What thought processes do atheists use that misguides them into not believing? What are they missing?

Good question. It's been something I've been meaning to write about for a while.

Speaking broadly (with many exceptions), atheists have a tendency towards skeptical empiricism, or the notion that one should only believe things one sees with one's own eyes, and ideally verified, and then maybe not even then.

The trouble with this thinking is that if you could somehow add up all the facts that one knows, only a very small percentage of these facts have been personally seen, let alone verified. We all have a cognitive bias which greatly overestimates how much we've tested for ourself. "See, this GPS device wouldn't work if relativity wasn't true!" doesn't count, for example. So it's a very fragile belief system to begin with, as these atheists, if they realized just how little they hold as true can actually be empirically confirmed, they'd lose the ability to speak.

This fragile foundation is further troubled by the question of what evidence means. If you follow that link above, he defines skeptical empiricism as "demand empirical evidence for everything". Logic is meaningless noise.

The problem, of course, is that evidence is not the same thing as saying proof. Evidence is anything that gives support to an argument. Proof is compelling evidence. Attorneys on both sides of a court case will introduce evidence that they think will influence the jury or judge to rule their way. The trouble is, evidence can often be presented for both sides: "Here's a picture, your honor, of my client's car in New York at 1100." vs. "That's nice, but here is the sworn testimony of two witnesses who saw him in New Jersey at 1100." There is no contradiction in having evidence for both sides. And people can entertain beliefs either way if evidence has been presented for both sides in an argument.

Proof, by contrast, is a clincher. Nobody who accepts the proof as legitimate (for example, a video of the defendant robbing a liquor store in New Jersey) could reasonably hold a belief contrary to the proof.

But atheists repeatedly, time and again, confuse this issue. They think evidence means proof, and proof means evidence. (If you'd like, imagine each word in that sentence is a link to another atheist misusing the terms, but I don't want to embarrass anyone or brigade them.)

So when they say, "There's no evidence for God", what they actually mean is that there's no proof for God. There's plenty of evidence. Even if you are an atheist and think it is weak evidence, you still can't deny that the gospels, to witnesses, logical arguments for God, and so forth exist.

So, in summary, atheist epistemology (again, this is a generalization) is this: 1. Only accept beliefs that have empirical evidence. Hold logic and such philosophizing in deep skepticism. 2. Ignore that this undermines most of your one's own beliefs. 3. Ignore the fact that we've thrown out logic, which is literally the set of all things that must be true. 4. When dealing with religion, change the standard from evidence to proof, but still call it evidence. 5. Claim there is no evidence for God, and therefore one cannot logically hold that belief. 6. Ignore the fact that we just made a logical argument.

I it's clear the flaws in this belief system are large enough to drive a truck through.

Even worse than this is the Scientism that a certain fraction of atheists here adhere to, which is even more fundamentally flawed, but that's a post for another time.