinity</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">):</span>
<span class="k">return</span> <span class="n">_infinite_like</span><span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">probs</span><span class="p">)</span>
<span class="nd">@register_kl</span><span class="p">(</span><span class="n">ContinuousBernoulli</span><span class="p">,</span> <span class="n">Exponential</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">_kl_continuous_bernoulli_exponential</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">):</span>
<span class="k">return</span> <span class="o">-</span><span class="n">p</span><span class="o">.</span><span class="n">entropy</span><span class="p">()</span> <span class="o">-</span> <span class="n">torch</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="n">q</span><span class="o">.</span><span class="n">rate</span><span class="p">)</span> <span class="o">+</span> <span class="n">q</span><span class="o">.</span><span class="n">rate</span> <span class="o">*</span> <span class="n">p</span><span class="o">.</span><span class="n">mean</span>
<span class="c1"># Note that the KL between a ContinuousBernoulli and Gamma has no closed form</span>
<span class="c1"># TODO: Add ContinuousBernoulli-Laplace KL Divergence</span>
<span class="nd">@register_kl</span><span class="p">(</span><span class="n">ContinuousBernoulli</span><span class="p">,</span> <span class="n">Normal</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">_kl_continuous_bernoulli_normal</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">):</span>
<span class="n">t1</span> <span class="o">=</span> <span class="o">-</span><span class="n">p</span><span class="o">.</span><span class="n">entropy</span><span class="p">()</span>
<span class="n">t2</span> <span class="o">=</span> <span class="mf">0.5</span> <span class="o">*</span> <span class="p">(</span><span class="n">math</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="mf">2.</span> <span class="o">*</span> <span class="n">math</span><span class="o">.</span><span class="n">pi</span><span class="p">)</span> <span class="o">+</span> <span class="n">torch</span><span class="o">.</span><span class="n">square</span><span class="p">(</span><span class="n">q</span><span class="o">.</span><span class="n">loc</span> <span class="o">/</span> <span class="n">q</span><span class="o">.</span><span class="n">scale</span><span class="p">))</span> <span class="o">+</span> <span class="n">torch</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="n">q</span><span class="o">.</span><span class="n">scale</span><span class="p">)</span>
<span class="n">t3</span> <span class="o">=</span> <span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">variance</span> <span class="o">+</span> <span class="n">torch</span><span class="o">.</span><span class="n">square</span><span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">mean</span><span class="p">)</span> <span class="o">-</span> <span class="mf">2.</span> <span class="o">*</span> <span class="n">q</span><span class="o">.</span><span class="n">loc</span> <span class="o">*</span> <span class="n">p</span><span class="o">.</span><span class="n">mean</span><span class="p">)</span> <span class="o">/</span> <span class="p">(</span><span class="mf">2.0</span> <span class="o">*</span> <span class="n">torch</span><span class="o">.</span><span class="n">square</span><span class="p">(</span><span class="n">q</span><span class="o">.</span><span class="n">scale</span><span class="p">))</span>
<span class="k">return</span> <span class="n">t1</span> <span class="o">+</span> <span class="n">t2</span> <span class="o">+</span> <span class="n">t3</span>
<span class="nd">@register_kl</span><span class="p">(</span><span class="n">ContinuousBernoulli</span><span class="p">,</span> <span class="n">Uniform</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">_kl_continuous_bernoulli_uniform</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">):</span>
<span class="n">result</span> <span class="o">=</span> <span class="o">-</span><span class="n">p</span><span class="o">.</span><span class="n">entropy</span><span class="p">()</span> <span class="o">+</span> <span class="p">(</span><span class="n">q</span><span class="o">.</span><span class="n">high</span> <span class="o">-</span> <span class="n">q</span><span class="o">.</span><span class="n">low</span><span class="p">)</span><span class="o">.</span><span class="n">log</span><span class="p">()</span>
<span class="k">return</span> <span class="n">torch</span><span class="o">.</span><span class="n">where</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">max</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">ge</span><span class="p">(</span><span class="n">q</span><span class="o">.</span><span class="n">low</span><span class="p">,</span> <span class="n">p</span><span class="o">.</span><span class="n">support</span><span class="o">.</span><span class="n">lower_bound</span><span class="p">),</span>
<span class="n">torch</span><span class="o">.</span><span class="n">le</span><span class="p">(</span><span class="n">q</span><span class="o">.</span><span class="n">high</span><span class="p">,</span> <span class="n">p</span><span class="o">.</span><span class="n">support</span><span class="o">.</span><span class="n">upper_bound</span><span class="p">)),</span>
<span class="n">torch</span><span class="o">.</span><span class="n">ones_like</span><span class="p">(</span><span class="n">result</span><span class="p">)</span> <span class="o">*</span> <span class="n">inf</span><span class="p">,</span> <span class="n">result</span><span class="p">)</span>
<span class="nd">@register_kl</span><span class="p">(</span><span class="n">Exponential</span><span class="p">,</span> <span class="n">Beta</span><span class="p">)</span>
<span class="nd">@register_kl</span><span class="p">(</span><span class="n">Exponential</span><span class="p">,</span> <span class="n">ContinuousBernoulli</span><span class="p">)</span>
<span class="nd">@register_kl</span><span class="p">(</span><span class="n">Exponential</span><span class="p">,</span> <span class="n">Pareto</span><span class="p">)</span>
<span class="nd">@register_kl</span><span class="p">(</span><span class="n">Exponential</span><span class="p">,</span> <span class="n">Uniform</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">_kl_exponential_infinity</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">):</span>
<span class="k">return</span> <span class="n">_infinite_like</span><span class="p">(</span><span class="n">p</span><span class="o">.</span><span class="n">rate</span><span class="p">)</span>
<span class="nd">@register_kl</span><span class="p">(</span><span class="n">Exponential</span><span class="p">,</span> <span class="n">Gamma</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">_kl_exponential_gamma</span><span class="p">(</span><span class="n">p</span><span class="p">,</span> <span class="n">q</span><span class="p">):</span>
<span class="n">ratio</span> <span class="o">=</span> <span class="n">q</span><span class="o">.</span><span class="n">rate</span> <span class="o">/</span> <span class="n">p</span><span class="o">.</span><span class="n">rate</span>