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=== Distribution regression ===
In distribution regression, the goal is to regress from probability distributions to reals (or vectors). Many important [[machine learning]] and statistical tasks fit into this framework, including [[Multiple-instance learning|multi-instance learning]], and [[point estimation]] problems without analytical solution (such as [[Hyperparameter (Bayesian statistics)|hyperparameter]] or [[entropy estimation]]). In practice only samples from sampled distributions are observable, and the estimates have to rely on similarities computed between ''sets of points''. Distribution regression has been successfully applied for example in supervised entropy learning, and aerosol prediction using multispectral satellite images.<ref name = "MERR">Z. Szabó, B. Sriperumbudur, B. Póczos, A. Gretton. [http://jmlr.org/papers/v17/14-510.html Learning Theory for Distribution Regression]. ''Journal of Machine Learning Research'', 17(152):1–40, 2016.</ref>
Given <math>{\left(\{X_{i,n}\}_{n=1}^{N_i}, y_i\right)}_{i=1}^\ell</math> training data, where the <math>\hat{X_i} := \{X_{i,n}\}_{n=1}^{N_i}</math> bag contains samples from a probability distribution <math>X_i</math> and the <math>i^\text{th}</math> output label is <math>y_i\in \R</math>, one can tackle the distribution regression task by taking the embeddings of the distributions, and learning the regressor from the embeddings to the outputs. In other words, one can consider the following kernel [[Tikhonov regularization|ridge regression]] problem <math>(\lambda>0)</math>
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