"Do not walk to the truth, but dance. On each and every step of that dance your foot comes down in exactly the right spot. Each piece of evidence shifts your beliefs by exactly the right amount, neither more nor less. What is exactly the right amount? To calculate this you must study probability theory. Even if you cannot do the math, knowing that the math exists tells you that the dance step is precise and has no room in it for your whims." -- from "Twelve Virtues of Rationality", by Eliezer Yudkowsky
One of the more useful mental tools I've found is the language Lojban ( http://www.lojban.org/tiki/Learning ), which makes explicit many of the implicit assumptions in languages. (There's also a sub-language based on Lojban, called Cniglic ( http://www.datapacrat.com/cniglic/ ), which can be added to most existing languages to offer some additional functionality.)
One of the things Lojban (and Cniglic) has are 'evidentials', words which can be used to tag other words and sentences to explain how the speaker knows them: "ja'o", meaning "I conclude", "za'a" meaning "I observe", "pe'i" meaning "It's my opinion", and more. However, there hasn't been any easy and explicit way to use this system to express Bayesian reasoning...
... until today.
Lojban not only allows for, but encourages, "experimental" words of certain sorts; and using that system, I have now created the word "bei'e" (pronounced BAY-heh), which allows a speaker to tag a word or sentence with how confident they are, in the Bayesian sense, of its truth. Taking an idea from the foundational text by E.T. Jaynes, "bei'e" is measured in decibels of logarithmic probability. This sounds complicated, but in many cases, is actually much easier to use than simple odds or probability; adding 10 decibels multiplies the odds by a factor of 10.
The current reftext for "bei'e" is at http://www.lojban.org/tiki/bei%27e , which basically amounts to adding Lojbannic digits to the front of the word:
| ni'uci'ibei'e | -oo | 0% | 1:oo | complete disbelief, paradox |
| ni'upabei'e | -1 | 44.3% | 4:5 | |
| ni'ubei'e | <0 | <50% | <1:1 | less than even odds, less likely than so |
| nobei'e | 0 | 50% | 1:1 | neither belief nor disbelief, agnosticism |
| ma'ubei'e | >0 | >50% | >1:1 | greater than even odds, more likely than not |
| pabei'e | 1 | 55.7% | 5:4 | preponderance of the evidence |
| rebei'e | 2 | 61.3% | 3:2 | |
| cibei'e | 3 | 66.6% | 2:1 | clear and convincing evidence |
| vobei'e | 4 | 71.5% | 5:2 | |
| mubei'e | 5 | 76.0% | 3:1 | beyond a reasonable doubt |
| xabei'e | 6 | 80.0% | 4:1 | |
| zebei'e | 7 | 83.3% | 5:1 | |
| bibei'e | 8 | 86.3% | 6:1 | |
| sobei'e | 9 | 88.8% | 8:1 | |
| panobei'e | 10 | 90.9% | 10:1 | |
| pacibei'e | 13 | 95.2% | 20:1 | |
| xarebei'e | 62 | 99.99994% | 1,500,000:1 | 5 standard deviations |
| ci'ibei'e | oo | 100% | oo:1 | complete belief, tautology |
| xobei'e | ? | ?% | ?:? | question, asking listener their level of belief |
By having this explicit mental tool, even if I don't use it aloud, I'm finding it much easier to remember to gauge how confident I am in any given proposition. If anyone else finds use in this idea, so much the better; and if anyone can come up with an even better mental tool after seeing this one, that would be better still.
.uo .ua .uisai .oinairo'e
Well, here's a first stab. We only need to cover 50-100% since English gives us negations: "unlikely" versus "likely", "unprobable" versus "probable". (If we can express 60% and we want to express 40%, we can just negate whatever we say for 60%.) Going by the above scheme, the 50-100% range requires 13 modifiers. If I replace the >99% for 99%, which I don't think is very useful, I need 13 or so. For infinity or 100%, I think it's better to signal a discontinuity by using a pair like "certain"/"impossible" (neg infinity or 0%) - since they aren't probabilities. I originally set had the range start at 50%, but then it was pointed out that negation was funny, so I realize I had to make that a special word as well, along with 0% and 100%. Half-way through, if I switch from "likely" to "probable" I can reuse the previous modifiers, so I only need to think of 5-6 modifiers.
It turns out that this is a really hard balancing act. I think I'm roughly satisfied with:
I'll think about this for a while more, and then I think I'll go through gwern.net and try to rationalize all uses of informal probability to use this scheme. I'm calibrated, so I might as well get some mileage out of it!
Translating numerical probabilities into verbal labels has been an active area of research. As an entry point into that literature, see the review article Teigen & Brun (2003, Verbal expressions of uncertainty and probability).
You might want to take a look at some of the other attempts out there to try to come up with labels that are more intuitive (I see "likely" and "probable" as equivalent, which would make this system where "somewhat probable" > "extremely likely" very unintuitive for me). Teigen and Bru... (read more)