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| <li class="toctree-l1 has-children"><a class="reference internal" href="https://scikit-learn.org/1.5/supervised_learning.html">1. Supervised learning</a><details><summary><span class="toctree-toggle" role="presentation"><svg class="svg-inline--fa fa-chevron-down" aria-hidden="true" focusable="false" data-prefix="fas" data-icon="chevron-down" role="img" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" data-fa-i2svg=""><path fill="currentColor" d="M233.4 406.6c12.5 12.5 32.8 12.5 45.3 0l192-192c12.5-12.5 12.5-32.8 0-45.3s-32.8-12.5-45.3 0L256 338.7 86.6 169.4c-12.5-12.5-32.8-12.5-45.3 0s-12.5 32.8 0 45.3l192 192z"></path></svg></span></summary><ul> |
| <li class="toctree-l2"><a class="reference internal" href="https://scikit-learn.org/1.5/modules/linear_model.html">1.1. Linear Models</a></li> |
| <li class="toctree-l2"><a class="reference internal" href="https://scikit-learn.org/1.5/modules/lda_qda.html">1.2. Linear and Quadratic Discriminant Analysis</a></li> |
| <li class="toctree-l2"><a class="reference internal" href="https://scikit-learn.org/1.5/modules/kernel_ridge.html">1.3. Kernel ridge regression</a></li> |
| <li class="toctree-l2"><a class="reference internal" href="https://scikit-learn.org/1.5/modules/svm.html">1.4. Support Vector Machines</a></li> |
| <li class="toctree-l2"><a class="reference internal" href="https://scikit-learn.org/1.5/modules/sgd.html">1.5. Stochastic Gradient Descent</a></li> |
| <li class="toctree-l2"><a class="reference internal" href="https://scikit-learn.org/1.5/modules/neighbors.html">1.6. Nearest Neighbors</a></li> |
| <li class="toctree-l2"><a class="reference internal" href="https://scikit-learn.org/1.5/modules/gaussian_process.html">1.7. Gaussian Processes</a></li> |
| <li class="toctree-l2"><a class="reference internal" href="https://scikit-learn.org/1.5/modules/cross_decomposition.html">1.8. Cross decomposition</a></li> |
| <li class="toctree-l2"><a class="reference internal" href="https://scikit-learn.org/1.5/modules/naive_bayes.html">1.9. Naive Bayes</a></li> |
| <li class="toctree-l2"><a class="reference internal" href="https://scikit-learn.org/1.5/modules/tree.html">1.10. Decision Trees</a></li> |
| <li class="toctree-l2"><a class="reference internal" href="https://scikit-learn.org/1.5/modules/ensemble.html">1.11. Ensembles: Gradient boosting, random forests, bagging, voting, stacking</a></li> |
| <li class="toctree-l2"><a class="reference internal" href="https://scikit-learn.org/1.5/modules/multiclass.html">1.12. Multiclass and multioutput algorithms</a></li> |
| <li class="toctree-l2"><a class="reference internal" href="https://scikit-learn.org/1.5/modules/feature_selection.html">1.13. Feature selection</a></li> |
| <li class="toctree-l2"><a class="reference internal" href="https://scikit-learn.org/1.5/modules/semi_supervised.html">1.14. Semi-supervised learning</a></li> |
| <li class="toctree-l2"><a class="reference internal" href="https://scikit-learn.org/1.5/modules/isotonic.html">1.15. Isotonic regression</a></li> |
| <li class="toctree-l2"><a class="reference internal" href="https://scikit-learn.org/1.5/modules/calibration.html">1.16. Probability calibration</a></li> |
| <li class="toctree-l2"><a class="reference internal" href="https://scikit-learn.org/1.5/modules/neural_networks_supervised.html">1.17. Neural network models (supervised)</a></li> |
| </ul> |
| </details></li> |
| <li class="toctree-l1 current active has-children"><a class="reference internal" href="https://scikit-learn.org/1.5/unsupervised_learning.html">2. Unsupervised learning</a><details open="open"><summary><span class="toctree-toggle" role="presentation"><svg class="svg-inline--fa fa-chevron-down" aria-hidden="true" focusable="false" data-prefix="fas" data-icon="chevron-down" role="img" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" data-fa-i2svg=""><path fill="currentColor" d="M233.4 406.6c12.5 12.5 32.8 12.5 45.3 0l192-192c12.5-12.5 12.5-32.8 0-45.3s-32.8-12.5-45.3 0L256 338.7 86.6 169.4c-12.5-12.5-32.8-12.5-45.3 0s-12.5 32.8 0 45.3l192 192z"></path></svg></span></summary><ul class="current"> |
| <li class="toctree-l2 current active"><a class="current reference internal" href="https://scikit-learn.org/1.5/modules/mixture.html#">2.1. Gaussian mixture models</a></li> |
| <li class="toctree-l2"><a class="reference internal" href="https://scikit-learn.org/1.5/modules/manifold.html">2.2. Manifold learning</a></li> |
| <li class="toctree-l2"><a class="reference internal" href="https://scikit-learn.org/1.5/modules/clustering.html">2.3. Clustering</a></li> |
| <li class="toctree-l2"><a class="reference internal" href="https://scikit-learn.org/1.5/modules/biclustering.html">2.4. Biclustering</a></li> |
| <li class="toctree-l2"><a class="reference internal" href="https://scikit-learn.org/1.5/modules/decomposition.html">2.5. Decomposing signals in components (matrix factorization problems)</a></li> |
| <li class="toctree-l2"><a class="reference internal" href="https://scikit-learn.org/1.5/modules/covariance.html">2.6. Covariance estimation</a></li> |
| <li class="toctree-l2"><a class="reference internal" href="https://scikit-learn.org/1.5/modules/outlier_detection.html">2.7. Novelty and Outlier Detection</a></li> |
| <li class="toctree-l2"><a class="reference internal" href="https://scikit-learn.org/1.5/modules/density.html">2.8. Density Estimation</a></li> |
| <li class="toctree-l2"><a class="reference internal" href="https://scikit-learn.org/1.5/modules/neural_networks_unsupervised.html">2.9. Neural network models (unsupervised)</a></li> |
| </ul> |
| </details></li> |
| <li class="toctree-l1 has-children"><a class="reference internal" href="https://scikit-learn.org/1.5/model_selection.html">3. Model selection and evaluation</a><details><summary><span class="toctree-toggle" role="presentation"><svg class="svg-inline--fa fa-chevron-down" aria-hidden="true" focusable="false" data-prefix="fas" data-icon="chevron-down" role="img" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" data-fa-i2svg=""><path fill="currentColor" d="M233.4 406.6c12.5 12.5 32.8 12.5 45.3 0l192-192c12.5-12.5 12.5-32.8 0-45.3s-32.8-12.5-45.3 0L256 338.7 86.6 169.4c-12.5-12.5-32.8-12.5-45.3 0s-12.5 32.8 0 45.3l192 192z"></path></svg></span></summary><ul> |
| <li class="toctree-l2"><a class="reference internal" href="https://scikit-learn.org/1.5/modules/cross_validation.html">3.1. Cross-validation: evaluating estimator performance</a></li> |
| <li class="toctree-l2"><a class="reference internal" href="https://scikit-learn.org/1.5/modules/grid_search.html">3.2. Tuning the hyper-parameters of an estimator</a></li> |
| <li class="toctree-l2"><a class="reference internal" href="https://scikit-learn.org/1.5/modules/classification_threshold.html">3.3. Tuning the decision threshold for class prediction</a></li> |
| <li class="toctree-l2"><a class="reference internal" href="https://scikit-learn.org/1.5/modules/model_evaluation.html">3.4. Metrics and scoring: quantifying the quality of predictions</a></li> |
| <li class="toctree-l2"><a class="reference internal" href="https://scikit-learn.org/1.5/modules/learning_curve.html">3.5. Validation curves: plotting scores to evaluate models</a></li> |
| </ul> |
| </details></li> |
| <li class="toctree-l1 has-children"><a class="reference internal" href="https://scikit-learn.org/1.5/inspection.html">4. Inspection</a><details><summary><span class="toctree-toggle" role="presentation"><svg class="svg-inline--fa fa-chevron-down" aria-hidden="true" focusable="false" data-prefix="fas" data-icon="chevron-down" role="img" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" data-fa-i2svg=""><path fill="currentColor" d="M233.4 406.6c12.5 12.5 32.8 12.5 45.3 0l192-192c12.5-12.5 12.5-32.8 0-45.3s-32.8-12.5-45.3 0L256 338.7 86.6 169.4c-12.5-12.5-32.8-12.5-45.3 0s-12.5 32.8 0 45.3l192 192z"></path></svg></span></summary><ul> |
| <li class="toctree-l2"><a class="reference internal" href="https://scikit-learn.org/1.5/modules/partial_dependence.html">4.1. Partial Dependence and Individual Conditional Expectation plots</a></li> |
| <li class="toctree-l2"><a class="reference internal" href="https://scikit-learn.org/1.5/modules/permutation_importance.html">4.2. Permutation feature importance</a></li> |
| </ul> |
| </details></li> |
| <li class="toctree-l1"><a class="reference internal" href="https://scikit-learn.org/1.5/visualizations.html">5. Visualizations</a></li> |
| <li class="toctree-l1 has-children"><a class="reference internal" href="https://scikit-learn.org/1.5/data_transforms.html">6. Dataset transformations</a><details><summary><span class="toctree-toggle" role="presentation"><svg class="svg-inline--fa fa-chevron-down" aria-hidden="true" focusable="false" data-prefix="fas" data-icon="chevron-down" role="img" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" data-fa-i2svg=""><path fill="currentColor" d="M233.4 406.6c12.5 12.5 32.8 12.5 45.3 0l192-192c12.5-12.5 12.5-32.8 0-45.3s-32.8-12.5-45.3 0L256 338.7 86.6 169.4c-12.5-12.5-32.8-12.5-45.3 0s-12.5 32.8 0 45.3l192 192z"></path></svg></span></summary><ul> |
| <li class="toctree-l2"><a class="reference internal" href="https://scikit-learn.org/1.5/modules/compose.html">6.1. Pipelines and composite estimators</a></li> |
| <li class="toctree-l2"><a class="reference internal" href="https://scikit-learn.org/1.5/modules/feature_extraction.html">6.2. Feature extraction</a></li> |
| <li class="toctree-l2"><a class="reference internal" href="https://scikit-learn.org/1.5/modules/preprocessing.html">6.3. Preprocessing data</a></li> |
| <li class="toctree-l2"><a class="reference internal" href="https://scikit-learn.org/1.5/modules/impute.html">6.4. Imputation of missing values</a></li> |
| <li class="toctree-l2"><a class="reference internal" href="https://scikit-learn.org/1.5/modules/unsupervised_reduction.html">6.5. Unsupervised dimensionality reduction</a></li> |
| <li class="toctree-l2"><a class="reference internal" href="https://scikit-learn.org/1.5/modules/random_projection.html">6.6. Random Projection</a></li> |
| <li class="toctree-l2"><a class="reference internal" href="https://scikit-learn.org/1.5/modules/kernel_approximation.html">6.7. Kernel Approximation</a></li> |
| <li class="toctree-l2"><a class="reference internal" href="https://scikit-learn.org/1.5/modules/metrics.html">6.8. Pairwise metrics, Affinities and Kernels</a></li> |
| <li class="toctree-l2"><a class="reference internal" href="https://scikit-learn.org/1.5/modules/preprocessing_targets.html">6.9. Transforming the prediction target (<code class="docutils literal notranslate"><span class="pre">y</span></code>)</a></li> |
| </ul> |
| </details></li> |
| <li class="toctree-l1 has-children"><a class="reference internal" href="https://scikit-learn.org/1.5/datasets.html">7. Dataset loading utilities</a><details><summary><span class="toctree-toggle" role="presentation"><svg class="svg-inline--fa fa-chevron-down" aria-hidden="true" focusable="false" data-prefix="fas" data-icon="chevron-down" role="img" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" data-fa-i2svg=""><path fill="currentColor" d="M233.4 406.6c12.5 12.5 32.8 12.5 45.3 0l192-192c12.5-12.5 12.5-32.8 0-45.3s-32.8-12.5-45.3 0L256 338.7 86.6 169.4c-12.5-12.5-32.8-12.5-45.3 0s-12.5 32.8 0 45.3l192 192z"></path></svg></span></summary><ul> |
| <li class="toctree-l2"><a class="reference internal" href="https://scikit-learn.org/1.5/datasets/toy_dataset.html">7.1. Toy datasets</a></li> |
| <li class="toctree-l2"><a class="reference internal" href="https://scikit-learn.org/1.5/datasets/real_world.html">7.2. Real world datasets</a></li> |
| <li class="toctree-l2"><a class="reference internal" href="https://scikit-learn.org/1.5/datasets/sample_generators.html">7.3. Generated datasets</a></li> |
| <li class="toctree-l2"><a class="reference internal" href="https://scikit-learn.org/1.5/datasets/loading_other_datasets.html">7.4. Loading other datasets</a></li> |
| </ul> |
| </details></li> |
| <li class="toctree-l1 has-children"><a class="reference internal" href="https://scikit-learn.org/1.5/computing.html">8. Computing with scikit-learn</a><details><summary><span class="toctree-toggle" role="presentation"><svg class="svg-inline--fa fa-chevron-down" aria-hidden="true" focusable="false" data-prefix="fas" data-icon="chevron-down" role="img" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" data-fa-i2svg=""><path fill="currentColor" d="M233.4 406.6c12.5 12.5 32.8 12.5 45.3 0l192-192c12.5-12.5 12.5-32.8 0-45.3s-32.8-12.5-45.3 0L256 338.7 86.6 169.4c-12.5-12.5-32.8-12.5-45.3 0s-12.5 32.8 0 45.3l192 192z"></path></svg></span></summary><ul> |
| <li class="toctree-l2"><a class="reference internal" href="https://scikit-learn.org/1.5/computing/scaling_strategies.html">8.1. Strategies to scale computationally: bigger data</a></li> |
| <li class="toctree-l2"><a class="reference internal" href="https://scikit-learn.org/1.5/computing/computational_performance.html">8.2. Computational Performance</a></li> |
| <li class="toctree-l2"><a class="reference internal" href="https://scikit-learn.org/1.5/computing/parallelism.html">8.3. Parallelism, resource management, and configuration</a></li> |
| </ul> |
| </details></li> |
| <li class="toctree-l1"><a class="reference internal" href="https://scikit-learn.org/1.5/model_persistence.html">9. Model persistence</a></li> |
| <li class="toctree-l1"><a class="reference internal" href="https://scikit-learn.org/1.5/common_pitfalls.html">10. Common pitfalls and recommended practices</a></li> |
| <li class="toctree-l1 has-children"><a class="reference internal" href="https://scikit-learn.org/1.5/dispatching.html">11. Dispatching</a><details><summary><span class="toctree-toggle" role="presentation"><svg class="svg-inline--fa fa-chevron-down" aria-hidden="true" focusable="false" data-prefix="fas" data-icon="chevron-down" role="img" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" data-fa-i2svg=""><path fill="currentColor" d="M233.4 406.6c12.5 12.5 32.8 12.5 45.3 0l192-192c12.5-12.5 12.5-32.8 0-45.3s-32.8-12.5-45.3 0L256 338.7 86.6 169.4c-12.5-12.5-32.8-12.5-45.3 0s-12.5 32.8 0 45.3l192 192z"></path></svg></span></summary><ul> |
| <li class="toctree-l2"><a class="reference internal" href="https://scikit-learn.org/1.5/modules/array_api.html">11.1. Array API support (experimental)</a></li> |
| </ul> |
| </details></li> |
| <li class="toctree-l1"><a class="reference internal" href="https://scikit-learn.org/1.5/machine_learning_map.html">12. Choosing the right estimator</a></li> |
| <li class="toctree-l1"><a class="reference internal" href="https://scikit-learn.org/1.5/presentations.html">13. External Resources, Videos and Talks</a></li> |
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| <section id="gaussian-mixture-models"> |
| <span id="gmm"></span><span id="mixture"></span><h1><span class="section-number">2.1. </span>Gaussian mixture models<a class="headerlink" href="https://scikit-learn.org/1.5/modules/mixture.html#gaussian-mixture-models" title="Link to this heading">#</a></h1> |
| <p><code class="docutils literal notranslate"><span class="pre">sklearn.mixture</span></code> is a package which enables one to learn |
| Gaussian Mixture Models (diagonal, spherical, tied and full covariance |
| matrices supported), sample them, and estimate them from |
| data. Facilities to help determine the appropriate number of |
| components are also provided.</p> |
| <figure class="align-center" id="id2"> |
| <a class="reference external image-reference" href="https://scikit-learn.org/1.5/auto_examples/mixture/plot_gmm_pdf.html"><img alt="../_images/sphx_glr_plot_gmm_pdf_001.png" src="./2.1. Gaussian mixture models — scikit-learn 1.5.2 documentation_files/sphx_glr_plot_gmm_pdf_001.png" style="width: 320.0px; height: 240.0px;"> |
| </a> |
| <figcaption> |
| <p><span class="caption-text"><strong>Two-component Gaussian mixture model:</strong> <em>data points, and equi-probability |
| surfaces of the model.</em></span><a class="headerlink" href="https://scikit-learn.org/1.5/modules/mixture.html#id2" title="Link to this image">#</a></p> |
| </figcaption> |
| </figure> |
| <p>A Gaussian mixture model is a probabilistic model that assumes all the |
| data points are generated from a mixture of a finite number of |
| Gaussian distributions with unknown parameters. One can think of |
| mixture models as generalizing k-means clustering to incorporate |
| information about the covariance structure of the data as well as the |
| centers of the latent Gaussians.</p> |
| <p>Scikit-learn implements different classes to estimate Gaussian |
| mixture models, that correspond to different estimation strategies, |
| detailed below.</p> |
| <section id="gaussian-mixture"> |
| <h2><span class="section-number">2.1.1. </span>Gaussian Mixture<a class="headerlink" href="https://scikit-learn.org/1.5/modules/mixture.html#gaussian-mixture" title="Link to this heading">#</a></h2> |
| <p>The <a class="reference internal" href="https://scikit-learn.org/1.5/modules/generated/sklearn.mixture.GaussianMixture.html#sklearn.mixture.GaussianMixture" title="sklearn.mixture.GaussianMixture"><code class="xref py py-class docutils literal notranslate"><span class="pre">GaussianMixture</span></code></a> object implements the |
| <a class="reference internal" href="https://scikit-learn.org/1.5/modules/mixture.html#expectation-maximization"><span class="std std-ref">expectation-maximization</span></a> (EM) |
| algorithm for fitting mixture-of-Gaussian models. It can also draw |
| confidence ellipsoids for multivariate models, and compute the |
| Bayesian Information Criterion to assess the number of clusters in the |
| data. A <a class="reference internal" href="https://scikit-learn.org/1.5/modules/generated/sklearn.mixture.GaussianMixture.html#sklearn.mixture.GaussianMixture.fit" title="sklearn.mixture.GaussianMixture.fit"><code class="xref py py-meth docutils literal notranslate"><span class="pre">GaussianMixture.fit</span></code></a> method is provided that learns a Gaussian |
| Mixture Model from train data. Given test data, it can assign to each |
| sample the Gaussian it most probably belongs to using |
| the <a class="reference internal" href="https://scikit-learn.org/1.5/modules/generated/sklearn.mixture.GaussianMixture.html#sklearn.mixture.GaussianMixture.predict" title="sklearn.mixture.GaussianMixture.predict"><code class="xref py py-meth docutils literal notranslate"><span class="pre">GaussianMixture.predict</span></code></a> method.</p> |
| <p>The <a class="reference internal" href="https://scikit-learn.org/1.5/modules/generated/sklearn.mixture.GaussianMixture.html#sklearn.mixture.GaussianMixture" title="sklearn.mixture.GaussianMixture"><code class="xref py py-class docutils literal notranslate"><span class="pre">GaussianMixture</span></code></a> comes with different options to constrain the |
| covariance of the difference classes estimated: spherical, diagonal, tied or |
| full covariance.</p> |
| <figure class="align-center"> |
| <a class="reference external image-reference" href="https://scikit-learn.org/1.5/auto_examples/mixture/plot_gmm_covariances.html"><img alt="../_images/sphx_glr_plot_gmm_covariances_001.png" src="./2.1. Gaussian mixture models — scikit-learn 1.5.2 documentation_files/sphx_glr_plot_gmm_covariances_001.png" style="width: 450.0px; height: 450.0px;"> |
| </a> |
| </figure> |
| <p class="rubric">Examples</p> |
| <ul class="simple"> |
| <li><p>See <a class="reference internal" href="https://scikit-learn.org/1.5/auto_examples/mixture/plot_gmm_covariances.html#sphx-glr-auto-examples-mixture-plot-gmm-covariances-py"><span class="std std-ref">GMM covariances</span></a> for an example of |
| using the Gaussian mixture as clustering on the iris dataset.</p></li> |
| <li><p>See <a class="reference internal" href="https://scikit-learn.org/1.5/auto_examples/mixture/plot_gmm_pdf.html#sphx-glr-auto-examples-mixture-plot-gmm-pdf-py"><span class="std std-ref">Density Estimation for a Gaussian mixture</span></a> for an example on plotting the |
| density estimation.</p></li> |
| </ul> |
| <details class="sd-sphinx-override sd-dropdown sd-card sd-mb-3" id="pros-and-cons-of-class-gaussianmixture" open=""> |
| <summary class="sd-summary-title sd-card-header"> |
| <span class="sd-summary-text">Pros and cons of class GaussianMixture<a class="headerlink" href="https://scikit-learn.org/1.5/modules/mixture.html#pros-and-cons-of-class-gaussianmixture" title="Link to this dropdown">#</a></span><span class="sd-summary-state-marker sd-summary-chevron-right sk-toggle-all" data-bs-toggle="tooltip" data-bs-placement="top" data-bs-offset="0,10" data-bs-title="Toggle all dropdowns"><svg class="svg-inline--fa fa-angles-right" aria-hidden="true" focusable="false" data-prefix="fas" data-icon="angles-right" role="img" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" data-fa-i2svg=""><path fill="currentColor" d="M470.6 278.6c12.5-12.5 12.5-32.8 0-45.3l-160-160c-12.5-12.5-32.8-12.5-45.3 0s-12.5 32.8 0 45.3L402.7 256 265.4 393.4c-12.5 12.5-12.5 32.8 0 45.3s32.8 12.5 45.3 0l160-160zm-352 160l160-160c12.5-12.5 12.5-32.8 0-45.3l-160-160c-12.5-12.5-32.8-12.5-45.3 0s-12.5 32.8 0 45.3L210.7 256 73.4 393.4c-12.5 12.5-12.5 32.8 0 45.3s32.8 12.5 45.3 0z"></path></svg></span><span class="sd-summary-state-marker sd-summary-chevron-right"><svg version="1.1" width="1.5em" height="1.5em" class="sd-octicon sd-octicon-chevron-right" viewBox="0 0 24 24" aria-hidden="true"><path d="M8.72 18.78a.75.75 0 0 1 0-1.06L14.44 12 8.72 6.28a.751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018l6.25 6.25a.75.75 0 0 1 0 1.06l-6.25 6.25a.75.75 0 0 1-1.06 0Z"></path></svg></span></summary><div class="sd-summary-content sd-card-body docutils"> |
| <p class="rubric">Pros</p> |
| <dl class="field-list simple"> |
| <dt class="field-odd">Speed<span class="colon">:</span></dt> |
| <dd class="field-odd"><p class="sd-card-text">It is the fastest algorithm for learning mixture models</p> |
| </dd> |
| <dt class="field-even">Agnostic<span class="colon">:</span></dt> |
| <dd class="field-even"><p class="sd-card-text">As this algorithm maximizes only the likelihood, it |
| will not bias the means towards zero, or bias the cluster sizes to |
| have specific structures that might or might not apply.</p> |
| </dd> |
| </dl> |
| <p class="rubric">Cons</p> |
| <dl class="field-list simple"> |
| <dt class="field-odd">Singularities<span class="colon">:</span></dt> |
| <dd class="field-odd"><p class="sd-card-text">When one has insufficiently many points per |
| mixture, estimating the covariance matrices becomes difficult, |
| and the algorithm is known to diverge and find solutions with |
| infinite likelihood unless one regularizes the covariances artificially.</p> |
| </dd> |
| <dt class="field-even">Number of components<span class="colon">:</span></dt> |
| <dd class="field-even"><p class="sd-card-text">This algorithm will always use all the |
| components it has access to, needing held-out data |
| or information theoretical criteria to decide how many components to use |
| in the absence of external cues.</p> |
| </dd> |
| </dl> |
| </div> |
| </details><details class="sd-sphinx-override sd-dropdown sd-card sd-mb-3" id="selecting-the-number-of-components-in-a-classical-gaussian-mixture-model" open=""> |
| <summary class="sd-summary-title sd-card-header"> |
| <span class="sd-summary-text">Selecting the number of components in a classical Gaussian Mixture model<a class="headerlink" href="https://scikit-learn.org/1.5/modules/mixture.html#selecting-the-number-of-components-in-a-classical-gaussian-mixture-model" title="Link to this dropdown">#</a></span><span class="sd-summary-state-marker sd-summary-chevron-right sk-toggle-all" data-bs-toggle="tooltip" data-bs-placement="top" data-bs-offset="0,10" data-bs-title="Toggle all dropdowns"><svg class="svg-inline--fa fa-angles-right" aria-hidden="true" focusable="false" data-prefix="fas" data-icon="angles-right" role="img" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" data-fa-i2svg=""><path fill="currentColor" d="M470.6 278.6c12.5-12.5 12.5-32.8 0-45.3l-160-160c-12.5-12.5-32.8-12.5-45.3 0s-12.5 32.8 0 45.3L402.7 256 265.4 393.4c-12.5 12.5-12.5 32.8 0 45.3s32.8 12.5 45.3 0l160-160zm-352 160l160-160c12.5-12.5 12.5-32.8 0-45.3l-160-160c-12.5-12.5-32.8-12.5-45.3 0s-12.5 32.8 0 45.3L210.7 256 73.4 393.4c-12.5 12.5-12.5 32.8 0 45.3s32.8 12.5 45.3 0z"></path></svg></span><span class="sd-summary-state-marker sd-summary-chevron-right"><svg version="1.1" width="1.5em" height="1.5em" class="sd-octicon sd-octicon-chevron-right" viewBox="0 0 24 24" aria-hidden="true"><path d="M8.72 18.78a.75.75 0 0 1 0-1.06L14.44 12 8.72 6.28a.751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018l6.25 6.25a.75.75 0 0 1 0 1.06l-6.25 6.25a.75.75 0 0 1-1.06 0Z"></path></svg></span></summary><div class="sd-summary-content sd-card-body docutils"> |
| <p class="sd-card-text">The BIC criterion can be used to select the number of components in a Gaussian |
| Mixture in an efficient way. In theory, it recovers the true number of |
| components only in the asymptotic regime (i.e. if much data is available and |
| assuming that the data was actually generated i.i.d. from a mixture of Gaussian |
| distribution). Note that using a <a class="reference internal" href="https://scikit-learn.org/1.5/modules/mixture.html#bgmm"><span class="std std-ref">Variational Bayesian Gaussian mixture</span></a> |
| avoids the specification of the number of components for a Gaussian mixture |
| model.</p> |
| <figure class="align-center"> |
| <a class="reference external image-reference" href="https://scikit-learn.org/1.5/auto_examples/mixture/plot_gmm_selection.html"><img alt="../_images/sphx_glr_plot_gmm_selection_002.png" src="./2.1. Gaussian mixture models — scikit-learn 1.5.2 documentation_files/sphx_glr_plot_gmm_selection_002.png" style="width: 320.5px; height: 250.0px;"> |
| </a> |
| </figure> |
| <p class="rubric">Examples</p> |
| <ul class="simple"> |
| <li><p class="sd-card-text">See <a class="reference internal" href="https://scikit-learn.org/1.5/auto_examples/mixture/plot_gmm_selection.html#sphx-glr-auto-examples-mixture-plot-gmm-selection-py"><span class="std std-ref">Gaussian Mixture Model Selection</span></a> for an example |
| of model selection performed with classical Gaussian mixture.</p></li> |
| </ul> |
| </div> |
| </details><details class="sd-sphinx-override sd-dropdown sd-card sd-mb-3" id="expectation-maximization" open=""> |
| <span id="estimation-algorithm-expectation-maximization"></span><summary class="sd-summary-title sd-card-header"> |
| <span class="sd-summary-text">Estimation algorithm expectation-maximization<a class="headerlink" href="https://scikit-learn.org/1.5/modules/mixture.html#estimation-algorithm-expectation-maximization" title="Link to this dropdown">#</a></span><span class="sd-summary-state-marker sd-summary-chevron-right sk-toggle-all" data-bs-toggle="tooltip" data-bs-placement="top" data-bs-offset="0,10" data-bs-title="Toggle all dropdowns"><svg class="svg-inline--fa fa-angles-right" aria-hidden="true" focusable="false" data-prefix="fas" data-icon="angles-right" role="img" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" data-fa-i2svg=""><path fill="currentColor" d="M470.6 278.6c12.5-12.5 12.5-32.8 0-45.3l-160-160c-12.5-12.5-32.8-12.5-45.3 0s-12.5 32.8 0 45.3L402.7 256 265.4 393.4c-12.5 12.5-12.5 32.8 0 45.3s32.8 12.5 45.3 0l160-160zm-352 160l160-160c12.5-12.5 12.5-32.8 0-45.3l-160-160c-12.5-12.5-32.8-12.5-45.3 0s-12.5 32.8 0 45.3L210.7 256 73.4 393.4c-12.5 12.5-12.5 32.8 0 45.3s32.8 12.5 45.3 0z"></path></svg></span><span class="sd-summary-state-marker sd-summary-chevron-right"><svg version="1.1" width="1.5em" height="1.5em" class="sd-octicon sd-octicon-chevron-right" viewBox="0 0 24 24" aria-hidden="true"><path d="M8.72 18.78a.75.75 0 0 1 0-1.06L14.44 12 8.72 6.28a.751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018l6.25 6.25a.75.75 0 0 1 0 1.06l-6.25 6.25a.75.75 0 0 1-1.06 0Z"></path></svg></span></summary><div class="sd-summary-content sd-card-body docutils"> |
| <p class="sd-card-text">The main difficulty in learning Gaussian mixture models from unlabeled |
| data is that one usually doesn’t know which points came from |
| which latent component (if one has access to this information it gets |
| very easy to fit a separate Gaussian distribution to each set of |
| points). <a class="reference external" href="https://en.wikipedia.org/wiki/Expectation%E2%80%93maximization_algorithm">Expectation-maximization</a> |
| is a well-founded statistical |
| algorithm to get around this problem by an iterative process. First |
| one assumes random components (randomly centered on data points, |
| learned from k-means, or even just normally distributed around the |
| origin) and computes for each point a probability of being generated by |
| each component of the model. Then, one tweaks the |
| parameters to maximize the likelihood of the data given those |
| assignments. Repeating this process is guaranteed to always converge |
| to a local optimum.</p> |
| </div> |
| </details><details class="sd-sphinx-override sd-dropdown sd-card sd-mb-3" id="choice-of-the-initialization-method"> |
| <summary class="sd-summary-title sd-card-header"> |
| <span class="sd-summary-text">Choice of the Initialization method<a class="headerlink" href="https://scikit-learn.org/1.5/modules/mixture.html#choice-of-the-initialization-method" title="Link to this dropdown">#</a></span><span class="sd-summary-state-marker sd-summary-chevron-right sk-toggle-all" data-bs-toggle="tooltip" data-bs-placement="top" data-bs-offset="0,10" data-bs-title="Toggle all dropdowns"><svg class="svg-inline--fa fa-angles-right" aria-hidden="true" focusable="false" data-prefix="fas" data-icon="angles-right" role="img" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" data-fa-i2svg=""><path fill="currentColor" d="M470.6 278.6c12.5-12.5 12.5-32.8 0-45.3l-160-160c-12.5-12.5-32.8-12.5-45.3 0s-12.5 32.8 0 45.3L402.7 256 265.4 393.4c-12.5 12.5-12.5 32.8 0 45.3s32.8 12.5 45.3 0l160-160zm-352 160l160-160c12.5-12.5 12.5-32.8 0-45.3l-160-160c-12.5-12.5-32.8-12.5-45.3 0s-12.5 32.8 0 45.3L210.7 256 73.4 393.4c-12.5 12.5-12.5 32.8 0 45.3s32.8 12.5 45.3 0z"></path></svg></span><span class="sd-summary-state-marker sd-summary-chevron-right"><svg version="1.1" width="1.5em" height="1.5em" class="sd-octicon sd-octicon-chevron-right" viewBox="0 0 24 24" aria-hidden="true"><path d="M8.72 18.78a.75.75 0 0 1 0-1.06L14.44 12 8.72 6.28a.751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018l6.25 6.25a.75.75 0 0 1 0 1.06l-6.25 6.25a.75.75 0 0 1-1.06 0Z"></path></svg></span></summary><div class="sd-summary-content sd-card-body docutils"> |
| <p class="sd-card-text">There is a choice of four initialization methods (as well as inputting user defined |
| initial means) to generate the initial centers for the model components:</p> |
| <dl class="simple"> |
| <dt>k-means (default)</dt><dd><p class="sd-card-text">This applies a traditional k-means clustering algorithm. |
| This can be computationally expensive compared to other initialization methods.</p> |
| </dd> |
| <dt>k-means++</dt><dd><p class="sd-card-text">This uses the initialization method of k-means clustering: k-means++. |
| This will pick the first center at random from the data. Subsequent centers will be |
| chosen from a weighted distribution of the data favouring points further away from |
| existing centers. k-means++ is the default initialization for k-means so will be |
| quicker than running a full k-means but can still take a significant amount of |
| time for large data sets with many components.</p> |
| </dd> |
| <dt>random_from_data</dt><dd><p class="sd-card-text">This will pick random data points from the input data as the initial |
| centers. This is a very fast method of initialization but can produce non-convergent |
| results if the chosen points are too close to each other.</p> |
| </dd> |
| <dt>random</dt><dd><p class="sd-card-text">Centers are chosen as a small perturbation away from the mean of all data. |
| This method is simple but can lead to the model taking longer to converge.</p> |
| </dd> |
| </dl> |
| <figure class="align-center"> |
| <a class="reference external image-reference" href="https://scikit-learn.org/1.5/auto_examples/mixture/plot_gmm_init.html"><img alt="../_images/sphx_glr_plot_gmm_init_001.png" src="./2.1. Gaussian mixture models — scikit-learn 1.5.2 documentation_files/sphx_glr_plot_gmm_init_001.png" style="width: 400.0px; height: 300.0px;"> |
| </a> |
| </figure> |
| <p class="rubric">Examples</p> |
| <ul class="simple"> |
| <li><p class="sd-card-text">See <a class="reference internal" href="https://scikit-learn.org/1.5/auto_examples/mixture/plot_gmm_init.html#sphx-glr-auto-examples-mixture-plot-gmm-init-py"><span class="std std-ref">GMM Initialization Methods</span></a> for an example of |
| using different initializations in Gaussian Mixture.</p></li> |
| </ul> |
| </div> |
| </details></section> |
| <section id="variational-bayesian-gaussian-mixture"> |
| <span id="bgmm"></span><h2><span class="section-number">2.1.2. </span>Variational Bayesian Gaussian Mixture<a class="headerlink" href="https://scikit-learn.org/1.5/modules/mixture.html#variational-bayesian-gaussian-mixture" title="Link to this heading">#</a></h2> |
| <p>The <a class="reference internal" href="https://scikit-learn.org/1.5/modules/generated/sklearn.mixture.BayesianGaussianMixture.html#sklearn.mixture.BayesianGaussianMixture" title="sklearn.mixture.BayesianGaussianMixture"><code class="xref py py-class docutils literal notranslate"><span class="pre">BayesianGaussianMixture</span></code></a> object implements a variant of the |
| Gaussian mixture model with variational inference algorithms. The API is |
| similar to the one defined by <a class="reference internal" href="https://scikit-learn.org/1.5/modules/generated/sklearn.mixture.GaussianMixture.html#sklearn.mixture.GaussianMixture" title="sklearn.mixture.GaussianMixture"><code class="xref py py-class docutils literal notranslate"><span class="pre">GaussianMixture</span></code></a>.</p> |
| <p id="variational-inference"><strong>Estimation algorithm: variational inference</strong></p> |
| <p>Variational inference is an extension of expectation-maximization that |
| maximizes a lower bound on model evidence (including |
| priors) instead of data likelihood. The principle behind |
| variational methods is the same as expectation-maximization (that is |
| both are iterative algorithms that alternate between finding the |
| probabilities for each point to be generated by each mixture and |
| fitting the mixture to these assigned points), but variational |
| methods add regularization by integrating information from prior |
| distributions. This avoids the singularities often found in |
| expectation-maximization solutions but introduces some subtle biases |
| to the model. Inference is often notably slower, but not usually as |
| much so as to render usage unpractical.</p> |
| <p>Due to its Bayesian nature, the variational algorithm needs more hyperparameters |
| than expectation-maximization, the most important of these being the |
| concentration parameter <code class="docutils literal notranslate"><span class="pre">weight_concentration_prior</span></code>. Specifying a low value |
| for the concentration prior will make the model put most of the weight on a few |
| components and set the remaining components’ weights very close to zero. High |
| values of the concentration prior will allow a larger number of components to |
| be active in the mixture.</p> |
| <p>The parameters implementation of the <a class="reference internal" href="https://scikit-learn.org/1.5/modules/generated/sklearn.mixture.BayesianGaussianMixture.html#sklearn.mixture.BayesianGaussianMixture" title="sklearn.mixture.BayesianGaussianMixture"><code class="xref py py-class docutils literal notranslate"><span class="pre">BayesianGaussianMixture</span></code></a> class |
| proposes two types of prior for the weights distribution: a finite mixture model |
| with Dirichlet distribution and an infinite mixture model with the Dirichlet |
| Process. In practice Dirichlet Process inference algorithm is approximated and |
| uses a truncated distribution with a fixed maximum number of components (called |
| the Stick-breaking representation). The number of components actually used |
| almost always depends on the data.</p> |
| <p>The next figure compares the results obtained for the different type of the |
| weight concentration prior (parameter <code class="docutils literal notranslate"><span class="pre">weight_concentration_prior_type</span></code>) |
| for different values of <code class="docutils literal notranslate"><span class="pre">weight_concentration_prior</span></code>. |
| Here, we can see the value of the <code class="docutils literal notranslate"><span class="pre">weight_concentration_prior</span></code> parameter |
| has a strong impact on the effective number of active components obtained. We |
| can also notice that large values for the concentration weight prior lead to |
| more uniform weights when the type of prior is ‘dirichlet_distribution’ while |
| this is not necessarily the case for the ‘dirichlet_process’ type (used by |
| default).</p> |
| <p class="centered"> |
| <strong><a class="reference external" href="https://scikit-learn.org/1.5/auto_examples/mixture/plot_concentration_prior.html"><img alt="plot_bgmm" src="./2.1. Gaussian mixture models — scikit-learn 1.5.2 documentation_files/sphx_glr_plot_concentration_prior_001.png" style="width: 676.8px; height: 384.0px;"></a> <a class="reference external" href="https://scikit-learn.org/1.5/auto_examples/mixture/plot_concentration_prior.html"><img alt="plot_dpgmm" src="./2.1. Gaussian mixture models — scikit-learn 1.5.2 documentation_files/sphx_glr_plot_concentration_prior_002.png" style="width: 676.8px; height: 384.0px;"></a></strong></p><p>The examples below compare Gaussian mixture models with a fixed number of |
| components, to the variational Gaussian mixture models with a Dirichlet process |
| prior. Here, a classical Gaussian mixture is fitted with 5 components on a |
| dataset composed of 2 clusters. We can see that the variational Gaussian mixture |
| with a Dirichlet process prior is able to limit itself to only 2 components |
| whereas the Gaussian mixture fits the data with a fixed number of components |
| that has to be set a priori by the user. In this case the user has selected |
| <code class="docutils literal notranslate"><span class="pre">n_components=5</span></code> which does not match the true generative distribution of this |
| toy dataset. Note that with very little observations, the variational Gaussian |
| mixture models with a Dirichlet process prior can take a conservative stand, and |
| fit only one component.</p> |
| <figure class="align-center"> |
| <a class="reference external image-reference" href="https://scikit-learn.org/1.5/auto_examples/mixture/plot_gmm.html"><img alt="../_images/sphx_glr_plot_gmm_001.png" src="./2.1. Gaussian mixture models — scikit-learn 1.5.2 documentation_files/sphx_glr_plot_gmm_001.png" style="width: 448.0px; height: 336.0px;"> |
| </a> |
| </figure> |
| <p>On the following figure we are fitting a dataset not well-depicted by a |
| Gaussian mixture. Adjusting the <code class="docutils literal notranslate"><span class="pre">weight_concentration_prior</span></code>, parameter of the |
| <a class="reference internal" href="https://scikit-learn.org/1.5/modules/generated/sklearn.mixture.BayesianGaussianMixture.html#sklearn.mixture.BayesianGaussianMixture" title="sklearn.mixture.BayesianGaussianMixture"><code class="xref py py-class docutils literal notranslate"><span class="pre">BayesianGaussianMixture</span></code></a> controls the number of components used to fit |
| this data. We also present on the last two plots a random sampling generated |
| from the two resulting mixtures.</p> |
| <figure class="align-center"> |
| <a class="reference external image-reference" href="https://scikit-learn.org/1.5/auto_examples/mixture/plot_gmm_sin.html"><img alt="../_images/sphx_glr_plot_gmm_sin_001.png" src="./2.1. Gaussian mixture models — scikit-learn 1.5.2 documentation_files/sphx_glr_plot_gmm_sin_001.png" style="width: 650.0px; height: 650.0px;"> |
| </a> |
| </figure> |
| <p class="rubric">Examples</p> |
| <ul class="simple"> |
| <li><p>See <a class="reference internal" href="https://scikit-learn.org/1.5/auto_examples/mixture/plot_gmm.html#sphx-glr-auto-examples-mixture-plot-gmm-py"><span class="std std-ref">Gaussian Mixture Model Ellipsoids</span></a> for an example on |
| plotting the confidence ellipsoids for both <a class="reference internal" href="https://scikit-learn.org/1.5/modules/generated/sklearn.mixture.GaussianMixture.html#sklearn.mixture.GaussianMixture" title="sklearn.mixture.GaussianMixture"><code class="xref py py-class docutils literal notranslate"><span class="pre">GaussianMixture</span></code></a> |
| and <a class="reference internal" href="https://scikit-learn.org/1.5/modules/generated/sklearn.mixture.BayesianGaussianMixture.html#sklearn.mixture.BayesianGaussianMixture" title="sklearn.mixture.BayesianGaussianMixture"><code class="xref py py-class docutils literal notranslate"><span class="pre">BayesianGaussianMixture</span></code></a>.</p></li> |
| <li><p><a class="reference internal" href="https://scikit-learn.org/1.5/auto_examples/mixture/plot_gmm_sin.html#sphx-glr-auto-examples-mixture-plot-gmm-sin-py"><span class="std std-ref">Gaussian Mixture Model Sine Curve</span></a> shows using |
| <a class="reference internal" href="https://scikit-learn.org/1.5/modules/generated/sklearn.mixture.GaussianMixture.html#sklearn.mixture.GaussianMixture" title="sklearn.mixture.GaussianMixture"><code class="xref py py-class docutils literal notranslate"><span class="pre">GaussianMixture</span></code></a> and <a class="reference internal" href="https://scikit-learn.org/1.5/modules/generated/sklearn.mixture.BayesianGaussianMixture.html#sklearn.mixture.BayesianGaussianMixture" title="sklearn.mixture.BayesianGaussianMixture"><code class="xref py py-class docutils literal notranslate"><span class="pre">BayesianGaussianMixture</span></code></a> to fit a |
| sine wave.</p></li> |
| <li><p>See <a class="reference internal" href="https://scikit-learn.org/1.5/auto_examples/mixture/plot_concentration_prior.html#sphx-glr-auto-examples-mixture-plot-concentration-prior-py"><span class="std std-ref">Concentration Prior Type Analysis of Variation Bayesian Gaussian Mixture</span></a> |
| for an example plotting the confidence ellipsoids for the |
| <a class="reference internal" href="https://scikit-learn.org/1.5/modules/generated/sklearn.mixture.BayesianGaussianMixture.html#sklearn.mixture.BayesianGaussianMixture" title="sklearn.mixture.BayesianGaussianMixture"><code class="xref py py-class docutils literal notranslate"><span class="pre">BayesianGaussianMixture</span></code></a> with different |
| <code class="docutils literal notranslate"><span class="pre">weight_concentration_prior_type</span></code> for different values of the parameter |
| <code class="docutils literal notranslate"><span class="pre">weight_concentration_prior</span></code>.</p></li> |
| </ul> |
| <details class="sd-sphinx-override sd-dropdown sd-card sd-mb-3" id="pros-and-cons-of-variational-inference-with-bayesiangaussianmixture"> |
| <summary class="sd-summary-title sd-card-header"> |
| <span class="sd-summary-text">Pros and cons of variational inference with BayesianGaussianMixture<a class="headerlink" href="https://scikit-learn.org/1.5/modules/mixture.html#pros-and-cons-of-variational-inference-with-bayesiangaussianmixture" title="Link to this dropdown">#</a></span><span class="sd-summary-state-marker sd-summary-chevron-right sk-toggle-all" data-bs-toggle="tooltip" data-bs-placement="top" data-bs-offset="0,10" data-bs-title="Toggle all dropdowns"><svg class="svg-inline--fa fa-angles-right" aria-hidden="true" focusable="false" data-prefix="fas" data-icon="angles-right" role="img" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" data-fa-i2svg=""><path fill="currentColor" d="M470.6 278.6c12.5-12.5 12.5-32.8 0-45.3l-160-160c-12.5-12.5-32.8-12.5-45.3 0s-12.5 32.8 0 45.3L402.7 256 265.4 393.4c-12.5 12.5-12.5 32.8 0 45.3s32.8 12.5 45.3 0l160-160zm-352 160l160-160c12.5-12.5 12.5-32.8 0-45.3l-160-160c-12.5-12.5-32.8-12.5-45.3 0s-12.5 32.8 0 45.3L210.7 256 73.4 393.4c-12.5 12.5-12.5 32.8 0 45.3s32.8 12.5 45.3 0z"></path></svg></span><span class="sd-summary-state-marker sd-summary-chevron-right"><svg version="1.1" width="1.5em" height="1.5em" class="sd-octicon sd-octicon-chevron-right" viewBox="0 0 24 24" aria-hidden="true"><path d="M8.72 18.78a.75.75 0 0 1 0-1.06L14.44 12 8.72 6.28a.751.751 0 0 1 .018-1.042.751.751 0 0 1 1.042-.018l6.25 6.25a.75.75 0 0 1 0 1.06l-6.25 6.25a.75.75 0 0 1-1.06 0Z"></path></svg></span></summary><div class="sd-summary-content sd-card-body docutils"> |
| <p class="rubric">Pros</p> |
| <dl class="field-list simple"> |
| <dt class="field-odd">Automatic selection<span class="colon">:</span></dt> |
| <dd class="field-odd"><p class="sd-card-text">When <code class="docutils literal notranslate"><span class="pre">weight_concentration_prior</span></code> is small enough and |
| <code class="docutils literal notranslate"><span class="pre">n_components</span></code> is larger than what is found necessary by the model, the |
| Variational Bayesian mixture model has a natural tendency to set some mixture |
| weights values close to zero. This makes it possible to let the model choose |
| a suitable number of effective components automatically. Only an upper bound |
| of this number needs to be provided. Note however that the “ideal” number of |
| active components is very application specific and is typically ill-defined |
| in a data exploration setting.</p> |
| </dd> |
| <dt class="field-even">Less sensitivity to the number of parameters<span class="colon">:</span></dt> |
| <dd class="field-even"><p class="sd-card-text">Unlike finite models, which will |
| almost always use all components as much as they can, and hence will produce |
| wildly different solutions for different numbers of components, the |
| variational inference with a Dirichlet process prior |
| (<code class="docutils literal notranslate"><span class="pre">weight_concentration_prior_type='dirichlet_process'</span></code>) won’t change much |
| with changes to the parameters, leading to more stability and less tuning.</p> |
| </dd> |
| <dt class="field-odd">Regularization<span class="colon">:</span></dt> |
| <dd class="field-odd"><p class="sd-card-text">Due to the incorporation of prior information, |
| variational solutions have less pathological special cases than |
| expectation-maximization solutions.</p> |
| </dd> |
| </dl> |
| <p class="rubric">Cons</p> |
| <dl class="field-list simple"> |
| <dt class="field-odd">Speed<span class="colon">:</span></dt> |
| <dd class="field-odd"><p class="sd-card-text">The extra parametrization necessary for variational inference makes |
| inference slower, although not by much.</p> |
| </dd> |
| <dt class="field-even">Hyperparameters<span class="colon">:</span></dt> |
| <dd class="field-even"><p class="sd-card-text">This algorithm needs an extra hyperparameter |
| that might need experimental tuning via cross-validation.</p> |
| </dd> |
| <dt class="field-odd">Bias<span class="colon">:</span></dt> |
| <dd class="field-odd"><p class="sd-card-text">There are many implicit biases in the inference algorithms (and also in |
| the Dirichlet process if used), and whenever there is a mismatch between |
| these biases and the data it might be possible to fit better models using a |
| finite mixture.</p> |
| </dd> |
| </dl> |
| </div> |
| </details><section id="the-dirichlet-process"> |
| <span id="dirichlet-process"></span><h3><span class="section-number">2.1.2.1. </span>The Dirichlet Process<a class="headerlink" href="https://scikit-learn.org/1.5/modules/mixture.html#the-dirichlet-process" title="Link to this heading">#</a></h3> |
| <p>Here we describe variational inference algorithms on Dirichlet process |
| mixture. The Dirichlet process is a prior probability distribution on |
| <em>clusterings with an infinite, unbounded, number of partitions</em>. |
| Variational techniques let us incorporate this prior structure on |
| Gaussian mixture models at almost no penalty in inference time, comparing |
| with a finite Gaussian mixture model.</p> |
| <p>An important question is how can the Dirichlet process use an infinite, |
| unbounded number of clusters and still be consistent. While a full explanation |
| doesn’t fit this manual, one can think of its <a class="reference external" href="https://en.wikipedia.org/wiki/Dirichlet_process#The_stick-breaking_process">stick breaking process</a> |
| analogy to help understanding it. The stick breaking process is a generative |
| story for the Dirichlet process. We start with a unit-length stick and in each |
| step we break off a portion of the remaining stick. Each time, we associate the |
| length of the piece of the stick to the proportion of points that falls into a |
| group of the mixture. At the end, to represent the infinite mixture, we |
| associate the last remaining piece of the stick to the proportion of points |
| that don’t fall into all the other groups. The length of each piece is a random |
| variable with probability proportional to the concentration parameter. Smaller |
| values of the concentration will divide the unit-length into larger pieces of |
| the stick (defining more concentrated distribution). Larger concentration |
| values will create smaller pieces of the stick (increasing the number of |
| components with non zero weights).</p> |
| <p>Variational inference techniques for the Dirichlet process still work |
| with a finite approximation to this infinite mixture model, but |
| instead of having to specify a priori how many components one wants to |
| use, one just specifies the concentration parameter and an upper bound |
| on the number of mixture components (this upper bound, assuming it is |
| higher than the “true” number of components, affects only algorithmic |
| complexity, not the actual number of components used).</p> |
| </section> |
| </section> |
| </section> |
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