In cosmology, we can't directly measure what goes on in black holes. For one thing, they're too far away and even if we could reach one, once we've penetrated its event horizon, no signal could escape its gravity. No signal could return to Earth.
Quantum physics has many measurement problems, one is called The Measurement Problem:
Even seemingly straightforward measurements aren't so straightforward. If you wanted to measure the length of a coastline, how would you do it? If you drove it and used your odometer, you might get a length of 100 miles, but if you walked it with a tape, so you could better measure each little inlet and protrusion, you might get a length of 300 miles.
Health physics has its measurements problems. I'll draw your attention to two of them, which are very similar. One is rarely denied, but the other seems to induce denialism, usually in those who are already biased.
The first involves a radioactive spill. Let's say one occurs. Invariably, people clean up what they can see. But there is a measurement problem, our eyes can only resolve so well. Since we have some grasp of fundamental physical properties like porosity, adhesion, absorption, etc. we don't conclude there is no more radioactivity on the surface just because we can't see it. We know better. We don't even think the chance is 50-50, there's almost certainly residual radioactivity there.
So we get a handheld radiation detector which has better resolving power than our eyes:
And we'll try to get the contamination levels down so that the detector shows no increase above background when we survey the area. Does that mean there's no radioactivity on the surface? Of course not, it's just what little is there can't be resolved with that detector. This isn't a 50-50 judgement either, there is almost certainly residual radioactivity there.
Get a more sensitive instrument, like a laboratory quality one, and you'll find some residual radioactivity. The burden of proof of NO excess radioactivity at any point in the spill cleanup is on the person making that claim, because we understand the fundamental physical properties.
Practically no one has any psychological issues with what I've just described. Now let's look at the second health physics measurement problem in this case it's dealing with LNT theory.
If we have a radiation exposure to many people (analogous to a spill) we can detect how many eventually get cancer (analogous to visualizing a spill). We expect an increase in cancer because we understand the fundamental physical principles like mechanisms of radiation interactions with DNA, DNA repair mechanisms, DNA damage leading to cancer, etc. So when we encounter an expected resolution problem, we don't claim there is no more cancer induction at low doses. We know better. It's not a 50-50 judgement either.
In order to get better resolution, we'd have to intentionally expose a much larger population to radiation, which would be unethical (because it would increase their risk of cancer). But because we're still speciesists (like "racists", but applied to non-humans) we don't consider it unethical to intentionally irradiate mice, where we can get better resolution. And that's what we do (analogous to using handheld detector) in order to develop better risk resolution at low doses.
And not surprisingly, we find at lower doses than we can see in humans, an increased risk of cancer in mice. We use the mice data to better inform us of the potential cancer risk with humans. But there is still a resolution problem. Does that mean there is zero chance of radiation causing cancer at even lower doses? Of course not. The burden of proof of NO excess cancer at any point is on the person making that claim, because we understand the fundamental physical properties.
To get even better cancer risk resolution (analogous to using a laboratory quality radiation detector) we look at molecular biology. The problem has been that the molecular biology is so complex there hasn't been any prominent phenomenon that helps inform us that we should further modify our cancer risk estimates. So we still await the evidence that will convince us to shift our risk estimates. And when (if?) we get that evidence we'll shift accordingly. Bizarrely some people have a problem accepting this measurement problem and shift into health physics denial.
It's just a measurement problem within a measurement problem.