Epsilon is not a probability, it's a cop-out

Have you calibrated the numbers that come out of this technique?   :whistles_innocently:

I tend to use ε for shorthand for cases where there aren't any sensible ways to update to, say, 5% probability of the thing happening.

Well, why don't you use zero instead, what is the difference, instrumentally?

I think there is (sometimes) value in distinguishing two separate probabilities for any given thing. There's the "naïve" probability that you estimate while ignoring the possibility that you've blundered, that you misread something critical, that some underlying assumption of yours is wrong in a way that never crossed your mind, etc. And then there's the "pessimistic" probability that tries to account for those things.

You want these to be separate because if you're doing a calculation using the various probabilities, sometimes it's better to do all the calculations using "naïve" probabilities and then do a final correction at the end for blunders, wrong fundamental assumptions, etc.

... Maybe. It depends on what the calculation is, what sort of out-of-model errors there might be, etc.

Of course this is a rough heuristic. I think what it's an approximation to is a more careful tracking of lots of conditional probabilities (people around here sometimes talk as if being a Bayesian means assigning probabilities to things, but it would be more precise to say that being a Bayesian means assigning conditional probabilities to things, and a lot of the information is in that extra structure). E.g., suppose there are 100 things, each of which you give naïve probability 10^-9 to, but there's a 10^-3 chance that some fundamental error in your model makes them actually happen 1/10 of the time. Then your "adjusted" probability for each one is about 10^-4, and if you use those to estimate the probability that at least one happens you get about 10^-2; but in this situation -- assuming that the "fundamental error in your model" is actually the only substantial cause of out-of-model errors -- that probability should actually be more like 10^-4. Of course, if you make a calculation like that then sometimes there's a fundamental error in your model of where the possible errors come from :-).

Weird, I'm totally in the other boat. I think we can use sub-1% or super-99% probabilities easily, all the time.

I just went on a long road trip. What probability should I have used that my car springs a brake fluid leak slow enough that it's going to be useful for me to have a can of brake fluid in the car? I'd guess it happens once every 250k miles or so, and I just drove about 1k, so that's about 1 in 250 (or let's say 1 in 500 to guesstimate at the effect of doing highway driving). Bam, sub-1% probability. Now, did I need to consciously evaluate the probabilities to decide that I should definitely bring engine oil, might as well bring brake fluid but it's not super important, and not need to bring a bicycle pump? No. But it you ask me, I don't see what's stopping me from giving totally reasonable probabilities for needing these things.

Yeah, being inaccurate about personal fallibility levels is something I was trying to gesture at in https://www.lesswrong.com/posts/3duptyaLKKJxcnRKA/you-are-way-more-fallible-than-you-think [LW · GW]

and I think your comment summarizes what I wanted to express.

That depends on the subject. People are not as fallible on life or death subjects or more people would accidentally walk in front of cars, trains or fall of high places. Anybody have an idea what that probability would be?

I mean more or less the same thing, but I think I'm lazier <3

I've noticed myself say epsilon as a placeholder for "that is not in an event space I've thought about and so I was probably acting like the probability was 0%"...

...then I think rigorously for an hour and end up with a probability as high as 10% or 7% or so. So for me I guess one hundred and one trillion are "roughly equally too big to really intuit without formal reasoning"?

Part of the challenge is having a habit that works with a range of possible situations and audiences.

Some people say 100% (or 0%) and are wrong about 25% of the time in such cases. 

This is so common that if I just say "100%" near another person who is probabilistically numerate and calibrated on normal humans, they might hear "100%" as "75% (and this person cannot number real good (though they might human very well))".

Some people say 99.999% as a way to say they are "as absolutely sure one can be, while pragmatically admitting that imperfect knowledge is impossible"... 

...but they usually don't think that they've just said that "in 49,000 tries the thing would probably never be observed, and with 100,000 tries ~1 case would occur, and if you were searching for true cases and each person was a case, then ~70,000 people (out of ~7 billion) are actually probably like this, and if they exist equally among the WEIRD and non-WEIRD then ~3500 people who are positive cases are in the US, and therefore an ad campaign could probably find 100 such people with only a small budget".

For these reason, I try to say "1 in 50" and 98% roughly interchangeably, and for probabilities more extreme than that I try to stick to a "1 in N" format with round number N.

Like "1 in a trillion" or "1 in 5000" or whatever is me explicitly trying to say that the main argument in favor of this involves some sort of "mass action" model based on known big numbers and imaginable handfuls of examples. 

A person has ~500M "3 second long waking thoughts" per life.  There are approximately ~50k municipalities in the US.  There are ~10M people in the greater LA region (and in 2010 I could only find ~4 people willing to drive an hour on a freeway to attend a LW meetup). And so on.

Example: There are ~10k natural languages left (and I think only Piraha lacks a word for "four" (Toki Pona is an edge case and calls it "nanpa tu tu" (but Toki Pona is designed (and this is one of the worst parts of its design))). So "based in informed priors" I tend to think that the probability of a language having no word for "four" is roughly 1 in 5000. If betting, I might offer 1 in 500 odds as a safety margin... so "99.8%" probability but better to say "between 1 in 500 and 1 in 15,000" so that if someone thinks it is 1 in 100,000 they can bet against me on one side and if someone else thinks it is 1 in 100, then I can think about taking both bets and arb them on average :-)

If I say "epsilon" in a mixed audience, some people will know that I'm signaling the ability to try to put high quality numbers on things with effort, and basically get sane results, but other people (like someone higher than me in a company hierarchy who is very non-technical) might ask "what do you mean by epsilon?" and then I can explain that mathematicians use it as a variable or placeholder for unknown small numbers on potentially important problems, and it sounds smart instead of "sounding ignorant". Then I can ask if it matters what the number actually might be, and it leads the conversation in productive directions :-)

My guess is that the work of quantifying small probabilities scales up with how small probability is, given the number of rare events that one has to take into consideration. I wonder if this can be estimated. It would be some function of 1/p, assuming a power law scaling of the number of disjunctive possibilities.

This is actually not a bad example, because the definition of "sun rising" is ambiguous. Does it mean that it doesn't get bright in the morning because of something? Does it mean that the Earth and the Sun are not in a particular arrangement? Would a solar eclipse count? Ash from a volcano? Going blind overnight from a stroke and not being able to see the sunrise? These are the disjunctive possibilities one has to think through that contribute to any unlikely event. If you reply is "I intuitively know what I mean by sun not rising" without actually going through the possibilities, then you don't know what you mean.