Precisely this:
That is, as I understand things, sampling of the model given the first data point (talking timeseries data here) should be informed by the priors (priors in the “I haven’t seen any data yet, just using domain expertise” sense), which yields a posterior (posterior_0) that reflects an update from the priors given the single data point. This posterior is then passed as starting points for the next round of sampling that includes the second data point, and since posterior_0 already contains the information/influence of the prior, sampling for the second data point and onwards shouldn’t add the prior yet again in the log-likelihood.