It is commonly argued that Hume's "in principle" argument (in section 1 of the Essay on Miracles) against miracles begs the question; on this view, Hume argues that there is completely uniform experience against miracles, and therefore there can't be miracles. The problem, of course, is that this begs the question: precisely what is at stake is whether there is uniform experience against miracles. To be broad enough to spawn 'laws of nature', 'uniform experience' has to include testimony of other people's experiences, and this cannot exclude experience of miracles (for which there is certainly testimony) without begging the question.
A common defense of the argument is to argue that Hume's real point is this: A miracle is a violation of a law of nature; law of nature is based on uniform experience; the defender of miracles denies uniform experience; therefore, Hume says, there is no law of nature to violate, and hence no miracle.
There are three obvious problems with this defense.
(1) If this is Hume's argument, it is trivial to the point of stupidity. Hume would have read more sophisticated accounts of what a miracle is (Butler's, for instance, or Malebranche's) that can easily evade this sort of semantic gerrymandering, because they don't define a miracle as a violation of natural law in this sense. Either Hume's argument applies in at least a general way to these more sophisticated accounts with which Hume would have had acquaintance, or it is an exercise in missing the point.
(2) It doesn't get us anywhere; on this view, Hume would constantly have to re-organize the bounds of the laws of nature to accommodate claims of miracles. If he doesn't, he still begs the question in the way the original objection suggested -- either the determination of the laws of nature is based on testimony that includes purported miracles, or it doesn't. If it doesn't, it can only be because the issue has already been decided, which means it begs the question.
(3) This can't actually be Hume's argument -- that is, neither the objectors nor the defenders are getting Hume's argument right. This is clear given that virtually all of Hume's actual argument drops out in both cases, and the defenders make some especially egregious slips. Hume divides grounds into three kinds:
demonstration -- necessity due to relations of ideas
proof -- no room for doubt, given experience
probability -- room for doubt, given experience; there are two kinds: probability of chances (which we can set aside for our purposes) and probability of causes.
Here are examples Hume gives of beliefs based on proof:
Fire burns. Water suffocates. Impulse produces motion. Gravity produces motion. All men die. The sun rises every day. Lead cannot of itself remain in the air. Fire is extinguished by water.
Here are examples Hume gives of beliefs based on probability:
Rhubarb purges. Opium induces sleep. In January there is frost.
In Section One, Hume assumes that laws of nature are grounded in proof. He also assumes that the purported miracle is. He is very explicit about this; and only goes on in Section Two to argue that in fact no purported miracle has this level of support. The determination of whether testimonial evidence amounts to proof or probability or neither, Hume thinks, always depends on the balancing of contrarieties: "We balance the opposite circumstances, which cause any doubt or uncertainty; and when we discover a superiority on any side, we incline to it; but still with a diminution of assurance, in proportion to the force of its antagonist."
In a case where you have the laws of nature, which are based on proof, balanced against a purported miracle, which ex hypothesi is also based on proof, Hume says, "there is proof against proof, of which the strongest must prevail, but still with a diminution of its force, in proportion to that of its antagonist."
It's in this context that Hume makes the famous pronouncement "that 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 endeavors to establish": on the suppositions of an argument, the falsehood of the testimony for the miracle would really be miraculous, for the same reasons a violation of natural law would be: they are both, ex hypothesi, supported by proof. As for which would be believed, we would, if wise, accept the least miraculous, i.e., the one that has more of its proof still left standing after the one proof is balanced against the other.
Now, not only does one find nothing of this in the defense, the defense very clearly tries to treat Hume's 'in-principle' argument as if the matter were one of demonstration, for it treats the matter as if it were simply a matter of relations of ideas. But this is contrary to Hume's explicit statements; it's a question of balancing experiences to form proofs and probabilities.
So what about the objection? Does Hume beg the question in the in-principle argument? The answer, I think, is this: There is no in-principle argument against miracles. Section One of the Essay on Miracles is not an argument against miracles; it is the groundwork for the argument. The only conclusion of the argument in Section One is that a purported miracle can only be accepted on testimony if that testimonial evidence amounts to a stronger proof than the proof for the law of nature. This is not an argument against miracles; it is an argument for a standard of evidence for admission of miracles, and that is a very different thing. The rest of the Essay presupposes this reasoning, and attempts to show that the evidence for miracles fails to meet the standards of evidence that would be required. Now, this argument for the standard of evidence raises a lot of questions; but it does not obviously beg the question against miracles. Indeed, it explicitly proceeds on the assumption that the purported miracle has enough evidence to be considered as based on proof.
There is one point, crucial to the argument, on which Hume might be begging the question against his opponents: He assumes that the proof for the laws of nature will be proof against miracles, so that they have to be balanced against each other. But it's by no means clear that the proponent of miracles has to accept this; indeed, I suspect most of Hume's contemporaries would not have accepted it, because on their view, 'events according to the laws of nature' and 'miracles' are compartmentalized -- they occupy two domains (both domains are usually regarded as based on laws) within a larger domain, and the evidence for one need not be evidence against the other (any more than evidence for minds need be treated as evidence against matter). But that, perhaps, is a post for another day.