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In this thesis we study recursion methods for making Bayesian inference on a class of multiple changepoint models, introduced in Fearnhead(2006). We present and implement recursion algorithms, and we evaluate how parameter relations in a changepoint model may affect method performance. We apply the methods to a number of single and multiple changepoint problems, with respectively normal and Poisson distributed observations and with varying model parameter relations. Finally, we test the methods on a set of coal-mining disasters from Jarrett(1979) before discussing results and proposing future method improvements and application areas.