Testing a Hypothesis about the Supernatural

[Angrboda]

We are left with only observing the natural effect. We can rationally infer a supernatural cause if the probability of there being a natural cause is sufficiently low. 

No, this is simply false. Our knowledge of the natural is necessarily incomplete, so we can never completely rule out a natural cause. We can determine the probability of the natural causes we know about being responsible, however we can never know the true probability that an event has a natural cause. Your argument is faulty. This follows from the apophatic way in which you've defined the supernatural. This is little more than an appeal to ignorance. If an event has no known natural cause then the event is "unexplained," not supernatural.