Let us take inductive inference to consist in extrapolating the broadest-according regularities of our experience to universal laws, each within some larger domain or context than the experiences themselves. On a small scale this is easy to envision, and we can appeal to such canonical examples as the inference that all swans are white from a large random sample of uniformly white swans. However, we must also take inductively-inferred laws to "accord broadly" with all of the regularities of our experience. For instance despite the fact that we have only ever personally encountered white swans, perhaps we have heard from a reliable source (wikipedia?) that there exist black swans. To infer that all swans are white under these circumstances might accord narrowly with our first-hand experiences of swans, but not broadly with our other non-swan experiences, namely the experiences which lead us to decide that our source for information on black swans is reliable. So inductive inferences must in this sense comport with the "big picture," so to speak, which is to say that wherever the narrow regularities of our experience conflict, induction must follow the most well-evidenced, i.e. the most well-represented, of these.
Suppose we have experiences which we wish to conceptually model according to induction, and consider the question, could God ever appear in such a model? More generally, suppose an actual object O causes us to have some set E of experiences. Let F denote the subset of experiences in E which follow some discernible order and/or regularity. Then in the best case scenario we can infer that there exists an object P which is like O in the sense that, in our conceptual model, it causes the sort of experiences contained in set F. (Here we are using some unspoken assumptions about uniqueness and maximality of subsets of regular experiences and the identity of objects whose existence we infer from them, but given the context this seems tolerable.) In this way, P is an object in the model which corresponds to the actual object O, i.e. P is a conceptual model of O. Now, O also causes the sorts of experiences contained in E-F, where E-F denotes the set of experiences contained in E but not F. Recalling that induction works only via regularities of our experience, we see that since the experiences in E-F follow no discernible order and/or regularity then we cannot inductively infer that P (nor any other object) is causing the experiences in E-F. So if it is an essential property of O that it causes experiences of a sort contained in E-F, then since we cannot infer that P causes those kinds of experiences it means that we have not inferred the existence of O when we infer the existence of P. Ideally, we would say that if a model object P whose existence has been inferred sufficiently resembles some object O which actually exists, then we have inferred the existence of O, thus permitting if not delivering knowledge of O. However in this case there is some essential property of O which is not shared by P, and so this prevents us from modeling O with P, and hence from inductively inferring that O exists.
To put it another way, when the object O in question has as an essential property that it transcends the regularities of our experience, then we can never inductively infer its existence. So we can never inductively infer the existence of an essentially omnipotent being like God, since omnipotence by definition involves the capacity to transcend the regularities of our experience. Notice, however, that we must use the qualifier "essentially," because it is nevertheless possible (in principle) to infer the existence of a being which possesses a non-essential property involving transcendence of the regularities of our experience. So, for example, it could be that Barack Obama can work a regularity-transcending form of magic. Although we could never inductively infer this hypothetical fact about him, we can nevertheless know that there is such a person as Mr. Obama---at least as long as we decline to take his supposed possession of magical powers to be one of his essential properties (which seems reasonable enough not to do).
In the case of God, we might possibly be able to infer the existence of an intelligent being B whose actions in the world correspond to some significant subset of God's own actions. However, we can never inductively infer that B has any of God's regularity-transcending abilities. Namely, we cannot attribute miracle-working to B, unless we water down our conception of what constitutes a miracle at least enough so that we can speak of miracles following the broadest regularities of our experience. So for instance, given that resurrections run afoul the regularities of our experience, we cannot currently infer through our experiences the existence of a being with (essentially) the power to raise the dead, unless those experiences change so dramatically that resurrections come to comport best with them.
Consider the following instructive example: Suppose we encounter a being B which has never failed despite extensive testing to deliver upon request perfectly truthful information. Perhaps we ask B a few dozen times for the winning lottery numbers in advance, and each time B tells us the winning numbers. Moreover, we could ask for descriptions of future events which we later observe to come to pass exactly as B has promised. And so on, every verifiable statement of B checks out to be true, despite the fact that we have no explanation for its remarkable truth-reporting abilities. Under these hypothetical circumstances, suppose further that B claims to be able to work a miracle, say raising Elvis Presley from the dead. In order to evaluate his claim, we must weigh the inductive evidence for his consistency in truth-reporting against the inductive evidence for the consistency of dead bodies to remain dead. If we decide that B's truth-reporting exhibitions have the greatest inductive weight between the two, then the "miracle" of raising Elvis will not in a broad sense transcend the regularities of our experience. If on the other hand we decide that the inductive evidence for B's truth-reporting is insufficient to outweigh the inductive evidence for dead bodies staying dead, then we shall be unable to infer that B has the power he claims to raise Elvis. So whichever turns out to be the case, we can never inductively infer that which runs against the broadest regularities of our experience.
In the preceding illustration we can see how it happens that miracles which we would currently regard as transcending the broadest regularities of our experience are not strictly epistemically off-limits, and we can infer their occurrence by accumulating sufficiently strong inductive evidence. To show that a miracle has occurred, then, we must weaken its status as a bona fide miracle, that is, we have to show that it broadly accords with the regularities of our experience. From a practical standpoint, though, this is as good as impossible; for example we cannot hope to prove through, say, examining historical documents and archaeological remains that the laws of physics were suspended in first-century Palestine. Moreover, in the case of omnipotence such weakening is impossible not just in practice but also in principle, since part of what it means to be omnipotent is to be able to transcend the regularities of our experience. So any being, namely God, for which omnipotence is an essential property can never be shown to exist by inductive inference.
To a great extent this is nothing new. Hume, for instance, and others after him have already dealt most brilliantly and succinctly with the subject of weighing inductive evidence for and against miracles, and I could not hope to surpass their marvelous eloquence. Rather, I intend for the preceding arguments and observations to show that we can go yet a step further from Hume and observe the consequences with respect to those hypothetical objects whose essential properties involve the transcendence of the regularities of our experiences. The existence of such objects can never be inductively inferred since by hypothesis they reside, so to speak, at least in part outside the context of those universal laws which extend the regularities of our experience; and through similar reasoning we see that neither can their nonexistence be inductively inferred. Immediately corollary to this conclusion, no causal argument for or against the existence of God shall ever succeed as long as we take omnipotence to be an essential property of God. Furthermore, no cumulative case can ever amass sufficient strength to show the existence or nonexistence of God as long as it depends on induction for its summed force. In fact, any justification we have for beliefs about the existence of God must have a non-inductive character. Given that induction appears necessary to learn about the world beyond our own present experience, internally-justified knowledge of the existence or nonexistence of God thus seems beyond our grasp.