Comment author: JoshuaZ 29 July 2011 02:12:42PM 11 points [-]

This actually gets even worse. Consider for example a hypothetical Bayesian version of Issac Newton, trying to estimate what exponent k the radius is raised to in F= GMm/R^k. There's an intuition that mathematically simple numbers should be more likely, such as say "2". A while ago jimrandomh and benelliiot discussed this with me. Ben suggested that in this sort of context you might just have a complicated distribution where part of the distribution arose from something continuous and the other part arose from discrete probabilities for simple numbers. This seems to do a decent job capturing our intuition but it seems to be very hard to actually use that sort of distribution.

Comment author: lucidfox 30 July 2011 11:17:56AM 4 points [-]

If Newton tried to derive his law purely from empirical measurements, then yes, he would never be exactly sure (ignoring general relativity for a moment) that the exponent is exactly 2. For all he would know, it could actually be 2.00000145...

But that would be like trying to derive the value of pi or the exponents in the Pythagorean theorem by measuring physical circles and triangles. If the law of gravity is derived from more general axioms, then its form can be computed exactly provided that these axioms are correct.

Comment author: handoflixue 28 July 2011 10:51:38PM 3 points [-]

Your examples all come from the Discussion area. Are there examples in the Sequences or Promoted posts that you feel still suffer from this? Have you run in to issues with "main" posts where the authors reaction is anything other than "oh, thanks, let me fix that"?

I don't think policing the Discussion area is a worthwhile community goal.

I'll go ahead and just quote my original response to you:

People in the US use imperial measures as their native units. I doubt anyone on this site uses arshins and sazhens as their primary day-to-day measure of objects. Asking someone in the casual discussion area to translate out of their native units, for your convenience, when probably half of this site uses those units, is selfish.

If half the site were dominated by pre-revolution Russians I would (a) be very confused and (b) once I accepted that this wasn't a hoax, I'd use Google to learn the local vernacular rather than expecting them to cater to me.

Comment author: lucidfox 30 July 2011 10:27:51AM 0 points [-]

I don't see discussion posts as being inherently of lesser value and lesser impact to readers than promoted posts. I judge posts based on their content and the points they bring up, not by their location on the site.

Comment author: lucidfox 30 July 2011 08:44:38AM 2 points [-]

galactic intellectual property law

Be precise. Do you mean galactic patent law, galactic copyright law, or galactic trademark law?

Comment author: lucidfox 30 July 2011 04:32:48AM 0 points [-]

whereupon if I'm playing WOW, I roleplay an elf. <...> If I'm on LessWrong, I roleplay a rationalist.

Or you can roleplay a rationalist elf in WoW. :)

A long time ago, back before I quit WoW, I roleplayed an atheist draenei who refused to believe in the night elf goddess Elune. The catch here is that we players know she actually exists in the setting, because Blizzard told us so, but the characters would have no way of verifying this since she never appeared in the world in person. From my character's point of view, the magical powers that priests of Elune attributed to their goddess were actually (unknown to them) given to them by other, non-personified sources of power followed by other priests in the setting.

Comment author: DanielLC 29 July 2011 08:17:06PM 0 points [-]

I feel like independence really is just a definition, or at least something close to it. I guess P(A|B) = P(A|~B) might be better. Independence is just another way of saying that A is just as likely regardless of B.

Comment author: lucidfox 30 July 2011 04:19:52AM -1 points [-]

P(A|B) = P(A|~B) is equivalent to the classic definition of independence, and intuitively it means that "whether B happens or not, it doesn't affect the likelihood of A happening".

I guess that since other basic probability concepts are defined in terms of set operations (union and intersection), and independence lacks a similar obvious explanation in terms of sets and measure, I wanted to find one.

Comment author: Alicorn 29 July 2011 06:50:30PM *  4 points [-]

This one isn't even a matter of neglecting to convert; it's a cultural divide - while I expect you knew what Matt meant, it's entirely possible he didn't know how to translate it for you. Presumably you don't round to the nearest 1.5875 millimeters. What do metric users round to when measuring lengths? Millimeters? Those are little - even littler than sixteenths of an inch! Do most metric rulers even mark them, or do they just mark halfway points between centimeter lines? I don't know.

Comment author: lucidfox 30 July 2011 03:56:13AM *  2 points [-]

What do metric users round to when measuring lengths? Millimeters?

Depends. In casual use, typically centimeters. But yes, as muflax said, metric rulers have individual millimeters marked, and typically they mark half-centimeters with slightly longer bars.

Comment author: Matt_Simpson 29 July 2011 04:04:42PM 6 points [-]

Let X be a random variable over the interval [0, 10]. Then, by the definition of probability over continuous domains, P(X = 1) = 0.

Only if you have a continuous probability distribution over that domain. It's quite possible to have a probability distribution with, for example, a point mass at 5 such that p(X=5)=0.5.

This is sometimes described as counterintuitive: surely, at any measurement, X must be equal to something, and thus its probability cannot be zero since its clearly happened.

Others have answered this below, but there is another aspect to this I'd like to discuss. All data are discrete. When you measure something, your measurement apparatus is only ever going to give you one of a discrete, finite set of values. (I'm pretty sure about finite, but willing to be corrected). Any probability distribution over the possible values that you might measure with your apparatus can easily satisfy p(X=x)>0 for all x.

Concretely, if you're measuring the length of something with a ruler, you probably just round to the nearest 1/16th of an inch. This means there are only 12*16=192 possible measurements you can make, so you can create any number of probability distributions of these points where each point has p(X=x)>0.

Comment author: lucidfox 29 July 2011 06:13:51PM *  2 points [-]

I implicitly meant a continuous distribution. Clarified that in the post now.

Concretely, if you're measuring the length of something with a ruler, you probably just round to the nearest 1/16th of an inch.

As someone who lives in the dangerous and uncharted part of the world called "outside the US', I prefer centimeters. ;)

Comment author: Matt_Simpson 29 July 2011 04:12:18PM *  0 points [-]

Yes, thats the concept to which I am refering. The concept comes from measure theory. If you're familiar with I'm not sure why you're confused about probability 0 events.

I think her confusion comes from the fact that if your prior probability that an event happened is 0, no amount of evidence will convince you that it did happen. Suppose your prior probability that some random variable X is equal to 1 is P(X=1)=0. Now suppose you find out that actually, X=1. Then using Baye's rule:

P(X=1|X=1) = P(X=1|X=1)*P(X=1) / denominator

I'll leave the denominator out because the numerator is 0 (the denominator won't be 0), so P(X=1|X=1)=0, which makes no sense.

I don't claim the calculation I did above is correct - I realize conditional probabilities a fraught with difficulties, and I probably violated some rule I don't know about or have forgotten from my measure theory class. However, this does give you intuition for why lucidfox or perhaps someone else would be confused despite having knowledge of measure theory (if this is in fact why it was confusing to him/her).

Comment author: lucidfox 29 July 2011 06:07:59PM 0 points [-]

Her confusion.

Comment author: [deleted] 29 July 2011 01:09:14PM 0 points [-]

Interestingly, the words "Almost surely" also has a Wikipedia article that represents some of these mathematical concepts, and there are also related articles on "Almost All" and "Almost Everywhere."

http://en.wikipedia.org/wiki/Almost_surely http://en.wikipedia.org/wiki/Almost_all http://en.wikipedia.org/wiki/Almost_everywhere

Comment author: lucidfox 29 July 2011 01:12:25PM 0 points [-]

When I read thakll's post, I thought they indeed meant the mathematical definition of "almost surely". The domain of an event with probability zero is indeed "almost nowhere" in the rigorous sense, since it is a measure-zero set.

P(X = exact value) = 0: Is it really counterintuitive?

8 lucidfox 29 July 2011 12:45PM

I'm probably not going to say anything new here. Someone must have pondered over this already. However, hopefully it will invite discussion and clear things up.

Let X be a random variable with a continuous distribution over the interval [0, 10]. Then, by the definition of probability over continuous domains, P(X = 1) = 0. The same is true for P(X = 10), P(X = sqrt(2)), P(X = π), and in general, the probability that X is equal to any exact number is always zero, as an integral over a single point.

This is sometimes described as counterintuitive: surely, at any measurement, X must be equal to something, and thus its probability cannot be zero since its clearly happened. It can be, of course, argued that mathematical probability is abstract function that does not exactly map to our intuitive understanding of probability, but in this case, I would argue that it does.

What if X is the x-coordinate of a physical object? If classical physics are in question - for example, we pointed a needle at a random point on a 10 cm ruler - then it cannot be a point object, and must have a nonzero size. Thus, we can measure the probability of the 1 cm point lying within the space the end of the needle occupies, a probability that is clearly defined and nonzero.

But even if we're talking about a point object, while it may well occupy a definite and exact coordinate in classical physics, we'll never know what exactly it is. For one, our measuring tools are not that precise. But even if they had infinite precision, statements like "X equals exactly 2.(0)" or "X equals exactly π" contain infinite information, since they specify all the decimal digits of the coordinate into infinity. We would have an infinite number of measurements to confirm it. So while X may objectively equal exactly 2 or π - again, under classical physics - measurers would never know it. At any given point, to measurers, X would lie in an interval.

Then of course there is quantum physics, where it is literally impossible for any physical object, including point objects, to have a definite coordinate with arbitrary precision. In this case, the purely mathematical notion that any exact value is an impossible event turns out (by coincidence?) to match how the universe actually works.

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