Here is a summary/commentary of Hume's chapter "Of Miracles" in Enquiries Concerning Human Understanding - lots of copies available online and on amazon e.g.

"Of Miracles" is important in the history of philosophy. The first part discusses general issues in epistemology (how we know) - Hume explains why we generally trust people as sources of information and that we do so because we have generally found them to be reliable in the past. However, we rely on people's evidence less if what they claim sounds more bizarre or contrary to our experience. In the second part, Hume applies these general principles to the Philosophy of Religion topic of miracles. He ultimately concludes that it is never rational to believe that an event has occurred that breaks the laws of nature (indeed he argues that, even if it were to happen, it is still rational to believe that is doesn't!).

Part I
I (Hume) have discovered an argument against “superstitious delusions” and accounts of miracles and prodigies.

We rely on experience to give us knowledge, but sometimes it leads us to make mistakes, e.g. experience leads us to expect the weather to be warmer in June than in December, but it might be the case that there is then a really warm day in December. Such a situation would not mean that we would reject experience and never again use it in our reasoning, because the experience on which we are basing our judgement in this case (our experience of the weather) is varied (“that contrariety of events”), and we are less confident in any inference/ conclusion if the evidence on which we are basing it is less uniform.

We should proportion our belief to the evidence. If our past evidence/ experience is uniform or infallible, we can be assured of our conclusion/ be confident in our prediction. Otherwise, we weigh up the evidence for and against, proceed cautiously, and believe our conclusion in proportion to the degree of evidence we have for it.

We use human testimony (accounts from e.g. eye-witnesses) as a source of evidence in our reasoning. We do this because we have found that people are generally reliable, that what they say is true and that it agrees with other sources of evidence. If we had experience of people being generally unreliable or lying frequently, we would not rely on human testimony. Indeed, we do not rely on people who are “delirious, or noted for falsehood and villany”.

However, human testimony is just like any other form of evidence – we will weigh it against other evidence (including other testimony, experience, observation), and are inclined towards a conclusion in proportion to the degree of evidence we have for it (we will be less confident in proportion to the strength of the opposing evidence).

We may be less confident in a piece of testimony if witnesses contradict each other, if there are few witnesses or if they have a “doubtful character”, when they have a vested interest in what they are claiming, when they are hesitant in giving their account or equally if they are overly insistent.

Also, if the testimony is proposing something extraordinary (as in, not ordinary at all) or marvellous (as in, something that one would consider to be a marvel), we are less inclined to believe the evidence in proportion to how unusual the purported fact would be. We only rely on testimony (witnesses and historians) because we have experience of such testimony being a generally reliable way of finding out about the world, because what people say can be checked using other evidence or, if not, at least sounds like the type of thing we find by using other evidence. Our experience inclines us to believe the testimony of others, yet also inclines us to believe that extraordinary things do not occur (we are inclined to believe that the types of things we normally observe to occur always occur and that crazy things do not happen). So, our experience should make us less inclined to believe a witness claiming an event has happened that is nothing like the types of thing we usually experience.

Even if someone has been really reliable in the past, we might not believe their testimony if what they now say is incredible (as in, not credible/ believable/ trustworthy). The incredibility of their claim outweighs their previous reliability.

An Indian prince, who has never seen frost, would be rational (be reasoning correctly) if he refused to believe that frost exists because he has not experienced it nor anything like it.
[Note, this implies an awkward conclusion for later: that, even if miracles occur, it is rational to believe that they don’t – it might be objected that this is a rather unfortunate consequence of using reason and being rational.]

Now, let’s consider a testimony claiming something miraculous, rather than merely marvellous, and that there is no evidence for the purported miracle other than this testimony:

“A miracle is a violation of the laws of nature”.
Events that happen in the normal course of nature are not judged to be miracles. e.g. we would not say that a seemingly healthy man dying suddenly is a miracle – we would think such an event unusual, but we have observed this type of event previously. However, we would deem a dead man coming to life a miracle because this has never been observed [this is surely begging the question? – in that people do claim to have observed such an unusual/ miraculous event].
The laws of nature are established (we accept them) due to repeated and consistent past experiences. All this evidence that supports/ establishes the laws of nature is evidence against the possibility that these laws can be broken. In order to accept the miracle, we require sufficient evidence to reject the law of nature and overcome all the evidence that supports it.
“A miracle may be accurately defined, a transgression of a law of nature by a particular volition of the Deity, or by the interposition of some invisible agent.” [note, a second definition]

“no testimony is sufficient to establish a miracle, unless the testimony be of such a kind, that its falsehood would be more miraculous, than the fact, which it endeavours to establish… When anyone tells me, that he saw a dead man restored to life, I immediately consider with myself, whether it be more probable, that this person should either deceive or be deceived, or that the fact, which he relates, should really have happened.”

Part II
Thus far, we have supposed that we should accept that a miracle has occurred if we had testimony/ evidence so reliable that it is more incredible (more unlikely) that the testimony is false than that the miracle didn’t happen. However, there has never been an instance of testimony or evidence for a miracle that was this reliable:

Firstly, there have been no purported miracles where the witnesses’ testimonies have been sufficiently reliable: where the witnesses were so reliable, educated, sensible and honest that we have no reason not to believe them.

Secondly, we normally think that things we have not experienced are pretty similar to the things we have. Given this, we usually reject any experience or experimental result which isn’t normal in this way (we might think we were wrong, or had made a mistake or that there was some interfering factor). However, when the purported experience is “utterly absurd and miraculous”, rather than rejecting it, people believe it strongly. [credo quia absurdum – ‘I believe because it is absurd’ – it is so ridiculous, I believe it – it couldn’t be made up] This is because they get excited and feel “surprize and wonder”, which feels nice. And people get excited and love to hear such stories and indeed get excited by telling other people about them.

The person reporting the purported miracle may have a vested interest in others believing. They may know the story to be false but may still propagate it with some higher goal in mind (for instance, to spread or support their religion). Or he may genuinely believe it, but still has a vested interest in other people believing it so he will propagate it passionately. If the hearers do not believe him, this encourages him to express his ideas ever more forcefully until they do believe him.

Many supposed miracles, prophecies and supernatural events have later been found to be forgeries, or have been dismissed. People have a strong inclination to (want to) believe extraordinary and marvellous things: we should be wary of the reliability of their judgement.

Thirdly, these purported miracles always occur amongst “ignorant and barbarous nations”. If we look at ancient texts, their world seems to operate differently from ours, with events like war and famine never having natural causes but magical or mystical explanations. As societies develop, and the people become more educated and enlightened about the world, the number of these magical events decreases, as we realise they have other explanations.

Isn’t it strange that such miraculous events never happen in our age?! But it is not strange that, at all times, men have lied or got things wrong.

Miracle stories start among ignorant people in remote countries, rather than in cities renowned for arts and knowledge.

Fourth reason: each religion in the history of the world has accounts of miracles which support it. However, any one religion rejects the supposed miracles that other religions claim to have occurred. In which case, for every purported miracle, there are many people who vociferously deny its occurrence. A miracle supporting one religion is evidence against all other religions. But in which case, all religions have a larger number of supposed miracles opposing them than supporting them.

There was an amputee who claimed to have grown a new leg by rubbing oil on his stump. The people of the town believed a miracle had occurred and many claimed to have witnessed it. A visiting cardinal did not believe it, even though he could not disprove the testimony by tracing how it started, given the “bigotry, ignorance, cunning and roguery of a great part of mankind.” It is more likely that people lie or are wrong than that what they say actually happened. Such reports of miracles carry “falsehood upon the very face of it.”

If we are reasonable, even if there are many witnesses, “the absolute impossibility or miraculous nature” of the event reported is sufficient evidence to reject that report.

It is wise to be wary of reports the truth of which would benefit the person reporting, or their family or their interests.

Even if we had evidence that seems reliable, we would believe an alternative explanation, which leaves the laws of nature intact, rather than believe that a miracle had occurred which violated them. “I should rather believe the most extraordinary events to arise from [“the knavery and folly of men”], than admit… a violation of the laws of nature.”

We should “compare the instances of the violations of truth in the testimony of men with those of the violation of the laws of nature by miracles, in order to judge which of them is most likely and probable.” Is it more likely that a law of nature has been broken or that the report of it is false?

Christianity is founded on faith, not on reason. Attempts to defend Christianity using reason are flawed. The Biblical accounts of miracles were written by “a barbarous and ignorant people”, with accounts of events entirely different from anything we experience now. Is it really possible that the “falsehood of such a book… would be more extraordinary and miraculous than all the miracles it relates”?

We cannot convince ourselves of the truth of Christianity or of the existence of miracles using reason. People need faith to believe in these; “And whoever is moved by Faith… is conscious of a continued miracle in his own person, which subverts all the principles of his understanding, and gives him a determination to believe what is most contrary to custom and experience.” A person needs a miracle to suspend his reasoning and judgement and to allow him to believe things that go against his experience.