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Divergent transitions are explained here (1) in the stan docs.

They occur when the posterior has curvature that is varying too much.

My thought was that maybe the posterior would vary a lot in regions where there was not sufficient evidence in the data to describe it. This could happen when you have a parameter that is sort of "free," because the data doesn't have enough information to tell you about it.

This however is based on my limited understanding of Hamiltonian MC, so I hoped that someone might know more and be able to explain.

(1) https://mc-stan.org/docs/2_19/reference-manual/divergent-transitions

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If a model is too complex for the data that is being conditioned on, then divergent transitions under Hamiltonian MCMC may be a symptom of that. Unfortunately, if the model is not complex enough for the data that is being conditioned on, then divergent transitions under Hamiltonian MCMC may also be a symptom of that. Moreover, if the adapt_delta tuning parameter is too small or the priors are weaker than what you actually believe before conditioning on the data, then divergent transitions under Hamiltonian MCMC may be a symptom of those as well. Thus, the existence of divergent transitions does not uniquely imply anything about why the divergent transitions occur, and you would need to obtain more information from pairs plots, parallel coordinate plots, etc.

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