One thing I forgot to add is people use “changepoint model” in at least two sense:
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A discrete change point (as in the Stan user’s guid example @mike-lawrence mentioned).
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Fitting a function that is defined piecewise (this is IMHO closer to what you care about). A general approach using sigmoids to smooth-out transitions between piecewise linear functions is described at: Piecewise Linear Mixed Models With a Random Change Point and Simpler Change Point Models?