In The Existence of God (2nd edition, hereafter: EOG) , Richard Swinburne presents a careful and systematic case for the existence of God.  Eight of the arguments (that he considers to be significant) are presented as bits of empirical data each of which increases the probability of the hypothesis that God exists a bit (with the exception of the Problem of Evil, which he believes decreases the probability a bit).

These eight inductive arguments are supposed to make the hypothesis of the existence of God roughly a 50/50 proposition:

…it is something like as probable as not that theism is true, on the evidence so far considered.  However, so far in this chapter, I have ignored one crucial piece of evidence, the evidence from religious experience.  (EOG, p.341)

The argument from religious experience (hereafter: AFR) is supposed to put the hypothesis of theism over the top,  making the hypothesis probable, i.e. more probable than not:

…unless the probability of theism on other evidence is very low, the testimony of many witnesses to experiences apparently of God suffices to make many of those experiences probably veridical.    That is, the evidence of religious experience is in that case sufficient to make theism overall probable.  (EOG, p.341)

So long as the evidence from the nature of the universe and from the nature of human life (other than religious experiences of humans) is sufficient to make the probability of theism greater than ‘very low’ that is enough to get AFR off the ground and boost theism to being more probable than not, according to Swinburne.

AFR is based on three principles that are concerned with experience, memory, and testimony:

EXPERIENCE

…(in the absence of special considerations), if it seems (epistemically) to a subject that x is present (and has some characteristic), then probably x is present (and has that characteristic)… (EOG, p. 303)

MEMORY

If it seems to a subject that in the past he perceived something  or did something, then (in the absence of special considerations), probably he did. (EOG, p.303)

TESTIMONY

…(in the absence of special considerations) the experiences of others are (probably) as they report them. (EOG, p.322) 

Many people have on many occasions throughout human history have reported having had a religious experience in which it seemed (epistemically) to them that God was present.  Based on Swinburne’s principle concerning testimony, it is probable in each case (where there was an absence of special considerations concerning testimony) that the reported experience did occur,  and that in each of those cases the experience which seemed (epistemically) to to the subject that God was present (where there was an absence of special considerations) God was probably present.

In the Chapter in which Swinburne presents AFR,  various possible special considerations are reviewed and Swinburne argues that they do not work in general against religious experiences that seem to be of the presence of God.  The main question, from Swinburne’s point of view, is what the degree of probability the existence of God has based on other relevant evidence besides religious experiences.  If this probability is greater than ‘very low’, if the probability of the existence of God is, for example, low but not very low, then the evidence against the existence of God is insufficient to outweigh the force of religious experience.  Swinburne believes that the evidence for and against God, setting aside religious experiences, makes the existence of God about as probable as not.

Here is how I would put these claims in terms of numbers:

x has a very low probability  means P(x) is greater than 0 AND P(x) is less than .2

x has a low (but not very low) probability means P(x) is greater than or equal to .2  AND  P(x) is less than .4

x is about as probable as not means P(x) is greater than or equal to .4  AND  P(x) is less than .6

x is more probable than not means P(x) is greater than or equal to  .6

Let u be the evidence in the premises of the arguments from the nature of the universe.

Let h be the evidence in the premises of the arguments from the nature of human life (except for religious experiences).

Let r be the evidence of religious experiences (including testimony of alleged experiences that seemed to be of the presence of God).

Let g be the hypothesis that God exists.

Swinburne’s argument in EOG can be summarized this way:

1.   P(g|u & h)  ≥  .4  AND  P(g|u & h)  <  .6

2.  IF  P(g|u & h)  ≥  .2  THEN  P(g|u & h & r)  ≥  .6

Therefore:

3.  P(g|u & h & r) ≥ .6

Premise (1) entails that P(g|u & h) is greater than .2 and thus that P(g|u & h) is greater than or equal to .2, which is the antecedent of the conditional claim in premise (2), so the conclusion is thus entailed by modus ponens.