The Dragon in my Garage
The principle that "extraordinary claims require extraordinary evidence" seems to sort out a lot of "anecdotal evidence" and get to the point at once.
An extraordinary claim is one, which is outside what science has discovered. The natural laws are connected. The evidence in one field is supported by evidence in another, take evolution as an example.
If an extraordinary claim were true, we would have to re-evaluate large parts of our worldview. To accept such a claim, the evidence for it must be at least as good as the evidence on which we constructed our worldview. The more extraordinary the claim the more evidence is required.
The difference between an ordinary an a "normal" claim is illustrated by Carl Sagans The Dragon in the Garage, a story from "The Demon Haunted World". Carl Sagan points out that believing an invisible dragon lives in my garage, doesn't make the claim to be true.
What does it mean to say that a Dragon exist if you aren't able to explain what is meant by that. Claims that aren't falsifiable - they cannot be proven wrong - are worthless, whatever value they might have to magical thinking.
In this article from Red Nova called Do Extraordinary Claims Really Require Extraordinary Evidence? they mention Humes argument about the proportionality between the extraordinary claims and the evidence:
Humes argument about the proportionality between the extraordinariness of claims and the evidence necessary to back them up can be interpreted as saying that the a priori probability we attach to miracles is much lower than the a priori probability we grant to any natural phenomenon, because we have daily experience of phenomena that can be explained naturally, and rarely, if ever, do we even need to consider supernatural causes.Maybe it's the process that needs to be extraordinary and not necessarily the final evidence.
Hume’s argument illuminates important points about the relationship between evidence and hypothesis in both science and pseudoscience. Sagan (and other scholars before and after him) was right: extraordinary claims do require extraordinary evidence.