Is there evidence for God's existence?

Notice the question isn't asking whether God's existence is certain, highly plausible, or just more plausible than not.  Rather, it's asking whether there is any evidence that would increase the probability that God exists than had that evidence not existed.  Here's how we can assess such evidence:

Let G = God exists
E = some specific evidence
B = our background information
P = probability

Now, God's existence increases in its probability so long as the following is met:

P(G/E&B) > P(G/B)

Richard Swinburne calls a successful instance of the above a correct C-inductive argument for God's existence.  A correct C-inductive argument would only make the probability of God's existence more plausible than it would have been sans the specific evidence.  It does not purport to demonstrate a correct P-inductive argument, which would make God's existence more probable than not.

To give an example of a correct-C inductive argument, let's take the law-like operations of nature.  If God exists, and more specifically designed the universe, would we expect nature to exhibit law-like behavior?  The answer is yes, so long as the God in question is the God of classical theism.

Might we expect nature to behave in a law-like manner without God's design?  Maybe, but that's no reason to discount the evidence as constituting a correct C-inductive argument.  For instance, let's suppose that there was a theft, and two sets of fingerprints are found on the safe.  The probability that one person is the thief is not undermined by the fact that another person's fingerprints were found.  Of course, both could have agreed to steal from the safe, but that's impertinent to the argument.  Moreover, what if one of the persons was witnessed to be in another location during the time of the theft?  That would undermine this person's guilt.

What we have, then, are the beginnings of what Swinburne calls a cumulative case for God's existence, all of which are based on the accumulation of correct C-inductive arguments.  After analyzing all of the evidence for and against God's existence, Swinburne concludes that we have a correct P-inductive argument for God's existence.  However, that conclusion is beyond the scope of this post.