Universal Probability and Its Applications
In modern statistical and data science applications, the probability distribution generating the data in question is unknown (or even absent) and decisions must be taken in a purely data-driven manner. Thus motivated, in this dissertation the information-theoretic approach of universal probability is revisited and expanded upon. This approach gives us general principles and guidelines for assigning sequential probabilities to data (based on which a decision can then be made), and has been used successfully over the years to problems in compression and estimation among others. The utility of this approach is then demonstrated through three example problems, motivated by the aforementioned modern statistical applications----universal compression of graphical data, sequential prediction with side information, and universal portfolio selection with side information.