In Tim Gallagher's recent seminar, he says that he believes the "Ivory-bill" that he saw was a female, since he didn't see any red (or any other feature) on the head. Sparling reported seeing red on the head of his bird, which means that if it was an Ivory-bill, it had to be a male.
Putting those observations together, we'd have a pair of Ivory-bills, one of each sex, seen within a quarter of a mile of each other. The area is accessible via a short paddle from a busy road. In 20,000 hours of intense searching, neither bird was ever seen well or photographed; each of the "tantalizing glimpses" was of a single bird, never a pair.
I still think it's possible that an Ivory-bill lived in Arkansas in 2004. However, to "believe" at this point, I think you've got to accept a mountain of implausibilities like the paragraph above.