In order to make this a ‘self-contained’ post, I will need to repeat some of what I’ve written elsewhere.
Two Types of Evidence and Corresponding Probabilities
Using this notation, it is possible to provide a mathematically rigorous definition of an extraordinary claim. An extraordinary claim may be defined as an explanatory hypothesis which is extremely improbable, conditional upon background information alone, i.e., Pr(H | B) <<< 0.5.
Because we are using the epistemic interpretation of probability, it follows that what counts as an “extraordinary claim” may vary from person to person and from time to time. For example, a healing miracle attributed to the Virgin Mary will be a very extraordinary claim to naturalists (and perhaps? non-Catholics), but not as extraordinary (or not extraordinary at all) to Catholics.
With “extraordinary claim” defined, let us now define what constitutes extraordinary evidence. Before we do, however, we must first review Bayes’s Theorem.
Proposal for Both Theists and Naturalists Regarding Extraordinary Claims
Because background information plays such a crucial role in determining the prior probability of any hypothesis, extraordinary or not, one thing that both theists and naturalists could do to improve communication with one another is to explicitly identify the propositions which make up their background information. Doing so will not magically resolve their disagreements, but it will greatly improve the chances that the two parties are at least understanding one another. For an example of what this might look like, see any of the individual arguments for naturalism in my series on evidential arguments for naturalism.
 I owe these definitions to Robert Greg Cavin in private correspondence.