Causal structure of the inflationary multiverse In the multiverse as it is described by eternal inflation, it is not clear to me what is its causal structure and in particular if the bubble-universes are causally connected. We start from a de-Sitter space-time and a scalar field whose almost constant potential plays the role of a positive cosmological constant, but after decay of the scalar field in certain regions it is not clear to me what becomes of the causal structure. If they are considered universes on their own, it should mean that the bubble are causally disconnected from everything else; however I already heard of bubble colliding and leaving hypothetical imprints in the CMB. There also the different probability measure defined in inflationary cosmology, are there probabilities seen by an observer in one of the bubbles, or by an outside (super)observer seeing the whole space-time foliation and every bubble, no matter the causal structure of the space-time?
 A: The short answer to this question is that we do not know. The subject of your question is still in early "speculative", theorizing, and researching stages. I can say this because collisions of bubble universes under the eternal inflation theory just happens to be my specific area of work.
Non-colliding bubble universes (and the local potential minima in the scalar field are considered other universes) are causally disconnected from each other due to both the potential wall between the false and true vacuum as well as the rate of expansion of each bubble (it is fast enough that no information from inside could ever escape the bubble and thus non-colliding bubbles could not influence one another).
As for collisions leaving imprints on the CMB, that is still a matter of research. At the moment, it is considered the most likely outcome of a bubble collision (at least, the most likely one that would be visible and not completely destroy us). However, physicists like myself are still running simulations to determine what these effect would look like. Also keep in mind that for an observer in one of the bubbles to observe the effects of the collision, the collision wall would necessarily have to have passed them already (which is why we believe if we have experienced a visible collision, it likely would not destroy us because it hasn't).  It is only in this way that the bubbles can be causally connected, because they come into contact and (in some theories) expand through each other.
It is impossible for me to offer a non-speculative answer for whether there are probabilities that can be seen no matter the causal structure. As an observer in just one bubble, my view is understandably biased. Furthermore, since we allow for the physical constants and laws to vary between different bubbles (except for the speed of light), it is equally impossible for me to comment on the probabilities observed by someone in another bubble.
It should be noted that this area of research is still mostly blackboard physics. There have been some papers published concerning bubble collisions, but without hard empirical data, the best we can do is run complicated computer simulations of hypotheticals.
