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<main class="content" id="quarto-document-content">
<section id="causalgem" class="level1">
<h1>causalGEM</h1>
<p><a href="https://pypi.org/project/causalGEM/"><img src="https://badge.fury.io/py/causalGEM.svg" class="img-fluid" alt="PyPI version"></a> <a href="https://opensource.org/licenses/MIT"><img src="https://img.shields.io/badge/License-MIT-yellow.svg" class="img-fluid" alt="License: MIT"></a> <a href="https://www.python.org/downloads/"><img src="https://img.shields.io/badge/python-3.8+-blue.svg" class="img-fluid" alt="Python 3.8+"></a></p>
<p><strong>Bivariate causal discovery via the Generative Exposure Model (GEM).</strong></p>
<p>This package implements the GEM framework for determining causal direction between pairs of continuous random variables. The method exploits entropy asymmetries that arise under causal relationships of the form Y = g(X) + ε.</p>
<section id="the-generative-exposure-model" class="level2">
<h2 class="anchored" data-anchor-id="the-generative-exposure-model">The Generative Exposure Model</h2>
<p>In the GEM framework, we model causal relationships as:</p>
<pre><code>Y = g(X) + ε</code></pre>
<p>where: - <strong>X</strong> is the cause (exposure variable) - <strong>g</strong> is a smooth generative function<br>
- <strong>ε</strong> is independent noise - <strong>Y</strong> is the effect (outcome variable)</p>
<p>The key theoretical result is that under this model:</p>
<pre><code>H(Y) - H(X) = E[log|g'(X)|]</code></pre>
<p>This creates detectable asymmetries in marginal entropies that can be used for causal discovery.</p>
</section>
<section id="installation" class="level2">
<h2 class="anchored" data-anchor-id="installation">Installation</h2>
<div class="sourceCode" id="cb3"><pre class="sourceCode bash code-with-copy"><code class="sourceCode bash"><span id="cb3-1"><a href="#cb3-1" aria-hidden="true" tabindex="-1"></a><span class="ex">pip</span> install causalGEM</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<p>For development installation:</p>
<div class="sourceCode" id="cb4"><pre class="sourceCode bash code-with-copy"><code class="sourceCode bash"><span id="cb4-1"><a href="#cb4-1" aria-hidden="true" tabindex="-1"></a><span class="fu">git</span> clone https://github.com/yourusername/causalGEM.git</span>
<span id="cb4-2"><a href="#cb4-2" aria-hidden="true" tabindex="-1"></a><span class="bu">cd</span> causalGEM</span>
<span id="cb4-3"><a href="#cb4-3" aria-hidden="true" tabindex="-1"></a><span class="ex">pip</span> install <span class="at">-e</span> <span class="st">".[dev]"</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</section>
<section id="quick-start" class="level2">
<h2 class="anchored" data-anchor-id="quick-start">Quick Start</h2>
<div class="sourceCode" id="cb5"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb5-1"><a href="#cb5-1" aria-hidden="true" tabindex="-1"></a><span class="im">import</span> numpy <span class="im">as</span> np</span>
<span id="cb5-2"><a href="#cb5-2" aria-hidden="true" tabindex="-1"></a><span class="im">from</span> causalgem <span class="im">import</span> estimate_causal_direction</span>
<span id="cb5-3"><a href="#cb5-3" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb5-4"><a href="#cb5-4" aria-hidden="true" tabindex="-1"></a><span class="co"># Generate data with causal relationship X → Y</span></span>
<span id="cb5-5"><a href="#cb5-5" aria-hidden="true" tabindex="-1"></a>np.random.seed(<span class="dv">42</span>)</span>
<span id="cb5-6"><a href="#cb5-6" aria-hidden="true" tabindex="-1"></a>x <span class="op">=</span> np.random.uniform(<span class="dv">0</span>, <span class="dv">1</span>, <span class="dv">500</span>)</span>
<span id="cb5-7"><a href="#cb5-7" aria-hidden="true" tabindex="-1"></a>y <span class="op">=</span> x<span class="op">**</span><span class="dv">2</span> <span class="op">+</span> np.random.normal(<span class="dv">0</span>, <span class="fl">0.1</span>, <span class="dv">500</span>)</span>
<span id="cb5-8"><a href="#cb5-8" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb5-9"><a href="#cb5-9" aria-hidden="true" tabindex="-1"></a><span class="co"># Estimate causal direction</span></span>
<span id="cb5-10"><a href="#cb5-10" aria-hidden="true" tabindex="-1"></a>result <span class="op">=</span> estimate_causal_direction(x, y)</span>
<span id="cb5-11"><a href="#cb5-11" aria-hidden="true" tabindex="-1"></a><span class="bu">print</span>(result)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<p>Output:</p>
<pre><code>CausalGEMResult(
C_{X→Y} = 0.2847 [0.1523, 0.4172],
decision = 'X → Y',
H(X) = 0.0012, H(Y) = -0.2835,
n = 500
)</code></pre>
</section>
<section id="core-statistic" class="level2">
<h2 class="anchored" data-anchor-id="core-statistic">Core Statistic</h2>
<p>The main statistic is <strong>C_{X→Y} = H(X) - H(Y)</strong>, where H(·) denotes differential entropy.</p>
<table class="caption-top table">
<thead>
<tr class="header">
<th>Value of C</th>
<th>Interpretation</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td>C > 0 (significant)</td>
<td>Evidence for X → Y</td>
</tr>
<tr class="even">
<td>C < 0 (significant)</td>
<td>Evidence for Y → X</td>
</tr>
<tr class="odd">
<td>C ≈ 0 (CI includes 0)</td>
<td>Inconclusive</td>
</tr>
</tbody>
</table>
</section>
<section id="features" class="level2">
<h2 class="anchored" data-anchor-id="features">Features</h2>
<section id="causal-direction-estimation" class="level3">
<h3 class="anchored" data-anchor-id="causal-direction-estimation">Causal Direction Estimation</h3>
<div class="sourceCode" id="cb7"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb7-1"><a href="#cb7-1" aria-hidden="true" tabindex="-1"></a><span class="im">from</span> causalgem <span class="im">import</span> estimate_causal_direction</span>
<span id="cb7-2"><a href="#cb7-2" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb7-3"><a href="#cb7-3" aria-hidden="true" tabindex="-1"></a>result <span class="op">=</span> estimate_causal_direction(x, y, alpha<span class="op">=</span><span class="fl">0.05</span>)</span>
<span id="cb7-4"><a href="#cb7-4" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb7-5"><a href="#cb7-5" aria-hidden="true" tabindex="-1"></a><span class="bu">print</span>(<span class="ss">f"Point estimate: </span><span class="sc">{</span>result<span class="sc">.</span>delta<span class="sc">:.4f}</span><span class="ss">"</span>)</span>
<span id="cb7-6"><a href="#cb7-6" aria-hidden="true" tabindex="-1"></a><span class="bu">print</span>(<span class="ss">f"95% CI: [</span><span class="sc">{</span>result<span class="sc">.</span>ci_lower<span class="sc">:.4f}</span><span class="ss">, </span><span class="sc">{</span>result<span class="sc">.</span>ci_upper<span class="sc">:.4f}</span><span class="ss">]"</span>)</span>
<span id="cb7-7"><a href="#cb7-7" aria-hidden="true" tabindex="-1"></a><span class="bu">print</span>(<span class="ss">f"Direction: </span><span class="sc">{</span>result<span class="sc">.</span>direction_str<span class="sc">}</span><span class="ss">"</span>)</span>
<span id="cb7-8"><a href="#cb7-8" aria-hidden="true" tabindex="-1"></a><span class="bu">print</span>(<span class="ss">f"Significant: </span><span class="sc">{</span>result<span class="sc">.</span>is_significant<span class="sc">}</span><span class="ss">"</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</section>
<section id="diagnostics-for-gem-assumptions" class="level3">
<h3 class="anchored" data-anchor-id="diagnostics-for-gem-assumptions">Diagnostics for GEM Assumptions</h3>
<p>Check model assumptions before applying the method:</p>
<div class="sourceCode" id="cb8"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb8-1"><a href="#cb8-1" aria-hidden="true" tabindex="-1"></a><span class="im">from</span> causalgem <span class="im">import</span> run_diagnostics</span>
<span id="cb8-2"><a href="#cb8-2" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb8-3"><a href="#cb8-3" aria-hidden="true" tabindex="-1"></a>diagnostics <span class="op">=</span> run_diagnostics(x, y, n_bootstrap<span class="op">=</span><span class="dv">500</span>)</span>
<span id="cb8-4"><a href="#cb8-4" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb8-5"><a href="#cb8-5" aria-hidden="true" tabindex="-1"></a><span class="co"># Check functional orthogonality assumption</span></span>
<span id="cb8-6"><a href="#cb8-6" aria-hidden="true" tabindex="-1"></a><span class="bu">print</span>(<span class="ss">f"Orthogonality holds: </span><span class="sc">{</span>diagnostics<span class="sc">.</span>orthogonality<span class="sc">.</span>assumption_holds<span class="sc">}</span><span class="ss">"</span>)</span>
<span id="cb8-7"><a href="#cb8-7" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb8-8"><a href="#cb8-8" aria-hidden="true" tabindex="-1"></a><span class="co"># Analyze function dynamics (contracting vs expanding)</span></span>
<span id="cb8-9"><a href="#cb8-9" aria-hidden="true" tabindex="-1"></a><span class="bu">print</span>(<span class="ss">f"Function type: </span><span class="sc">{</span>diagnostics<span class="sc">.</span>dynamics<span class="sc">.</span>conclusion<span class="sc">}</span><span class="ss">"</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</section>
<section id="stratified-analysis" class="level3">
<h3 class="anchored" data-anchor-id="stratified-analysis">Stratified Analysis</h3>
<p>For data with multiple strata (e.g., by sex or study site):</p>
<div class="sourceCode" id="cb9"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb9-1"><a href="#cb9-1" aria-hidden="true" tabindex="-1"></a><span class="im">from</span> causalgem <span class="im">import</span> estimate_stratified</span>
<span id="cb9-2"><a href="#cb9-2" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb9-3"><a href="#cb9-3" aria-hidden="true" tabindex="-1"></a><span class="co"># sex: 0 = male, 1 = female</span></span>
<span id="cb9-4"><a href="#cb9-4" aria-hidden="true" tabindex="-1"></a>result <span class="op">=</span> estimate_stratified(x, y, strata<span class="op">=</span>sex)</span>
<span id="cb9-5"><a href="#cb9-5" aria-hidden="true" tabindex="-1"></a><span class="bu">print</span>(result)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</section>
<section id="simulation-tools" class="level3">
<h3 class="anchored" data-anchor-id="simulation-tools">Simulation Tools</h3>
<p>Generate synthetic data for validation:</p>
<div class="sourceCode" id="cb10"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb10-1"><a href="#cb10-1" aria-hidden="true" tabindex="-1"></a><span class="im">from</span> causalgem <span class="im">import</span> generate_gem_data, generate_causal_pair</span>
<span id="cb10-2"><a href="#cb10-2" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb10-3"><a href="#cb10-3" aria-hidden="true" tabindex="-1"></a><span class="co"># Simple causal pair with additive noise</span></span>
<span id="cb10-4"><a href="#cb10-4" aria-hidden="true" tabindex="-1"></a>x, y, y_true <span class="op">=</span> generate_causal_pair(</span>
<span id="cb10-5"><a href="#cb10-5" aria-hidden="true" tabindex="-1"></a> n<span class="op">=</span><span class="dv">500</span>, </span>
<span id="cb10-6"><a href="#cb10-6" aria-hidden="true" tabindex="-1"></a> func<span class="op">=</span><span class="st">'square'</span>, <span class="co"># Y = X² + noise</span></span>
<span id="cb10-7"><a href="#cb10-7" aria-hidden="true" tabindex="-1"></a> noise_sd<span class="op">=</span><span class="fl">0.1</span></span>
<span id="cb10-8"><a href="#cb10-8" aria-hidden="true" tabindex="-1"></a>)</span>
<span id="cb10-9"><a href="#cb10-9" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb10-10"><a href="#cb10-10" aria-hidden="true" tabindex="-1"></a><span class="co"># Full GEM simulation with correlated errors</span></span>
<span id="cb10-11"><a href="#cb10-11" aria-hidden="true" tabindex="-1"></a>data <span class="op">=</span> generate_gem_data(</span>
<span id="cb10-12"><a href="#cb10-12" aria-hidden="true" tabindex="-1"></a> n<span class="op">=</span><span class="dv">500</span>,</span>
<span id="cb10-13"><a href="#cb10-13" aria-hidden="true" tabindex="-1"></a> func<span class="op">=</span><span class="st">'exp'</span>,</span>
<span id="cb10-14"><a href="#cb10-14" aria-hidden="true" tabindex="-1"></a> rho<span class="op">=</span><span class="fl">0.2</span>, <span class="co"># Correlation between X and error</span></span>
<span id="cb10-15"><a href="#cb10-15" aria-hidden="true" tabindex="-1"></a> noise_sd<span class="op">=</span><span class="fl">0.1</span></span>
<span id="cb10-16"><a href="#cb10-16" aria-hidden="true" tabindex="-1"></a>)</span>
<span id="cb10-17"><a href="#cb10-17" aria-hidden="true" tabindex="-1"></a>x, y <span class="op">=</span> data[<span class="st">'x'</span>], data[<span class="st">'y'</span>]</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<p><strong>Available generative functions:</strong> <code>'linear'</code>, <code>'square'</code>, <code>'cube'</code>, <code>'sqrt'</code>, <code>'cbrt'</code>, <code>'exp'</code>, <code>'log'</code>, <code>'sin'</code>, <code>'tanh'</code>, <code>'sigmoid'</code></p>
</section>
</section>
<section id="method-details" class="level2">
<h2 class="anchored" data-anchor-id="method-details">Method Details</h2>
<section id="estimation-procedure" class="level3">
<h3 class="anchored" data-anchor-id="estimation-procedure">Estimation Procedure</h3>
<ol type="1">
<li><strong>Sample Splitting</strong>: Data is split into two halves for cross-fitting</li>
<li><strong>Density Estimation</strong>: Kernel density estimation via fastKDE</li>
<li><strong>Entropy Calculation</strong>: H(X) = -E[log f_X(X)]</li>
<li><strong>Variance Estimation</strong>: Delta method using covariance of log-densities</li>
<li><strong>Inference</strong>: Asymptotic confidence intervals and hypothesis testing</li>
</ol>
</section>
<section id="key-assumptions" class="level3">
<h3 class="anchored" data-anchor-id="key-assumptions">Key Assumptions</h3>
<ol type="1">
<li><strong>GEM Structure</strong>: Y = g(X) + ε with independent ε</li>
<li><strong>Functional Orthogonality</strong>: E[log|g’(X)|] equals its population average</li>
<li><strong>Smooth Densities</strong>: Both marginal densities are sufficiently smooth</li>
<li><strong>Bounded Support</strong>: Best performance when data is in bounded range</li>
</ol>
<p>Use <code>run_diagnostics()</code> to check these assumptions.</p>
</section>
</section>
<section id="api-reference" class="level2">
<h2 class="anchored" data-anchor-id="api-reference">API Reference</h2>
<section id="core-functions" class="level3">
<h3 class="anchored" data-anchor-id="core-functions">Core Functions</h3>
<table class="caption-top table">
<thead>
<tr class="header">
<th>Function</th>
<th>Description</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td><code>estimate_causal_direction(x, y)</code></td>
<td>Main estimation function</td>
</tr>
<tr class="even">
<td><code>estimate_entropy_difference(x, y)</code></td>
<td>Alias for main function</td>
</tr>
<tr class="odd">
<td><code>estimate_stratified(x, y, strata)</code></td>
<td>Stratified estimation</td>
</tr>
<tr class="even">
<td><code>estimate_with_decision(x, y)</code></td>
<td>Legacy tuple-returning interface</td>
</tr>
</tbody>
</table>
</section>
<section id="diagnostics" class="level3">
<h3 class="anchored" data-anchor-id="diagnostics">Diagnostics</h3>
<table class="caption-top table">
<thead>
<tr class="header">
<th>Function</th>
<th>Description</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td><code>check_orthogonality(x, y)</code></td>
<td>Check functional orthogonality</td>
</tr>
<tr class="even">
<td><code>analyze_dynamics(x, y)</code></td>
<td>Determine expanding/contracting</td>
</tr>
<tr class="odd">
<td><code>run_diagnostics(x, y)</code></td>
<td>Run all diagnostic checks</td>
</tr>
</tbody>
</table>
</section>
<section id="simulation" class="level3">
<h3 class="anchored" data-anchor-id="simulation">Simulation</h3>
<table class="caption-top table">
<thead>
<tr class="header">
<th>Function</th>
<th>Description</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td><code>generate_gem_data(n, func, rho, ...)</code></td>
<td>Full GEM simulation</td>
</tr>
<tr class="even">
<td><code>generate_causal_pair(n, func, ...)</code></td>
<td>Simple Y = f(X) + ε</td>
</tr>
<tr class="odd">
<td><code>generate_copula_data(n, ...)</code></td>
<td>Arbitrary copula data</td>
</tr>
</tbody>
</table>
</section>
</section>
<section id="examples" class="level2">
<h2 class="anchored" data-anchor-id="examples">Examples</h2>
<section id="benchmark-on-tuebingen-dataset" class="level3">
<h3 class="anchored" data-anchor-id="benchmark-on-tuebingen-dataset">Benchmark on Tuebingen Dataset</h3>
<div class="sourceCode" id="cb11"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb11-1"><a href="#cb11-1" aria-hidden="true" tabindex="-1"></a><span class="im">from</span> causalgem <span class="im">import</span> estimate_causal_direction</span>
<span id="cb11-2"><a href="#cb11-2" aria-hidden="true" tabindex="-1"></a><span class="im">from</span> cdt.data <span class="im">import</span> load_dataset <span class="co"># pip install cdt</span></span>
<span id="cb11-3"><a href="#cb11-3" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb11-4"><a href="#cb11-4" aria-hidden="true" tabindex="-1"></a>data, labels <span class="op">=</span> load_dataset(<span class="st">'tuebingen'</span>)</span>
<span id="cb11-5"><a href="#cb11-5" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb11-6"><a href="#cb11-6" aria-hidden="true" tabindex="-1"></a>correct <span class="op">=</span> <span class="dv">0</span></span>
<span id="cb11-7"><a href="#cb11-7" aria-hidden="true" tabindex="-1"></a><span class="cf">for</span> i <span class="kw">in</span> <span class="bu">range</span>(<span class="bu">len</span>(data)):</span>
<span id="cb11-8"><a href="#cb11-8" aria-hidden="true" tabindex="-1"></a> pair <span class="op">=</span> data.iloc[i]</span>
<span id="cb11-9"><a href="#cb11-9" aria-hidden="true" tabindex="-1"></a> result <span class="op">=</span> estimate_causal_direction(pair[<span class="st">'A'</span>], pair[<span class="st">'B'</span>])</span>
<span id="cb11-10"><a href="#cb11-10" aria-hidden="true" tabindex="-1"></a> <span class="cf">if</span> result.decision <span class="op">==</span> <span class="bu">int</span>(labels.iloc[i]):</span>
<span id="cb11-11"><a href="#cb11-11" aria-hidden="true" tabindex="-1"></a> correct <span class="op">+=</span> <span class="dv">1</span></span>
<span id="cb11-12"><a href="#cb11-12" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb11-13"><a href="#cb11-13" aria-hidden="true" tabindex="-1"></a><span class="bu">print</span>(<span class="ss">f"Accuracy: </span><span class="sc">{</span>correct<span class="op">/</span><span class="bu">len</span>(data)<span class="sc">:.1%}</span><span class="ss">"</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</section>
<section id="real-data-analysis-with-diagnostics" class="level3">
<h3 class="anchored" data-anchor-id="real-data-analysis-with-diagnostics">Real Data Analysis with Diagnostics</h3>
<div class="sourceCode" id="cb12"><pre class="sourceCode python code-with-copy"><code class="sourceCode python"><span id="cb12-1"><a href="#cb12-1" aria-hidden="true" tabindex="-1"></a><span class="im">import</span> pandas <span class="im">as</span> pd</span>
<span id="cb12-2"><a href="#cb12-2" aria-hidden="true" tabindex="-1"></a><span class="im">from</span> causalgem <span class="im">import</span> estimate_causal_direction, run_diagnostics</span>
<span id="cb12-3"><a href="#cb12-3" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb12-4"><a href="#cb12-4" aria-hidden="true" tabindex="-1"></a><span class="co"># Load your data</span></span>
<span id="cb12-5"><a href="#cb12-5" aria-hidden="true" tabindex="-1"></a>df <span class="op">=</span> pd.read_csv(<span class="st">'your_data.csv'</span>)</span>
<span id="cb12-6"><a href="#cb12-6" aria-hidden="true" tabindex="-1"></a>x, y <span class="op">=</span> df[<span class="st">'exposure'</span>].values, df[<span class="st">'outcome'</span>].values</span>
<span id="cb12-7"><a href="#cb12-7" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb12-8"><a href="#cb12-8" aria-hidden="true" tabindex="-1"></a><span class="co"># Check assumptions first</span></span>
<span id="cb12-9"><a href="#cb12-9" aria-hidden="true" tabindex="-1"></a>diag <span class="op">=</span> run_diagnostics(x, y)</span>
<span id="cb12-10"><a href="#cb12-10" aria-hidden="true" tabindex="-1"></a><span class="bu">print</span>(<span class="ss">f"Assumptions satisfied: </span><span class="sc">{</span>diag<span class="sc">.</span>all_passed<span class="sc">}</span><span class="ss">"</span>)</span>
<span id="cb12-11"><a href="#cb12-11" aria-hidden="true" tabindex="-1"></a><span class="bu">print</span>(<span class="ss">f"Function dynamics: </span><span class="sc">{</span>diag<span class="sc">.</span>dynamics<span class="sc">.</span>conclusion<span class="sc">}</span><span class="ss">"</span>)</span>
<span id="cb12-12"><a href="#cb12-12" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb12-13"><a href="#cb12-13" aria-hidden="true" tabindex="-1"></a><span class="co"># Run estimation</span></span>
<span id="cb12-14"><a href="#cb12-14" aria-hidden="true" tabindex="-1"></a>result <span class="op">=</span> estimate_causal_direction(x, y)</span>
<span id="cb12-15"><a href="#cb12-15" aria-hidden="true" tabindex="-1"></a><span class="bu">print</span>(result)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</section>
</section>
<section id="citation" class="level2">
<h2 class="anchored" data-anchor-id="citation">Citation</h2>
<p>If you use this package in your research, please cite:</p>
<div class="sourceCode" id="cb13"><pre class="sourceCode bibtex code-with-copy"><code class="sourceCode bibtex"><span id="cb13-1"><a href="#cb13-1" aria-hidden="true" tabindex="-1"></a><span class="va">@article</span>{<span class="ot">yourname2025gem</span>,</span>
<span id="cb13-2"><a href="#cb13-2" aria-hidden="true" tabindex="-1"></a> <span class="dt">title</span>={Your Paper Title},</span>
<span id="cb13-3"><a href="#cb13-3" aria-hidden="true" tabindex="-1"></a> <span class="dt">author</span>={Your Name and Coauthors},</span>
<span id="cb13-4"><a href="#cb13-4" aria-hidden="true" tabindex="-1"></a> <span class="dt">journal</span>={Journal Name},</span>
<span id="cb13-5"><a href="#cb13-5" aria-hidden="true" tabindex="-1"></a> <span class="dt">year</span>={2025}</span>
<span id="cb13-6"><a href="#cb13-6" aria-hidden="true" tabindex="-1"></a>}</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</section>
<section id="license" class="level2">
<h2 class="anchored" data-anchor-id="license">License</h2>
<p>MIT License - see <a href="LICENSE">LICENSE</a> for details.</p>
</section>
<section id="contributing" class="level2">
<h2 class="anchored" data-anchor-id="contributing">Contributing</h2>
<p>Contributions are welcome! Please see <a href="CONTRIBUTING.md">CONTRIBUTING.md</a> for guidelines.</p>
</section>
</section>
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