Let H be the hypothesis that God exists and ~H the negation of that hypothesis. A Bayesian would say that the probability of H being true is given by Bayes rule:
P(H|X) = P(X|H)P(H)/P(X) = P(X|H)P(H)/[P(X|H)P(H) + P(X|~H)P(~H)]
where P(H) is your prior belief in the existence of God and X represents our observations of God. However, we will instead look at the odds ratio - the ratio of the probability of God's existence and non-existence,
P(H|X)/P(~H|X) = P(X|H)P(H)/[P(X|~H)P(~H)] = P(X|H)/P(X|~H) * P(H)/P(~H)
The second term in the final expression is called the Bayes factor and it basically tells us how much the observations affect our relative belief in the existence and non-existence of God.
If it describes a God that regularly makes appearances - say "Q" in Star Trek the Next Generation, then P(X|H) and P(X|~H) will be very different and not observing "Q" (at least once a series) will give reasonable evidence that "Q" doesn't exist. In this case, the Bayes factor will be substantially less than one. In this case, absence of evidence is to some degree evidence of absence.
Now, if our hypothesis of God describes a God that rarely makes an appearance then P(X|H) and P(X|~H) are going to be very similar in the case of the absence of evidence and the Bayes factor will be close to 1. In this case, our posterior belief in God is basically just a restatement of our prior beliefs and absence of evidence is not evidence of absence.
For most theological beliefs, we will be in the latter case, and pretty much all evidence based arguments for and against the existence of God will be just restating our prior beliefs, and probably not a good use of time.
This is essentially formalising @MarcoOcram's answer (+1)