It would be possible to at least improve the situation. An approach common in machine learning is, before a lot of directly-on-point data comes in, to bootstrap predictions using a background dataset of existing data, which can at least give you hints about the structure of the space, and where in it you might fall.
In this case, if you (or some service) had a large data set of user interactions collected from many other websites, then even though the first N data points on your site might not be enough to make strong conclusions on their own, they could be used to match your website up to scenarios in the background dataset that are most similar. There are a lot of different approaches, ranging from Bayesian ones where the background dataset is used as a prior in one way or another, to more clustering-based ones, where your small initial dataset is used to match you to a set of "similar, as far as user-experience lessons go" websites in the existing dataset.
In this case, if you (or some service) had a large data set of user interactions collected from many other websites, then even though the first N data points on your site might not be enough to make strong conclusions on their own, they could be used to match your website up to scenarios in the background dataset that are most similar. There are a lot of different approaches, ranging from Bayesian ones where the background dataset is used as a prior in one way or another, to more clustering-based ones, where your small initial dataset is used to match you to a set of "similar, as far as user-experience lessons go" websites in the existing dataset.