Early this week, Bernhard and I started to discuss about future direction of our research. It is quite difficult to decide because, on the one hand, I think there still be a number of open questions along the line of kernel mean embedding and its applications. Kernel methods have become one of the most important tools in machine learning and I am certain it will still be. On the other hand, this might be a good opportunity to learn something new.
One of the possibilities that we discussed about is "causal inference". Causal inference has been one of the main research directions of our department in Tuebingen (see http://webdav.tuebingen.mpg.de/causality/ for people who work in this area and their contributions). I have to admit that this topic is new to me. I have very little knowledge about causal inference, which is why I am quite excited about it.
In a sense, the goal of causal inference is rather different from standard statistical inference. In statistical inference, given random variables X and Y, the goal is to discover association patterns between them encoded in the joint distribution P(X,Y). On the other hand, causal inference aims to discover the casual relations between X and Y, i.e., either X causes Y or Y causes X or the there is a common cause between X and Y. Since revealing causal relation involves an intervention on one of the variables, it is not trivial how to do so on the non-experimental data. Moreover, there is an issue of identifiability, i.e., several causal models could have generated the same P(X,Y). As a result, certain assumptions about the model are necessary.