The posterior of the experiment is a Bayesian term of art, which we co-opt without remorse for all experiments. Simply put, it's the current result state.
In Bayesian testing, the current result state influences the likelihood of the next draw (that is, variants with higher conversion rates automatically get picked more often). Updating the posterior is the process of grabbing the number of pulls and the number of successes of each variant to redraw the Beta distribution for the next random selection.
Ideally, pulls and successes are updated in real-time. Our systems automatically update the posterior information with great frequency (roughly every hour). And there are additional API endpoints exposed where you can explicitly request that the posterior information be updated. While you shouldn't need to explicitly update the posterior, we do give you that level of control.
Each experiment type (e.g., landing page) will define the API for updating the underlying experiment's posterior.