Kernel embedding of distributions: Difference between revisions

Revision as of 06:45, 9 November 2023

Revision as of 07:18, 9 November 2023

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:<math>\mu_{S^{t+1} \mid h^{t+1}} = \mathcal{C}_{S^{t+1} O^{t+1}}^\pi \left(\mathcal{C}_{O^{t+1} O^{t+1}}^\pi \right)^{-1} \varphi(o^{t+1}) </math>

by computing the embeddings of the prediction step via the [[#Kernel Sumsum Rulerule|kernel sum rule]] and the embedding of the conditioning step via [[#Kernel Bayes' Rulerule|kernel Bayes' rule]]. Assuming a training sample <math>(\widetilde{s}^1, \dots, \widetilde{s}^T, \widetilde{o}^1, \dots, \widetilde{o}^T) </math> is given, one can in practice estimate

:<math>\widehat{\mu}_{S^{t+1} \mid h^{t+1}} = \sum_{i=1}^T \alpha_i^t \varphi(\widetilde{s}^t)</math>