A common technique of theists is to exploit the problem of induction. This takes the form :
1) Point out that fact X is a standalone fact which must be interpreted by some system.
2) Argue that atheists interpret X under a naturalistic/materialistic worldview which excludes God or the supernatural by presupposition.
3) Attempt to demonstrate that X can be interpreted just as validly under a worldview which does accept the supernatural.
Example: 'The evidence of God is all around you. You are just blinded by naturalism and science'.
So do theists have a point? Are atheists ignoring the evidence by being closed minded and unfairly privileging the philosophical position of naturalism ? I do not believe we are and to understand why, we must first talk briefly about the problem of induction.
The problem of Induction
With inductive reasoning we move from premises about objects we have examined to conclusions about objects we haven't examined. For example :
The first six cartons of milk from the crate were sour.
All the cartons have the same use-by date.
Therefore the seventh carton will be rotten too.
We haven't examined the seventh carton but we can use induction to conclude that that cartoon is also sour.
The danger of induction is that we can easily move from true premises to a false conclusion without violating any logical laws. Deduction on the other hand is safer and therefore more reliable because if we start with true premises, then the conclusion should be true if we obey the laws of logic.
But despite the problems with induction we cannot live without it. When I press the power button on my PC, I expect the PC to turn on even though I have no logical grounds for believing this apart than past experience. I expect the sun to rise tomorrow because it has always done so in the past.
Induction according to the 18th-century Scottish philosopher David Hume cannot be rationally justified because inductive reasoning presupposes what Hume called the 'uniformity of nature': that objects we haven't examined will be similar in the relevant aspects to objects of the same sort of have examined.
To understand the power of the argument, imagine trying to persuade somebody who refused to accept induction. You might say 'My computer has always worked when I pressed the power button, the sun has always risen, the laws of physics appear to behave consistently'. But such arguments will fail because they themselves use inductive reasoning and are therefore circular because you are attempting to use inductive reasoning to prove the validity of inductive reasoning. And because science relies heavily on induction and induction has not been rationally justified, then science itself is not as rational as scientists would like us to believe. Or so the argument goes.
This is the problem of induction theists like to exploit. It has yet to be satisfactorily answered.
Inference to the best explanation
An inference to the best explanation (IBE) can be defined as an inductive inference where we move from examined to unexamined instances. For example, suppose I left a closed cookie jar alone with a child and a dog. I returned to find several cookies missing from the jar and the jar is undamaged. The best explanation is clearly that the child has eaten the cookies and replaced the lid. This is non-deductive but reasonable. Dogs do not have the ability to unscrew the lid, remove several cookies then replace the lid.
If we want to use IBE, and we clearly use it frequently in every day life, we must ask : which of the competing hypotheses provides the best explanation? Is X evidence of design by a creator or is X evidence of natural selection ? One possible technique is to deploy Occam Razor to remove the unnecessary premises and explanations involving unproven entities. Another is to use probability and statistics. Probability may appear appealing but all forms of probability fail to solve Hume problem of induction. (This will need another post to explain).
So induction cannot be fully rationally explained and neither can IBE which is itself a form of induction (or if you prefer: IBE is an non-deductive argument).
It's also worth pointing out that these are problems of philosophy and not of science itself : if it fits the evidence and it's testable and it has predictive powers, then well and good.
Deus ex Machina
The Latin term Deux (God) ex machina refers to a classical plot device of swinging a deity onto stage to wrap up a plot and bring the play to an end. But Even during classical times such techniques were considered sloppy :
It is obvious that the solutions of plots too should come about as a result of the plot itself, and not from a contrivance, as in the "Medea" and in the passage about sailing home in the "Iliad". A contrivance must be used for matters outside the drama—either previous events which are beyond human knowledge, or later ones that need to be foretold or announced.Aristotle, Poetics.
IBE and Deus ex Machina
Deploying a God or Gods to solve a problem is the philosophical equivalent of lowering a God figure onto a stage to wrap up a plot. In theater, it's considered almost cheating because the creditably of the plot is scarified for a convenient ending. In philosophy deploying God to solve a problem we cannot solve for ourselves is both cheating and disingenuous because it answers nothing, gives us no new information and lacks explanatory power. The entire premise of science and philosophy goes back to the Greeks and the principle of sufficient reason.
So when theists argue 'X is evidence of God' they are exploiting the problem of induction. Most problems have a number of possible answers or ways of interpreting the information and inference to the best explanation is itself a form of induction and so cannot be rationally proven. But not all answers are equally reasonable and deploying a deus ex machina will leave your audience feeling dissatisfied and cheated, whether you are a scientist, a philosopher, a playwright or all three. The theist paradigm was exposed then destroyed during the scientific revolution.