What is Bayes’ Theorem, and What Does It Have to Do with Arguments for God?

The strength of the evidence for a proposition is best measured in terms of the ratio of two probabilities, P(E|H) and P(E|~H) -- that is, the probability of the evidence (E) given that the hypothesis (H) is true, and the probability of E given that H is false. That ratio may be top heavy (in which case E favors H), bottom heavy, or neither (in which case E favors neither hypothesis, and we would not call it evidence for or against H). Bayes’ Theorem is a mathematical tool for modelling our evaluation of evidences to appropriately apportion the confidence in our conclusions to the strength of the evidence.

To take an example, suppose that P(E1|H) = 0.2, but P(E1|~H) = 0.04. Then the ratio P(E1|H)/P(E1|~H) has the value of 5 to 1, or just 5. If there are multiple pieces of independent evidence of the same sort, their power accumulates exponentially. Five such pieces would yield a cumulative ratio of 3125 to 1. If the initial ratio were 2 to 1, ten pieces of independent evidence would have a cumulative power of more than 1000 to 1. By expressing it in mathematical terms like this, hopefully you can see how small pieces of evidence, no single piece by itself of very great weight, can combine to create a massive cumulative case.

The equation given below represents the odds form of Bayes theorem, which is used in developing cumulative cases. Translated, it states that the posterior probability of your hypothesis (H) given the available evidence (E) is equal to the prior probability (defined as intrinsic plausibility) of the hypothesis being true (expressed as a ratio) multiplied by the ratio of the evidence given the hypothesis against the probability of the evidence given the antithesis.

Dividing the probability of the evidence given the hypothesis by the probability of the evidence given the antithesis gives you what is referred to in probability theory as the Bayes Factor. The Bayes Factor is a measure of the strength of the evidence, and indicates how many times more likely it is that you will observe this evidence given that your hypothesis is true than if it were false. For instance, a Bayes Factor of one hundred indicates that your evidence is one hundred times more likely if your hypothesis is true than if it were false.

This form of reasoning is used routinely in the discipline of forensic science. For instance, the presence of a defendant’s finger prints on a murder weapon may be taken as evidence for the hypothesis of guilt over the hypothesis of non-guilt because the probability of the defendant’s finger prints being on the murder weapon is much higher on the hypothesis that the defendant is guilty than on the hypothesis that he is not guilty.

How might we make a powerful case for the existence of God based on what we have just learned about Bayes Theorem? We can begin by giving an estimate of the probability of the evidence given theism and the probability of the evidence given atheism, in order to calculate the Bayes Factor.

A Defense of Theological Rationalism

Theological rationalism is the view that, generally speaking, the only way to have rational confidence in the truth of Christianity is by looking at the public evidence. What distinguishes a theological rationalist from a secular rationalist is that the former believes there is sufficient evidence to rationally warrant, or even compel, belief in the truth of the gospel for the man who is fully informed, whereas the latter does not. This stands in stark contrast to the paradigm, often referred to as “reformed epistemology”, which holds that one can be rationally warranted in believing Christianity to be true wholly apart from public evidence and argument. Reformed epistemology is not at all limited to fideists and presuppositional apologists. Even many classical apologists, notably William Lane Craig, frequently draw a distinction between how you know that your faith is true, and how you show to others that your faith is true. On how Dr. Craig personally knows his faith to be true, he is quite candid in saying that he is not an evidentialist. Dr. Craig states that he knows his faith is true because of the internal witness of the Holy Spirit, which imparts to him confidence that Christianity is true. Since other people do not have access to his own personal subjective experience, Dr. Craig appeals to public evidences to demonstrate to others that his faith is true – but, importantly, Dr. Craig’s own personal faith apparently does not rest on those evidences.

Even among apologists, sadly, theological rationalism is very much a minority position, and I would like to see it promoted more widely. In this article, I want to address some popular concerns about theological rationalism and provide clarity about what my view is and contrast it with popular misconceptions. I will begin by offering a brief positive case for adopting theological rationalism, and will then offer responses to 12 popular concerns or objections that I encounter frequently from other Christians.

Warranted Christian Belief and the Internal Witness of the Holy Spirit

Among many apologists, I often hear a distinction offered between "knowing" and "showing" that the Christian faith is true. On such a view, it is supposed that one can have a warranted knowledge in the truth of Christianity wholly apart from public evidence. Such an approach has long troubled me for a number of reasons, the foremost being that I have no internal witness to the truth of Christianity -- at least not of the kind that, in my judgment, would warrant belief. It would thus be rather dishonest for me to assert such an internal witness when I have no such subjective experience to speak of.

Now, I think in principle this could be a valid approach. For instance, suppose that whenever someone became a believer, there was a voice from heaven, only heard by the new believer, that said "Welcome to the family." That, I think, would be rational warrant for affirming Christianity to be true -- even though you would need to appeal to public evidence in order to demonstrate to others that Christianity is true.

I am not prepared to lie about my own experience when talking to non-believers, or for that matter other believers. I have not had some sort of tangible subjective experience that I would consider to impart to me rational warrant for my beliefs. Yes, I believe myself to have a relationship with God (which is absolutely necessary for any believer to have) -- but when I pray, I have confidence that God hears and answers my prayer because of the public evidence I have studied. By that, I do not mean to rule out the possibility that other believers have had such an experience that would rationally warrant belief.

Answered prayer is another piece of subjective evidence often asserted to warrant belief, and again, I think this could be a valid approach in principle. However, to use it as belief-warranting evidence, one would have to demonstrate a statistical significance to answered prayer, in order to distinguish it from mere coincidence. All Christians who pray can speak of times where they have requested something in prayer where they have not received what they asked for. There are a number of explanations given for this in Scripture. For example, prayer can be hindered by sin (Proverbs 28:9, 1 Peter 3:7) or by selfish-intent (James 4:3), and sometimes God knows that what we ask for is not good for us, and often his will and purpose is different from ours. All these potential variables make it difficult to use answered or unanswered prayer as evidence for or against the Christian faith. If fulfilled prayer is to be used as evidence for the truth of Christianity, one must be able to specify a hypothetical outcome which in principle could be dis-confirmatory evidence. This makes arguing from fulfilled prayer complicated.

If one were to ask me why I myself am a follower of Jesus, I would have to say "My faith rests entirely on the public evidence." The cumulative force of the evidence for Christianity gives me a robust basis for believing Christianity to be true -- and the extent of the evidence in which my faith is grounded means that my faith is not immediately perturbed by encountering fresh counter-evidence or arguments that I have not previously been exposed to (in much the same way that a well supported scientific theory is seldom overturned by a single anomalous observation) -- or, indeed, if some of the evidence on which my faith rests turns out, in the course of time, to be less strong than I presently believe (in much the same way that the discovery that some of the evidence for the earth's vast age was weaker than I previously thought would not seriously cause me to doubt the conclusion, which would still be supported by significant other public evidence).

For these reasons, even though I ground my personal faith in the public evidence (and not in subjective internal experience) my confidence in my faith is not tossed to and fro by the shifting sands of evidence.

An objection that I frequently encounter in this regard is that basing one's Christian convictions on the public evidence diminishes the role of the Holy Spirit in conversion. But this isn't so. Rather, it is my position that the Holy Spirit uses arguments and evidence to draw men into the kingdom of God.

If the appeal to subjective personal experience does not warrant the Mormons in their belief, why should it warrant the Christian? The way I know Christianity to be true is exactly the same as how I show Christianity to be true -- by means of appeal to the public evidence.