 The Likelihood Ratio Convergence Theorem merely provides some sufficient conditions for probable convergence. But likelihood ratios may well converge towards 0 (in the way described by the theorem) even when the antecedent conditions of the theorem are not satisfied. This theorem overcomes many of the objections raised by critics of Bayesian convergence results. First, this theorem does not employ second-order probabilities; it says noting about the probability of a probability. It only concerns the probability of a particular disjunctive sentence that expresses a disjunction of various possible sequences of experimental or observational outcomes. The theorem does not require evidence to consist of sequences of events that, according to the hypothesis, are identically distributed (like repeated tosses of a die). The result is most easily expressed in cases where the individual outcomes of a sequence of experiments or observations are probabilistically independent, given each hypothesis. So that is the version that will be presented in this section. However, a version of the theorem also holds when the individual outcomes of the evidence stream are not probabilistically independent, given the hypotheses. (This more general version of the theorem will be presented in a supplement on the  Probabilistic Refutation Theorem,  below, where the proof of both versions is provided.) In addition, this result does not rely on supposing that the probability functions involved are countably additive. Furthermore, the explicit lower bounds on the rate of convergence provided by this result means that there is no need to wait for the infinitely long run before convergence occurs (as some critics seem to think).