Billy seemed to have something of a sixth sense regarding the Predator, becoming aware of the Jungle Hunter's presence, even though he could not see the creature. In the novelization he is said to be actively psychic, able to access the memories of his ancestors and the ancient Mayans who used to live in the jungle, and can sense the Predator's presence. When Hawkins and Blain were killed by the Predator, Billy accepted that they were all going to die by the creature's hand and insisted that the hunter was not human.
Similar themes are brought up again in Predator 2 with King Willie. While Harrigan was convinced the suspect was merely a hitman, King Willie believed otherwise; through his beliefs and superstitions, he came to believe that a "demon" was responsible, and that the unstoppable creature would likely be the death of them all. He was very superstitious and a practitioner of fortune-telling, specifically cleromancy, which he practiced using a handful of small animal bones. From this he claimed to gain knowledge of the future and other hidden events. It was supposedly through this fortune-telling that he became aware of the other-worldly nature of the City Hunter stalking Los Angeles.
The sequence where King Willie is killed is very similar to Billy's death in Predator, whereby viewers see the build-up to the fight and its aftermath, yet the battle itself occurs off-screen. Both Billy's and King Willie's skulls are taken as trophies.
In Predator: Big Game, Nakai faces a Predator who he believes is a Navajo legend come back to life: the return of the Horned Monster, who was destroyed by the hero Nayenezgani, the mythical Monster Slayer. To defeat this hunter he rediscovers his Navajo roots by following in the Monster Slayer's footsteps. Using the wisdom of his ancestors, Nakai confronts the Predator in a final battle. In this story, Nakai receives visions from the spirit world and there is a fulfilment of an ancient prophecy. Nakai's story is continued in Predator: Blood on Two-Witch Mesa, in which he finds out that his great-grandfather also witnessed a Predator attack 75 years earlier. Nakai suggests that it could have been the same Predator that attacked him.
Another example of the fulfilment of a prophecy is in Predator: Blood Feud. Shogun Tokugawa Tsunayoshi and his forces face sudden attack by a group of Yautja, ending in a final duel between one Yautja and a blind swordsman, chosen by Shinto monks. The monks make prophecy that the duel will continue in the future where the swordsman, in some form, will have to face the Yautja again in another duel. In the future, an age-old Yautja returns to modern day Japan to hunt a blind martial arts instructor — the descendant of an adversary it faced in feudal Japan centuries earlier.
So what does this mean for the Predator Universe? There evidently appears to be an otherworldly aspect playing in the background of these stories. Are there ancient prophecies being fulfilled? Are there psychics who can use the spirit world to access the memories of their ancestors to help them sense the presence of a Predator? Or is it simply these characters rationalising the events with their beliefs. Evidence seems to point to the former.
Credit to Xenopedia for the information.