Drawing and Recolouring

Abstract

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.