In a previous post, I defined a miracle as a religiously significant event that is caused by God and exceeds the productive powers of nature. I also argued (contrary to David Hume) that evidence for the laws of nature has no direct bearing on the antecedent probability of a miracle. Such evidence only has indirect bearing, insofar as there is reason to believe that God does not exist or that he would not cause a miracle on some (or any) particular occasion.
In this week’s post, I will briefly outline which probabilistic variables are relevant to the question of whether it is rational to believe in a miracle claim. To do this, I will need to explain some key terms such as ‘evidence’, ‘background information’, and ‘Bayes’ theorem’.
For the purposes of this post, I define ‘evidence’ as any fact, event, or experience that raises the probability of a hypothesis in relation to its competitors. This definition has several advantages. First, it is broad enough to encompass the many ways in which hypotheses can be supported: whether inferentially (via. deduction, induction, or abduction), or non-inferentially, by experiences that spontaneously produce basic beliefs within us. Second, it is compatible with different sources of evidential support, including testimony, memory, perception, and so on. Third, it is open to many kinds of hypotheses. In the case of miracles, hypotheses are probabilistically related to the evidence by virtue of the causal role they play in increasing or decreasing the likelihood of the relevant data. As long as the entities postulated by the hypotheses play such a role, my definition permits them to be directly observable, indirectly observable, theoretical, or some other kind.
By using my definition, it becomes clearer how data becomes evidence for a miracle: evidence is data which we’d expect to observe if the miracle happened, but which is not similarly expected by naturalistic hypotheses.
Most of us have had the familiar experience of interpreting evidence in many different ways. Because evidence can be evaluated from different perspectives, not everyone is going to agree with the conclusions we draw. This is because our background beliefs about the world (prior to considering any evidence) shape the way we evaluate theories. Our prior beliefs partly decide what counts as evidence, what sources of evidence allowable, and how much force the evidence has for us.
For example, a naturalist will expect much more historical evidence for a miracle than a theist will. Why? Because the naturalist already assigns a low probability to God’s existence, and that background belief makes miracles exceedingly improbable, even before any evidence is weighed. But for a theist who assigns a higher probability to God’s existence, and who thinks that God may have reasons for acting in nature, the prior probability of a miracle will be higher. Consequently, less historical evidence (albeit a substantial amount) will be required for the theist to accept that a miracle has occurred.
It is one thing to acknowledge that background information must be taken into account when assessing probabilities. It is quite another to understand how this works! Let’s delve into some probability theory for further clarification…
Thomas Bayes (1701-1761) was an English mathematician who discovered a theorem for calculating the probability of a hypothesis on the basis of evidence and background information. Bayes’ theorem is a mathematical representation of how the prior probability of a hypothesis should be updated when new evidence is taken into account. It also shows us which variables must be considered when assessing the likelihood of an event which (apart from the evidence) is antecedently improbable. Unlike abductive reasoning, Bayes’ theorem does not assess theories in terms of their scope, power, or parsimony in explaining a broad range of data; neither does it tell us which values to plug into the variables of the calculus. But the theorem does work as a handy tool for seeing how the variables for assessing an antecedently improbable hypothesis must be weighed. As such, it can be very useful for thinking about the case for and against miracles.
In its original form, the theorem states that the posterior probability of a hypothesis (h) given some evidence (e), can be calculated in two steps: first, multiplying the probability of the evidence (e) given the hypothesis (h), by the probability of the hypothesis (h) apart from evidence (e); second, dividing the first result by the probability of the evidence (e) apart from the hypothesis (h). To put it more formally,
According to John Earman (2000), the original form of the theorem can be recast to include symbols for background information (b) and competing hypotheses (~h). This new form is called “the principle of total probability” as seen below:
VARIABLES FOR MIRACLES
Let us assume that (h) represents the hypothesis that God raised Jesus from the dead – the central miracle claim of orthodox Christianity. Let (e) stand for the historical data offered as evidence for a resurrection; let (b) symbolize the background information relevant to that event independent of (e); and finally, let (~h) stand for all the naturalistic explanations of the evidence. With these definitions in place, we can clearly see which variables must be weighed in order to assess the total probability of the resurrection hypothesis (h).
Contrary to popular opinion, it will not do simply to focus on the probability of the competing hypotheses relative to the background information only: namely, P(h/b) and P(~h/b). We must also weigh the evidential power of the competing hypotheses. Their power is determined by how expected the evidence is if a resurrection actually occurred, and by how likely we’d have the same evidence if a resurrection did not occur: namely, Pr (e/h.b) and Pr (e/~h.b).
What do you, the reader, think about using Bayes’ theorem in the context of miracle claims? Does it clarify or complicate matters even more?
 I am not endorsing the view that probabilistic theorizing must involve causal relations – only that causation is most salient in the discussion of miracles.
 Abductive arguments are fallible in nature. They rely on general principles that cannot be applied in a mechanical way (e.g. Explanatory Scope & Power, Least Ad Hoc, Simplicity, Coherence, Logical Consistency, Intellectual Fertility, etc). They depend on personal judgments that are trained by experience and cultivated by attention to a wide range of factors that can’t be fully articulated. And their plausibility is influenced by one’s expectations and background beliefs. But abductive arguments can still be reliable, even if fallible. In fact, we already use them to evaluate hypotheses in almost every domain of life. We use them in medical diagnoses, jurisprudence, language-acquisition, ethical reasoning, scientific theorizing, and even metaphysics! Indeed, much of how we cope with everyday situations depends on the kind of reasoning I’ve just described. So we mustn’t disregard abductive reasoning just because it can sometimes go wrong.
 Some philosophers like Lipton (2004) have argued that abductive reasoning can supplement bayesian confirmation theory by helping to decide which prior probabilities to start with. This can be done by determining whether a hypothesis exhibits theoretical virtues such as explanatory scope, power, and simplicity with respect to the background information apart from E. See P. Lipton (2004) Inference to the Best Explanation. 2nd Ed. London: Routledge.
 John Earman. (2000). Hume’s Abject Failure: the Argument against Miracles. Oxford University Press, p.27.