Richard Swinburne is an Oxford Professor of Philosophy, famous for his Christian apologetics (defence of religion). He wrote a trilogy of books in the late '70s called The Coherence of Theism, The Existence of God, and Faith and Reason. In these books he set out his inductive or "cumulative" approach to the argument for God's existence and his own focus on religious experience.
Swinburne's inductive argument is unusual because he only attempts to prove that God probably exists - and that this probability is greater than 50% (more likely than not). He uses a very mathematical approach to do this.
On our total evidence, theism is more probable than not - Richard Swinburne
Swinburne's ideas turn up throughout the Philosophy of Religion course but he is a key scholar for the argument for God's existence based on religious experience.
an experience of God or of some other supernatural thing - Richard Swinburne
we ought to believe that things are as they seem to be, until we have evidence that we are mistaken - Richard Swinburne
If you say ... never trust appearances until it is proved that they are reliable, you will never have any beliefs at all - Richard Swinburne
Just as you must trust your five ordinary senses, so it is equally rational to trust your religious sense - Richard Swinburne
However, Swinburne argues that we trust our ordinary senses even when no one else is about to back them up. Sometimes people disagree with what we see or hear and then we have to decide who is mistaken. If the other person is colour blind, they will not see the same colours that we see, but that doesn't mean we should doubt what we are seeing. In the same way, people who don't have religious experiences might be spiritually blind.
Check to see if you're colour blind by reading the numbers. Now, how could you check to see if you're spiritually blind?
(in the absence of special considerations), if it seems ... to a subject that x is present (and has some characteristic), then probably x is present (and has that characteristic) - Richard Swinburne
(in the absence of special considerations) the experiences of others are (probably) as they report them - Richard SWinburne
Swinburne argues that we have an irrational bias against religious experiences. We are inclined to distrust our own senses when we have religious experiences and distrust other people when they report them - but we don't do this with other experiences that are far weirder and less likely (such as UFOs traveling light years to abduct people for bizarre experiments).
Swinburne is not arguing that we should trust every religious experience - or even most. Lots of religious experiences are clearly hoaxes or products of hysteria or confusion. Swinburne admits that the "special circumstances" discredit a religious experience. But he believes that there are many left for which there is no rational reason to disbelieve.
Swinburne is arguing, not for credulity, but for gullibility - believing in irrational things without investigating the evidence. For centuries people believed superstitious tales, myths and legends but they only made progress when science came along and demanded evidence to back up beliefs. Blind faith in visions or supernatural testimony leads to witch-hunts and fanaticism.
The idea of a God who can intervene in human affairs to produce religious experiences leads to absurd conclusions. Why would such an interventionist God send visions to people like Bernadette Soubirous but not prevent evil and suffering? If things truly are as they seem, then it seems as if God is not interested in preventing innocent suffering.