How to "prove" a supernatural entity exists?

I know the innate contradiction present here: one cannot "prove" the supernatural exists, for a variety of reasons. But for the sake of argument I want you to consider this:

A person is convinced that they have been haunted and oppressed by an entity (a ghost, a demon, an inter-dimensional horror as of yet undetermined, etc.) for the majority of their life. This thing has made their life a living hell: scaring them, inconveniencing them, making them a social pariah, basically making it impossible for the person in question to live a normal life. It finally comes to the point that this individual sets out to prove that this entity exists, if only in the vain hope of demonstrating to people that they aren't simply crazy. Let's say they ignore or are unaware of the futility inherent in a task like this and they pursue it with all of their being.

So...how does one go about doing this?

Let's get some obvious things out of the way:

-Naturally, I figure they might go to religious figures to see what they would have to say on the matter. Christianity is not necessarily the only religion they would look into.

-Similarly, they might approach someone claiming to be a medium or a psychic to see what they had to say.

-To get a scholarly opinion, they may call a collegiate or university professor, or a similar "expert" in the field of the paranormal. Maybe popular demonology?

-Maybe for a more personal, down-to-earth approach, they might speak to people they've found who have had similar experiences?

-They might go so far as to seek out a ghost-hunting team of paranormal investigators, for better or worse.

-They may invest in camera equipment to film their everyday lives in the hopes of capturing paranormal activity. Maybe audio equipment as well.

-They may, regardless of their belief to the contrary, get a psychiatric evaluation, to at least see what a psychological explanation might be.

...and now maybe the not-so-obvious things:

-I'm very interested how someone might apply the scientific method to this situation. How would someone logically try to apply this to a haunting to prove if it was true, at least in their mind? Like: could an experiment actually be done to prove or disprove a hypothesis that this thing is real? How would that work?

-I'm also curious about applying philosophical argument to this (abductive, inductive, etc.) and how it could be "proved" - or disproved - the reasoning behind this event is that it's supernatural.

-If, say, the person being haunted wasn't willing to admit this thing was a demon or a ghost, BUT ALSO WASN'T WILLING TO SAY THAT THEY WERE CRAZY...then what would their explanation for this thing might be? How far might they go into saying the thing is, say, inter-dimensional? From a higher plane of existence? Would they dabble heavily into fringe science and pseudo-science? What would that person try to look for outside of these more traditional ideas as an explanation?

-I know there are minutiae and intricacies to the above sections I have overlooked and missed. Any fleshing out of those, or inclusions of things I haven't even thought of, would be greatly appreciated.