The cumulative case uses the prime principle of confirmation: Whenever we are considering two competing hypotheses, an observation counts as evidence in favor of the hypothesis under which the observation has the highest probability. This principle is sound under all interpretations of probability. Each argument must be taken on its own grounds and one cannot arrive at “God” at the end of each argument. The conjunction of arguments is what is needed to make a cumulative case for the existence of God.
The Likelihood Principle of Confirmation theory states as follows. Let h1 and h2 be two be competing hypothesis (in this case the existence of X and ~X, with X being a first cause, fine-tuner, etc.). According to the Likelihood Principle, an observation e counts as evidence in favor of hypothesis h1 over h2 if the observation is more probable under h1 than h2. Thus, e counts in favor of h1 over h2 if P(e|h1) > P(e|h2), where P(e|h1) and P(e|h2) depict a conditional probability of e on h1 and h2, respectively. The degree to which the evidence counts in favor of one hypothesis over another is proportional to the degree to which e is more probable under h1 than h2: particularly, it is proportional to P(e|h1)/P(e|h2) . The Likelihood Principle seems to be sound under all interpretations of probability. This form is concerned with epistemic probability.