In your opinion what causes christians to believe in Jesus

[Sheldon]

Is this the kind of thing you're thinking of? I'm curious about what you mean.

openness to evidence requires openness to possibility. 

What does "openness to possibility" mean? It has already been explained that not believing a deity is possible, is not a belief or claim a deity is impossible. I am not closed minded, I treat all claims to the same standard. 

the very act of seeking evidence for X implies that X is treated as possible. 

No it doesn't, this seems like begging the question to me. 

But of course, many here do not think the existence of God is even a possibility, and therefore I think them asking for evidence is illogical.

My threshold for possibility is simple: If a proposition is at least conceivable and coherent, and not contradictory, then I'm open to its possibility. 

Does this mean you presuppose it is possible, or that you simply don't rule it out a priori, as no one here would I suspect do either of those. I certainly would not. 

But people want me to give objective evidence for possibility itself. And I think that's a weird request. 

What's weird is coming to an atheist forum, making claims about a deity, and then expecting people not to insist you support those claims. 

So, with your boar example, I would say the possibility is already there, and the evidence you showed simply makes the case for its actuality.

If something can be demonstrated to exist, then it is demonstrably possible. 

I don't think there is such a thing as objective evidence of possibility itself, 

Of course there is, existence of X would be a pretty obvious example of objective empirical evidence that the existence of X is possible. One could demonstrate that steel is possible, by testing the process of manufacturing it. There must be countless other ways to demonstrate possibility objectively and empirically. 

possibility is abstract, it is rational rather than empirical. 

No it isn't, again the existence of an elephant is empirical evidence an elephant is possible. It literally can't exist if it is not possible. 

I would even say it's wholly subjective, it is part of your personal value judgment.

Oh dear John, of course it isn't, don't be silly. it's easy to empirically test possibility, and the results would be objectively verifiable, simply by designing a test that could be replicated, and testing to see if it produced the same results every time. The possibility that amlodipine meds will lower blood pressure, can and has been empirically tested, there is nothing subjective about the evidence.