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Various tasks encountered in real-world surveillance can be addressed by determining posteriors (e.g. by Bayesian inference or machine learning), based on which critical decisions must be taken. However, the surveillance domain (acquisition device, operating conditions, etc.) is often unknown, which prevents any possibility of scene-specific optimization. In this paper, we define a probabilistic framework and present a formal proof of an algorithm for the unsupervised many-to-infinity domain adaptation of posteriors. Our proposed algorithm is applicable when the probability measure associated with the target domain is a convex combination of the probability measures of the source domains. It makes use of source models and a domain discriminator model trained off-line to compute posteriors adapted on the fly to the target domain. Finally, we show the effectiveness of our algorithm for the task of semantic segmentation in real-world surveillance. The code is publicly available at https://github.com/rvandeghen/MDA.