Dear @ded,
this seems to be a broad question, but I think you understood a lot.
Since you write that data is generated, and that the means are off, I would guess your model is off. Note that this could be both the model being faultily specified, or the generation not generating what you want. Could you provide the code, please?
Now regarding your bigger question, how to interpret model results. This depends on the model context. I am not well trained in non-Bayesian statistics (apologies too my stats professors, but this is what all evidence suggests), so I have a hard time understanding what you try to achieve with “logp”. Note that some terms (such as “likelihood”) are ambiguous in different statistical schools, there’s a great talk by Richard McElreath here (youtube). Not having “any prior” might also be something to doubt, as I think is also discussed in the video.
In general, authors argue to not talk about “confidence intervals”. Instead, people report “credible intervals” (I got that first from Gelman’s book). This usually describes the 95% interval of highest probability density of the posterior.
Again, more can be said on inspection of your model.
Cheers,
Falk