What do you mean by Pascal's mugging?

None of the above are Pacal's Mugging, as stated. While some people have taken to using Pascal's Mugging as a generic term for anything that's very-low-probability and very-high-impact, I think that's missing the point of the original thought experiment. Pascal's Mugging is a scenario in which the size of the impact and the smallness of the probability are entangled together. It shows that, if your utility function and epistemology are broken in a particular technical way, then no degree of improbability, no matter how astronomical, will suffice to let you ignore something.

Given our ignorance we cannot rationally give zero probability to this possibility, and probably not even give it less than 1% (since that is about the natural lowest error rate of humans on anything)

I am pretty sick of 1% being given as the natural lowest error rate of humans on anything. It's not.

In this particular case, we've made balls of stuff much colder than this, though smaller. So not only does this killer effect have to exist, but it also needs to be size-dependent like fission.

If you give me 100 theories as far-fetched as this, I'd be more confident that all of them are false, than that any are true.

I keep seeing people responding to Pascal's Wager / Mugging by saying we just shouldn't pay attention to very low probabilities. (IIRC Eliezer said something similar as well). But intuitively I don't think this is true.

Imagine that a stranger comes up to you and offers you a cup of water and tells you, "Please drink this to humor me. It has a chemical I invented which seems to kill a tiny fraction of people in fascinatingly gruesome ways. But don't worry, I've already tested 10,000 people and only two of them have been affected." I have a hard time imagining that anybody would drink it.

The way I usually look at Pascal's Wager / Mugging is that this is just one of those paradoxes we seem to get into when we take decision theory and probability theory to extremes. There may or may not be a good decision-theoretic response to what precisely is the answer to the paradox, but in the meantime we need to go with our intuition that it's just wrong, even if we don't know exactly why.

So ultimately I think we just need to go with our intuitions on this. Is 1:10,000 too small a probability? What does your heart tell you?

Side note: I noticed that Nick Bostrom seems to invoke these sort of arguments several times in his book. I think he mentioned it regarding how much we should worry about x-risks especially given our astronomically large potential cosmic endowment. I also think he alluded to it when he mentions that a superintelligence would first take steps to prevent all sorts of catastrophic risks before proceeding with some new technologies.

In my current (fuzzy) thinking, it can be considered Pascal's Mugging if the probability is several orders of magnitude too low for you to even attempt to calculate, and your 'mugger' therefore needs depend on the payoff being large enough to try to outweigh in your mind the not actually calculated tininess of the probability.

Your scenarios 1, 2 and 4 fall in that category where I'm concerned. Your scenario 3 does not -- the payoff being limited to merely the global ecosystem means that I can try to evaluate the probability I assign to the systemic risk of GMOs, and evaluate the benefit vs the risk in a normal fashion. Other technologies that may increase existential risks because of weaponisation or accident (nuclear fission, AI, nanotechnology, advanced bioengineering) can be treated similarly to GMOs with normal benefit-vs-risks calculations, they're not Pascal's muggings.

I use the term for any case where people try to multiply a high utility by a low probability and get a substantial number. I think this is an error in general, and it's useful to have a term for it, rather than reserving the term for a specific technical scenario which is no longer especially interesting.

Well, with 4, as long as you're just floating the hypothesis rather than attempting a Pascal's Mugging and claiming you have knowledge from outside the Matrix that this is true, we have no evidence at all indicating it's more likely than the equal-and-opposite hypothesis that it causes 3^^^^3 negative singularities - this probably also applies to 1, but not 2 and 3, it seems if they have extreme outcomes they're more likely to go one way than the other - even considering that they might somehow produce FAI, it's more likely that they'd produce uFAI.