According to Bayesian confirmation theory, evidence E (incrementally) confirms (or supports) a hypothesis H (roughly) just in case E and H are positively probabilistically correlated (under an appropriate probability function Pr). There are many logically equivalent ways of saying that E and H are correlated under Pr. Surprisingly, this leads to a plethora of non-equivalent quantitative measures of the degree to which E confirms H (under Pr). In fact, many non-equivalent Bayesian measures of the degree to which E confirms (or (...) supports) H have been proposed and defended in the literature on inductive logic. I provide a thorough historical survey of the various proposals, and a detailed discussion of the philosophical ramifications of the differences between them. I argue that the set of candidate measures can be narrowed drastically by just a few intuitive and simple desiderata. In the end, I provide some novel and compelling reasons to think that the correct measure of degree of evidential support (within a Bayesian framework) is the (log) likelihood ratio. The central analyses of this research have had some useful and interesting byproducts, including: (i ) a new Bayesian account of (confirmationally) independent evidence, which has applications to several important problems in con- firmation theory, including the problem of the (confirmational) value of evidential diversity, and (ii ) novel resolutions of several problems in Bayesian confirmation theory, motivated by the use of the (log) likelihood ratio measure, including a reply to the Popper-Miller critique of probabilistic induction, and a new analysis and resolution of the problem of irrelevant conjunction (a.k.a., the tacking problem). (shrink)