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It happens all the time. Someone says, "Pray to God and he will answer your prayer." One may take that advice, try it, and see what happens. The outcome, whatever it is, would not qualify as scientific evidence. But it would be observational evidence.

Only within a very loose description of 'observational evidence'. Evidence is usually defined as a body of facts or information bearing on the truth of a proposition. If you collected a series of instances, you might have a body of information. But then, if you analysed that information properly, you'd be doing science; as was seen recently, when they analysed the effect of prayer on medical recovery.

So it appears your your test of other explanations seems to be concerned with cases where the evidence is scanty or not analyzed by scientifically adequate means. It's not so much that they are other than scientific; they are less than scientific.

More important, science cannot "rule out specific claimed interventions" of God. Just the opposite: Science can state that in general God does not appear to intervene or contravene natural law; but science cannot rule out the occasional miracle.

Sure it can. If you claim specific instance of a phenomenon A is an example of supernatural intervention, we can examine it to see if it followed deterministically from natural law, or if in fact supernatural intervention is a plausible explanation. So, for example, disease used to be attributed to supernatural action. We now know that it is caused by specific etiological agents. We have ruled out divine or Satanic action as a general cause of disease.

But many people believe that Jesus Christ came back from the dead. I know of no scientific test that can disprove that claim. It is a miracle, and therefore outside the realm of science.

We know that when people die, irreversible processes proceed that rapidly make resuscitation impossible. We can't rule out that some specific instance in time a miracle happened; but then we can't rule out any miracle, or any supernatural explanation. The world could have been created yesterday, with all our memories intact. So specific instances of 'miracles' are really inseparable from the general category of 'omphalism', and by adopting naturalism - and we are all naturalists in practice - we have discarded this category of explanations.

Miraculous explanations are nothing more than examples of spasmodic omphalism. 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?

I'm pinging thatcherite because he introduced me this morning to this magnificent and useful new word.

If you claim specific instance of a phenomenon A is an example of supernatural intervention, we can examine it to see if it followed deterministically from natural law, or if in fact supernatural intervention is a plausible explanation. So, for example, disease used to be attributed to supernatural action. We now know that it is caused by specific etiological agents. We have ruled out divine or Satanic action as a general cause of disease.

Not only disease but all manner of phenomena that we now attribute to natural agency used to be attributed to the will of God. When lightning rods were first invented they were resisted by the established church. Clearly it was gross hubris to seek to deflect the will of God. But the church elders soon noticed that the lightning-rod-equipped cathouse would be spared, while the neighbouring church would be the target of God's wrath. It would seem that God was weak indeed, to have His manifest will deflected by $1 of metal. Perhaps, after all, God was not directing every lightning strike with His will, but rather allowing the random happenstance of the natural world to strike all equally, sinner and saint.

Worship the God that resides in the gaps of our ignorance of the natural world if you will, but be aware that that particular God is a grossly shrunken God from the deity of a few-hundred years ago.

Evidence is usually defined as a body of facts or information bearing on the truth of a proposition. If you collected a series of instances, you might have a body of information. But then, if you analysed that information properly, you'd be doing science; as was seen recently, when they analysed the effect of prayer on medical recovery.

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.