Essay talk:A new approach to probability

This website is amazing.

Where else would you find both a guy who "really" understands how "gravitation" works, AND a guy who discovered n entirely new way of understanding a mathematical concept like probability -- and who also discovered what God is really all about. B♭maj7 “We are moving too fast for any label to stick.”-CLRJ 03:36, 28 August 2011 (UTC)

It's not an "entirely new way of understanding a mathematical concept like probability" - it is simply a new way of writing probabilities. (And maybe it isn't even new, probably someone has thought about it before.) I can't see anything in it which a mathematician could disagree with, it is not rejecting or denying any mainstream mathematics, it is simply proposing an alternate notation. (((Zack Martin)))^™ 04:19, 28 August 2011 (UTC)

This is already a pretty standard thing when programming, I think -- you don't want to store a really small probability as a double so you store the log. Calling a notational convenience a "new approach" is a bit strange though. --MarkGall (talk) 04:27, 28 August 2011 (UTC)

It's a new approach to the notation of probability. (And, as you say, maybe not that new, but it's new to me... I thought it up myself from scratch, but I'm unsurprised others have thought of the same approach before, because it is a reasonably obvious idea.) (((Zack Martin)))^™ 04:29, 28 August 2011 (UTC)

I guess it's probably faster in some applications too, depending on what you're doing: addition runs faster than multiplication. I'm not convinced that making up four new words for "take the log" is really a good notation though. --MarkGall (talk) 04:35, 28 August 2011 (UTC)

Well, my approach is slightly more complex than just taking the log... rather than four concepts, my more recent thought is reducing it down to two, "certainty" and "uncertainty":

${\begin{alignedat}{2}certainty(p)&={\begin{cases}-\log _{2}(p)&{\mbox{if }}p<{\frac {1}{2}}\\\log _{2}(1-p)&{\mbox{if }}p\geq {\frac {1}{2}}\end{cases}}\\uncertainty(p)&={\begin{cases}-\log _{2}({\frac {1}{2}}-p)&{\mbox{if }}p<{\frac {1}{2}}\\\log _{2}(p-{\frac {1}{2}})&{\mbox{if }}p\geq {\frac {1}{2}}\end{cases}}\end{alignedat}}$

If it's not clear why I do the above, I'd suggest graphing them, then the thought might be clearer. (((Zack Martin)))^™ 04:58, 28 August 2011 (UTC)

.5

Why on Earth is .5 so special that you consider it the difference between certainty and uncertainty? Tetronian ^{you're clueless} 12:34, 19 September 2011 (UTC)

If p(X) = .5, then we have 0 bits of information about whether X. The more we know about X, p(X) will tend to move further away from 0.5, be that in a positive or negative direction. (((Zack Martin)))^™ 06:58, 22 September 2011 (UTC)

If p(X) = 0.5 you have a great deal of information about whether X. You know it will happen roughly half the time. It's because we know that the probability of a coin flip is 0.5 that it can be used to decide, for example, who kicks off first at a footie match. A p(X) = 0.1 coin would be less useful. So, bollocks as usual, then. Bad Faith (talk) 07:27, 22 September 2011 (UTC)

You may have a lot of information about X in general, but no information about whether X will occur in any particular case. By X here I mean a proposition, not an event out of some repeatable series of possible events; I suppose the closest analogue in terms of events would be a single event only. (Think Bayesianism not frequentism.) Note that I said "we have 0 bits of information about whether X", not "we have 0 bits of information about X". The word whether is crucial. (((Zack Martin)))^™ 09:30, 22 September 2011 (UTC)

I assume it's just uniform prior. But I'm always suspicious of using that in any real sense as I don't think there's a genuine practical choice that can use it (though most priors are asspulls anyway, so at least uniform is consistent). I wouldn't risk platonifying the idea that 0.5 is the cutoff between your "certainty" and "uncertaint"y except as an entirely arbitrary labelling system - like something being statistically significant about 0.95 confidence, there's nothing special about it and 0.94 is just as valid, it's quantitative rather than qualitative. People who know what they're doing with that sort of thing tend to avoid making such calls, and so just leave it as the raw number. ADK^{...I'll incarcerate your needle!} 09:47, 22 September 2011 (UTC)

In my system, 0.5 is not a cut-off between "certainty" and "uncertainty". It is simply the point at which certainty equals uncertainty. (((Zack Martin)))^™ 09:54, 22 September 2011 (UTC)

Actually, the above isn't true, I was tired and not thinking straight when I wrote that. Graph the two functions above, all will be clear then. (((Zack Martin)))^™ 09:37, 23 September 2011 (UTC)

Yes, but it's a transition and your maths has it as a singularity. That doesn't map to the world as I know it. Bad Faith (talk) 10:34, 22 September 2011 (UTC)

It's been a long time since I looked at my math stat book, but I suspect a statistic based on your version of probability would be biased and utterly not robust, and hence not worth much. Ironically, you would probably take a logarithm somewhere to show that. Furthermore, that your intitution at assigning ad hoc probabilities is poor is a laughable reason for introducing a new stat. sterile^talk 00:25, 23 September 2011 (UTC)

I think statistics based on my version of probability would work pretty similar to standard stats. It's fundamentally a difference of notation, you'd have to adjust your formula appropriately, and the adjusted versions might possibly not be as elegant as the originals, but it should not make any fundamental difference to the outcomes. (((Zack Martin)))^™ 09:37, 23 September 2011 (UTC)