Collective Reasoning under Uncertainty and Inconsistency

In this thesis we investigate some global desiderata for probabilistic knowledge merging given several possibly jointly inconsistent, but individually consistent knowledge bases. We show that the most naive methods of merging, which combine applications of a single expert inference process with the application of a pooling operator, fail to satisfy certain basic consistency principles. We therefore adopt a different approach. Following recent developments in machine learning where Bregman divergences appear to be powerful, we define several probabilistic merging operators which minimise the joint divergence between merged knowledge and given knowledge bases. In particular we prove that in many cases the result of applying such operators coincides with the sets of fixed points of averaging projective procedures - procedures which combine knowledge updating with pooling operators of decision theory. We develop relevant results concerning the geometry of Bregman divergences and prove new theorems in this field. We show that this geometry connects nicely with some desirable principles which have arisen in the epistemology of merging. In particular, we prove that the merging operators which we define by means of convex Bregman divergences satisfy analogues of the principles of merging due to Konieczny and Pino-Perez. Additionally, we investigate how such merging operators behave with respect to principles concerning irrelevant information, independence and relativisation which have previously been intensively studied in case of single-expert probabilistic inference. Finally, we argue that two particular probabilistic merging operators which are based on Kullback-Leibler divergence, a special type of Bregman divergence, have overall the most appealing properties amongst merging operators hitherto considered. By investigating some iterative procedures we propose algorithms to practically compute them