Given two probabilistic distributions (red and blue) it is well known that a linear interpolation between them is well defined (see this).
For example, by the Wasserstein metric we have the following interpolation:
When I first saw this approach, it crossed my mind that the Bayes 's posterior could be explained by a similar geometric argument. Perhaps, the mean between these distributions by the Wassertein metric. However, when I plot the Bayes's posterior (in green) we see that it does not belong to this linear interpolation.
My question is: is there a geometrical reasoning that we can use to interpret the Bayes 's posterior? If there isn't a geometrical reason , can there be a variational one (the Bayes's posterior is the minimum of some energy in this space)?