The solution to this problem is Bayes’ theorem. The theorem’s conclusion takes into account our background knowledge, and the evidence we have for the hypothesis we make. Where as before we were jumping to the conclusion that the probability of a miracle is low based on our prior knowledge of how the world works, we will now take into account the actual evidence of the miracle. Quantitatively speaking, the theorem makes us express our premises in terms of degrees as opposed to absolutes by forcing us to numerically label them as probabilities. This is important because most claims in life, especially about historical events, can only be discussed in terms of probabilities, with us often saying things like “most likely” or “more likely”, for example. So, those that object to doing math in history, think again, because everyday we already speak in terms of probabilities. Moreover, the theorem forces you to think of alternative hypothesis, reducing confirmation bias. This formalized and systematic approach to viewing your hypothesis allows for a clarity unrivaled by other methods. I offer two quotes below that explain its history and importance.
In simple terms, Bayes’s Theorem is a logical formula that deals with cases of empirical ambiguity, calculating how confident we can be in any particular conclusion, given what we know at the time. The theorem was discovered in the late eighteenth century and has since been formally proved, mathematically and logically, so we now know its conclusions are always necessarily true if its premises are true (probabilities). [Richard Carrier]
Bayes’s theorem is at the heart of everything from genetics to Google, from health insurance to hedge funds. It is a central relationship for thinking concretely about uncertainty, and–given quantitative data, which is sadly not always a given–for using mathematics as a tool for thinking clearly about the world. [Chris Wiggins, Scientific American]
The specification of the prior is often the most subjective aspect of Bayesian probability theory, and it is one of the reasons statisticians held Bayesian inference in contempt. But closer examination of traditional statistical methods reveals that they all have their hidden assumptions and tricks built into them. Indeed, one of the advantages of Bayesian probability theory is that one’s assumptions are made up front, and any element of subjectivity in the reasoning process is directly exposed. [ Olshausen]