LW-specific: “probability epsilon” when one cannot or doesn’t want to estimate more accurately, but wants to avoid using “zero”, because “zero is not a probability”.
One of Feynman’s stories is about trying to get a NASA manager to quantify the risk that the liquid-fueled shuttle rocket would explode during a launch. After initially claiming zero, he finally allowed it might be “epsilon.” “Now we are getting somewhere” Feynman happily exclaimed. “What is epsilon equal to?”
Some of Feynman’s best stories are about running through semantic stop signs.
I don’t see how this is worse than any other probability estimate when a simple calculation with good numbers isn’t available. Do you consider “probably”, “unlikely”, or “almost certainly” to be semantic stopsigns as well? Would it be better to say “very unlikely” or “with extremely low probability”?
When people act as if the probability of a certain event were zero but try to pacify the reader by using a fake zero called epsilon instead (“Oh, epsilon is very small but non-zero, so I suppose it’s OK”), they consciously or subconsciously throw a semantic stop sign (“I don’t want to be called on this”).
I think they’re saying they don’t want to be nit picked with irrelevant objections. “Huh, you could be living in the Matrix. You can’t rule that out. There’s not zero probability of that, blah blah blah...”
Ah, my social circle’s idiom seems to use epsilon as anything that is tiny but not infinitesimal (for instance, children are called epsilons because they are small). If LW usage is closer to its roots and tries to sneak in the notion of an arbitrarily small quantity instead of a merely tiny one, that’s different, but I’m not sure I’d notice the difference in practice.
I’d say that’s just good practice, as demonstrated by a recent thread by Phil Goetz, and generally advocated by ET Jaynes. Low probability events can be resurrected into consideration by evidence, while zero probability events can’t.
“The probability of X is lower than the disjunction of the probabilities that I have misunderstood the question or you have misunderstood my response.”
In other words, as far as I can usefully communicate to you, the probability is, well, epsilon.
To expand on Kalium’s comment. Sometimes it takes massive amounts of cognitive effort to estimate a small probability, and it just isn’t worth it for the purposes of some discussion. For example, it is extremely unlikely that the Illuminati have been secretly running the world governments for the last 300 years. It is extremely unlikely that ZPP is not contained in P^R where R is an oracle for the finite ring isomorphism problem. It is extremely unlikely that RIPD is going to win any Oscars. I don’t need to work out exact probabilities for any of these to recognize that they are very small, other than to note that the first one is probably less likely than the second, which is probably less likely than the third.
In that context, saying epsilon to mean a small but hard to precisely estimate probability is reasonable.
While I agree with the reasonability of such a shorthand, what bothers me about this choice of term is that in mathematical usage, ‘epsilon’ generally stands for a variable, i.e. is bound by a quantifier: “for every epsilon > 0...”, etc. Thus, although one informally thinks of such an “epsilon” as “a small number”, the real point is that it’s a number that “moves”, usually “hitting” every positive number in some open interval containing 0. This is quite different from standing for a fixed but unknown number.
That seems like a not very persuasive complaint since even professional mathematicians will “epsilon” to mean a very small difference in an informal setting. To use a recent example from discussing a calculus midterm that we were going to make two versions, one of the professors said something like “the midterms will be within epsilon of each other” and no one batted an eye.
Because professional mathematicians understand and depend on the technical usage, there’s little risk of the technical sense becoming diluted by such quasi-humorous, figurative allusions to the technical jargon, which can serve as a means of in-group bonding. When outsiders do it, hover, it’s no longer clearly an allusion to something else, and risks being mistaken for a distinct technical usage in its own right, in addition to losing the slight humor/bonding value.
Another mathematical in-term that has been subject to similar abuse by outsiders is the word “isomorphic”. When a mathematician speaks to a colleague of all the local cafeterias being isomorphic, this is clearly hyperbole—but it’s only clear if one understands the actual meaning and normal context of the word.
From what I’ve seen of cafeterias on large college campuses, it isn’t actually hyperbole to say that “all the local cafeterias are isomorphic”. They’re technically distinct, but under a transformation that preserves all relevant properties, they can all be mapped to each other; they are the same up to isomorphism.
You should be peeved by “negligible probability” unless in the context of the discussion you had previously established what quantity would be below which things were negligible. Negligible literally means “not worth considering,” a value judgement, not a quantity.
One question about semantic stopsigns is the degree to which a given topic is worth pursuing in more detail… just how many sig figs are we going to need on this probability estimate?
LW-specific: “probability epsilon” when one cannot or doesn’t want to estimate more accurately, but wants to avoid using “zero”, because “zero is not a probability”.
One of Feynman’s stories is about trying to get a NASA manager to quantify the risk that the liquid-fueled shuttle rocket would explode during a launch. After initially claiming zero, he finally allowed it might be “epsilon.” “Now we are getting somewhere” Feynman happily exclaimed. “What is epsilon equal to?”
Some of Feynman’s best stories are about running through semantic stop signs.
I don’t see how this is worse than any other probability estimate when a simple calculation with good numbers isn’t available. Do you consider “probably”, “unlikely”, or “almost certainly” to be semantic stopsigns as well? Would it be better to say “very unlikely” or “with extremely low probability”?
When people act as if the probability of a certain event were zero but try to pacify the reader by using a fake zero called epsilon instead (“Oh, epsilon is very small but non-zero, so I suppose it’s OK”), they consciously or subconsciously throw a semantic stop sign (“I don’t want to be called on this”).
I think they’re saying they don’t want to be nit picked with irrelevant objections. “Huh, you could be living in the Matrix. You can’t rule that out. There’s not zero probability of that, blah blah blah...”
Ah, my social circle’s idiom seems to use epsilon as anything that is tiny but not infinitesimal (for instance, children are called epsilons because they are small). If LW usage is closer to its roots and tries to sneak in the notion of an arbitrarily small quantity instead of a merely tiny one, that’s different, but I’m not sure I’d notice the difference in practice.
I’d say that’s just good practice, as demonstrated by a recent thread by Phil Goetz, and generally advocated by ET Jaynes. Low probability events can be resurrected into consideration by evidence, while zero probability events can’t.
Something to consider:
“The probability of X is lower than the disjunction of the probabilities that I have misunderstood the question or you have misunderstood my response.”
In other words, as far as I can usefully communicate to you, the probability is, well, epsilon.
To expand on Kalium’s comment. Sometimes it takes massive amounts of cognitive effort to estimate a small probability, and it just isn’t worth it for the purposes of some discussion. For example, it is extremely unlikely that the Illuminati have been secretly running the world governments for the last 300 years. It is extremely unlikely that ZPP is not contained in P^R where R is an oracle for the finite ring isomorphism problem. It is extremely unlikely that RIPD is going to win any Oscars. I don’t need to work out exact probabilities for any of these to recognize that they are very small, other than to note that the first one is probably less likely than the second, which is probably less likely than the third.
In that context, saying epsilon to mean a small but hard to precisely estimate probability is reasonable.
While I agree with the reasonability of such a shorthand, what bothers me about this choice of term is that in mathematical usage, ‘epsilon’ generally stands for a variable, i.e. is bound by a quantifier: “for every epsilon > 0...”, etc. Thus, although one informally thinks of such an “epsilon” as “a small number”, the real point is that it’s a number that “moves”, usually “hitting” every positive number in some open interval containing 0. This is quite different from standing for a fixed but unknown number.
That seems like a not very persuasive complaint since even professional mathematicians will “epsilon” to mean a very small difference in an informal setting. To use a recent example from discussing a calculus midterm that we were going to make two versions, one of the professors said something like “the midterms will be within epsilon of each other” and no one batted an eye.
Because professional mathematicians understand and depend on the technical usage, there’s little risk of the technical sense becoming diluted by such quasi-humorous, figurative allusions to the technical jargon, which can serve as a means of in-group bonding. When outsiders do it, hover, it’s no longer clearly an allusion to something else, and risks being mistaken for a distinct technical usage in its own right, in addition to losing the slight humor/bonding value.
Another mathematical in-term that has been subject to similar abuse by outsiders is the word “isomorphic”. When a mathematician speaks to a colleague of all the local cafeterias being isomorphic, this is clearly hyperbole—but it’s only clear if one understands the actual meaning and normal context of the word.
From what I’ve seen of cafeterias on large college campuses, it isn’t actually hyperbole to say that “all the local cafeterias are isomorphic”. They’re technically distinct, but under a transformation that preserves all relevant properties, they can all be mapped to each other; they are the same up to isomorphism.
Do you have evidence that people are actually misunderstanding what either of these terms mean?
Would you be equally peeved by “negligible probability”?
You should be peeved by “negligible probability” unless in the context of the discussion you had previously established what quantity would be below which things were negligible. Negligible literally means “not worth considering,” a value judgement, not a quantity.
One question about semantic stopsigns is the degree to which a given topic is worth pursuing in more detail… just how many sig figs are we going to need on this probability estimate?