When I have prayed to God, he has answered. That certainly counts as "facts or information bearing on the truth of a proposition" and is therefore evidence by your definition. This has happened more than once, which would be "series of instances." Now, I do not know what it means to analyze that information "properly." As I said, it is not scientific evidence. Nevertheless, it is evidence.
Miraculous explanations are nothing more than examples of spasmodic omphalism. . . .
Not so. One does not have to buy into omphalism (spasmodic or otherwise) to accept the idea of miracles. As a matter of fact, omphalism strikes me as silly; but I do believe, for instance, in the Resurrection.
If you adopt naturalism as a useful general policy, why would you abandon it in special instances, those instances remarkable only because they are the ones that survived from an earlier, less naturalistic age, by being more difficult to challenge?
If you have a useful hammer, why would you abandon it when you need to drill holes or saw wood?
Look, I assume that by naturalism you mean the theory or belief that scientific laws are adequate to describe all phenomena. In the context of dealing with the natural world, naturalism is almost always the best approach. But naturalism cannot deal with miracles except by ignoring or discounting them. Nor can it deal with things that are not phenomena, such as questions of morality.
You need to have someone else reproduce your results, or observe you in your prayers and subsequent events.
Individual experiences are highly subjective. Shared experiences, when taken together have a tendency toward objectivity.
Give me specifics of these answers, and I'll be happy to advise.
One does not have to buy into omphalism (spasmodic or otherwise) to accept the idea of miracles.
Please identify how a specific invocation of the supernatural to explain one purported event of religious significance is different in kind from a general invocation of the supernatural to explain all phenomena in religious terms. In other words, why stop at one, or a million little miracles? Why not an all-encompassing, big miracle?
Look, I assume that by naturalism you mean the theory or belief that scientific laws are adequate to describe all phenomena.
There is no need for the word 'all'. Naturalism is the idea that physical explanations are useful to describe the world. One can be more or less naturalistic, although I'd argue we're all >99% naturalistic. A metaphysical naturalist would go further than practical natrualists and say, looking at the success of naturalism in replacing non-naturalistic explanations, and the repeated failure of the latter, that naturalism is indeed adequate to explain all phenomena.
But naturalism cannot deal with miracles except by ignoring or discounting them. Nor can it deal with things that are not phenomena, such as questions of morality. It can disprove miracles, by providing substitute natural explanations. And it can deal with questions of morality, by asking, if we adopt X as a moral principle, what will be the consequences. One can certainly have a naturalistic examination of the consequences of the principle 'homosexuality is wrong' for example. Whether morality could be dealt with in an exclusively naturalistic way is another issue, but let's get a foot in the door first :-)