What's the difference ? I mean, obviously your neighbour could be entirely irrational and refuse to listen to anything you say.

Let us say that I have gone fishing. I return from my fishing trip, and describe to my neighbour how I hooked a six-foot-long great white shark, but my fishing line snapped and it got away. Unfortunately, I failed to get a photograph or any other piece of hard evidence.

Assume that my neighbour is rational, intelligent, and patient. Would he be convinced by my account?

after you've explained that, he asks you, "ok, and why do you believe that ?". What's your answer ?

Short version; I started with a high prior, and certain experiences in my life have caused me to update that original prior in an upward direction.

This is our world; let's call it Alpha. The other world is called Beta, and it contains no gods at all. The two worlds are completely identical; except that, whenever something happens in Alpha, sometimes the god is responsible, and sometimes it just happens for mundane natural causes. When the same thing also happens in Beta, it's always due to mundane natural causes.

...hold on a minute. You are postulating that there is some way to set up the natural laws of a universe such that everything that God would want to do in Alpha happens anyway, even without direct involvement. Should that be the case, an omniscient being would know how to set up the physical laws in such a way; and an omnipotent being would be able to do that, and it would probably be much less effort than having to go back and fiddle with the universe every now and then.