Single Sample Statistics: exercises in learning from just one example

What can be learned from a single example? If we are faced with some complex process, producing large intricate constructs, but we are only given one example of its output, can we still draw conclusions about the source, or ascribe meaning to the patterns we find? This is by no means just an academic exercise: we only have one Internet, for example, only one climate system and only one global financial system. What assumptions must we make about the processes that generated them, in order to learn about their structure? What can we do if we make no assumptions at all? Each chapter of this dissertation addresses an aspect of this question, starting with a high-level theoretical approach, and gradually working towards more practical aspects.