2026-08-17T15:55:22Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:23567 2025-07-28T05:57:18Z ddc:004 open_access

Drawing and Recolouring Jacobs, Bart Széles, Márk Markov-chain Urn model Multiset Drawing a ball from an urn filled with balls of different colours is one of the basic models in probability theory. The probability of drawing a ball of a particular colour is determined by the proportion / fraction of balls of that colour. This paper introduces a new stochastic model for such urns: draw a ball, recolour (repaint) it, and put it back into the urn. One can distinguish four modes of drawing-and-recolouring, namely whether done proportionally or uniformly (both for drawing and recolouring). These modes can be reformulated in financial terms as redistribution of wealth or in biological terms as evolutionary drift. The resulting four operations form a coalgebra for the distribution monad, on the set of multisets of a fixed size. In fact they form a Markov chain and even a hidden Markov model, in combination with the frequentist learning map as emission channel. This paper identifies fixed points, capturing stable situations, for these four modes of drawing-and-recolouring. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Bart Jacobs and Márk Széles 2025 Is Part Of LIPIcs, Volume 342, 11th Conference on Algebra and Coalgebra in Computer Science (CALCO 2025) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.CALCO.2025.8 urn:nbn:de:0030-drops-235674 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CALCO.2025.8 eng https://creativecommons.org/licenses/by/4.0/legalcode