1236: "Seashell"

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1236: "Seashell"

Postby dalcde » Wed Jul 10, 2013 11:02 am UTC

Image
Title Text: This is roughly equivalent to 'number of times I've picked up a seashell at the ocean' / 'number of times I've picked up a seashell', which in my case is pretty close to 1, and gets much closer if we're considering only times I didn't put it to my ear.

Is it just me, or has he got it the other way round?

EDIT: The image only. The title text seems to be correct.

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Re: 1236: "Seashell"

Postby Cres » Wed Jul 10, 2013 11:06 am UTC

Yeah P(I picked up a seashell) and P(I'm near the ocean) need to be swapped

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Re: 1236: "Seashell"

Postby Calluin » Wed Jul 10, 2013 11:27 am UTC

I was puzzled by that, too. I think that you just need to swap the P(seashell) and P(ocean), not the conditional probabilities, because you have to calculate the probability of hearing the ocean when picking up seashells :-)

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Re: 1236: "Seashell"

Postby unratito » Wed Jul 10, 2013 12:44 pm UTC

You're right. That's Bayes' theorem, and the formula is reversed.

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Re: 1236: "Seashell"

Postby cellocgw » Wed Jul 10, 2013 12:47 pm UTC

Pete Peels P(shells) near the P(shore)
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Re: 1236: "Seashell"

Postby keldor » Wed Jul 10, 2013 12:52 pm UTC

I've picked up seashells, but never at the seashore. Which makes sense, since I've only visited a seashore about twice in my life, and never actually found any shells while there.

Not many seas in Colorado.

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Re: 1236: "Seashell"

Postby Whizbang » Wed Jul 10, 2013 1:02 pm UTC

I get the humor (in a gist-y sort of way. I am not edumacated on statistics and probabilities and whatnot), but the whole point of the seashell is not to listen to the sound of the ocean the moment you pick it up, but to take the shell home with you and listen to the calming white noise when you are nowhere near the ocean.

'Course, cupping your hand does just as well, but the shell, as a physical representation of the sea, makes the experience more real and/or less silly. Though one could argue that putting the exoskeleton of a sea creature to your ear is a bit silly.

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Re: 1236: "Seashell"

Postby rmsgrey » Wed Jul 10, 2013 1:04 pm UTC

I'm confused by the "roughly" in the title-text - P(S|O)*P(O) is exactly P(S&O) (and P(S) is obviously the same as P(S)) so there doesn't seem to be much scope for the two to be slightly different...

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Re: 1236: "Seashell"

Postby orthogon » Wed Jul 10, 2013 1:14 pm UTC

rmsgrey wrote:I'm confused by the "roughly" in the title-text - P(S|O)*P(O) is exactly P(S&O) (and P(S) is obviously the same as P(S)) so there doesn't seem to be much scope for the two to be slightly different...

I had the same thought, but then again the ratio of the number of occurrences is only an estimate of the ratios of the statistical probabilities of the events. Maybe that is what he was getting at?
xtifr wrote:... and orthogon merely sounds undecided.

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Re: 1236: "Seashell"

Postby prof_bart » Wed Jul 10, 2013 1:19 pm UTC

Rev. Bayes Says: P(A|B) P(B) = P(B|A) P(A)

[insert link to xkcd comic 386 here.... can't figure out how...]

Wait a minute.... have we just been trolled?

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Re: 1236: "Seashell"

Postby Erminaz » Wed Jul 10, 2013 1:38 pm UTC

Bayes' theorem in a nutshell, err... seashell :mrgreen:

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Re: 1236: "Seashell"

Postby shababo » Wed Jul 10, 2013 1:39 pm UTC

Agreed. He has the prior in the denominator and the evidence in the numerator... should be switched:

$p(\theta | X) = \frac{p(X|\theta)p(theta)}{p(X)}$

...does latex work in here?

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Re: 1236: "Seashell"

Postby Viadd » Wed Jul 10, 2013 2:08 pm UTC

He got Bayes' theorem wrong.

P(B|A) = P(A|B) P(B) /P(A)

P(ocean|shell) = P(shell|ocean) * P(ocean)/P(shell)
Last edited by Viadd on Wed Jul 10, 2013 2:16 pm UTC, edited 1 time in total.

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Re: 1236: "Seashell"

Postby DarkVeracity » Wed Jul 10, 2013 2:15 pm UTC

Bayes Theorem is that
[math]P(A|X) = ( P(X|A) * P(A) ) / ( P(X|A) * P(A) + P(X|~A) * P(~A) )[/math]
Which also equals
[math]P(X&A) / ( P(X&A) + P(X&~A) )[/math]
Since
[math]P(X|A) * P(A) = P(X&A)[/math]
By this theorem
[math]P(I'm near the ocean|I picked up a seashell)[/math]
Should equal
[math]( P(S|O) * P(O) ) / ( P(S|O) * P(O) + P(S|~O) * P(~O)[/math]

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Re: 1236: "Seashell"

Postby Flammifer » Wed Jul 10, 2013 2:15 pm UTC

Dammit, I came here to post that, but got beat to it by everybody ... I still have a lingering suspicion it might be a clever underhanded trick, though a mistake is more likely.

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Re: 1236: "Seashell"

Postby Ann_xkcd » Wed Jul 10, 2013 2:32 pm UTC

I've been to the seashore lots (living in a coastal town helps), but when does one ever just "find" a seashell like that on the beach? Maybe I just live on the wrong part of the ocean or the wrong ocean. On the contrary, my grandparents had one in their house. Hence lots of listening in a seashell while not at the sea and zero times picking up that type of seashell while at the sea.

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Re: 1236: "Seashell"

Postby Hiferator » Wed Jul 10, 2013 2:44 pm UTC

rmsgrey wrote:I'm confused by the "roughly" in the title-text - P(S|O)*P(O) is exactly P(S&O) (and P(S) is obviously the same as P(S)) so there doesn't seem to be much scope for the two to be slightly different...

Roughly, because the quantities in the alt text are not propabilities but absolute frequencies resulting in the quotient being a relative frequency. As such it only approximates the probability or (almost surely) converges to it when time or number of experiments aproaches infinity.
orthogon wrote:I had the same thought, but then again the ratio of the number of occurrences is only an estimate of the ratios of the statistical probabilities of the events. Maybe that is what he was getting at?

Exactly.

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Re: 1236: "Seashell"

Postby WolfieMario » Wed Jul 10, 2013 4:25 pm UTC

It's been corrected; for those who want context, this was the original:

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Re: 1236: "Seashell"

Postby rhomboidal » Wed Jul 10, 2013 4:51 pm UTC

I've stopped picking up seashells because of the probability there's a cone snail inside them. Thanks, math.

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Re: 1236: "Seashell"

Postby statismensch » Wed Jul 10, 2013 5:13 pm UTC

As a statistician who uses Bayesian methods, I have to disagree with the caption in the comic:

"Statistically speaking, if you pick up a seashell and don't hold it to your ear, you can probably hear the ocean."

I think this is a misunderstanding of Bayes rule. Let's consider the corrected version of the equation in the comic,

P(near ocean | seashell) ~ p(seashell | near ocean) * P(near ocean),

where "~" means proportional to. (I'm treating "seashell", the known quantity, as constant, and "near ocean", the unknown quantity, as stochastic.) If you're likely to be far away from the ocean to begin with, then P(near ocean | seashell) will be controlled mostly by your low prior probability, P(near ocean). If you're extremely unlikely to be near the ocean to begin with, then it won't make a difference whether you pick up the shell or not. P(near ocean | seashell) will still be low because your prior belief of being far away is so strong.

Doctors have a similar problem testing for rare diseases. Let's say I test positive for the Siberian Swazi-flu (a rare condition that I'm making up). My posterior probability of having the disease is

P(sick | +) ~ P(+ | sick) P(sick)

Since the disease is so rare, P(sick) is extremely low, and so the chances of my having the disease are still low even though I tested positive. Epidemiologists think about this sort of thing a lot.

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Re: 1236: "Seashell"

Postby drkslvr » Wed Jul 10, 2013 6:39 pm UTC

keldor wrote:I've picked up seashells, but never at the seashore.... Not many seas in Colorado.


Same for Kansas. I think I'm most likely to find a sea shell at Hobby Lobby, maybe?
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Re: 1236: "Seashell"

Postby statismensch » Wed Jul 10, 2013 7:43 pm UTC

Sorry about the repeat post. I'm new.
Last edited by statismensch on Thu Jul 11, 2013 4:04 pm UTC, edited 1 time in total.

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Re: 1236: "Seashell"

Postby cream wobbly » Wed Jul 10, 2013 8:00 pm UTC

keldor wrote:I've picked up seashells, but never at the seashore. Which makes sense, since I've only visited a seashore about twice in my life, and never actually found any shells while there.

Not many seas in Colorado.


He said oceans. Seas are different. But hey, not many oceans in Arizona either.

Only this morning, I picked up a seashell from the floor of my son's school. We also have a terrarium fitted out for a scraggly and entertaining bunch of decapods. They like to fling rejected shells around, which means I get to pick them up. Also, back when I lived in England, I used to pick up sea shells from the beach. This was on the North Sea. No ocean in sight.

I think to correct the expression, the following need to be integrated: P(domicile is located <60 minutes' journey from an ocean), P(has children), P(has hermit crabs), and perhaps change "ocean" to "saltwater".

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Re: 1236: "Seashell"

Postby Klear » Wed Jul 10, 2013 8:30 pm UTC

Hmm.. yeah... I've never been at an ocean, and I don't think I'm alone.

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Re: 1236: "Seashell"

Postby grafzero » Wed Jul 10, 2013 9:36 pm UTC

Calculations should take under consideration current level of ocean drain :D

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Re: 1236: "Seashell"

Postby greymatters » Wed Jul 10, 2013 11:00 pm UTC

Better title for this comic: "Bays" Theorem

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Re: 1236: "Seashell"

Postby The Moomin » Thu Jul 11, 2013 1:06 am UTC

So he moved the P around with the seashell?

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Re: 1236: "Seashell"

Postby addams » Thu Jul 11, 2013 5:26 am UTC

Ann_xkcd wrote:I've been to the seashore lots (living in a coastal town helps), but when does one ever just "find" a seashell like that on the beach? Maybe I just live on the wrong part of the ocean or the wrong ocean. On the contrary, my grandparents had one in their house. Hence lots of listening in a seashell while not at the sea and zero times picking up that type of seashell while at the sea.

Yep. You live by the Wrong part of the water.
Those things need warm water.

Do you want to know why?
I can't remember.
Those large shells are destroyed by the Rough Northern Waters.

It is nice to hear The Sea.
I first heard the Sea in a Sea Shell.
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Re: 1236: "Seashell"

Postby Klear » Thu Jul 11, 2013 7:19 am UTC

I think Randall doesn't know how to use three seashells...

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Re: 1236: "Seashell"

Postby garaden » Thu Jul 11, 2013 8:10 pm UTC

Is it just me, or has Randall been invoking the Bayes a lot lately?

statismensch wrote:I think this is a misunderstanding of Bayes rule. Let's consider the corrected version of the equation in the comic,

P(near ocean | seashell) ~ p(seashell | near ocean) * P(near ocean),

where "~" means proportional to. (I'm treating "seashell", the known quantity, as constant, and "near ocean", the unknown quantity, as stochastic.) If you're likely to be far away from the ocean to begin with, then P(near ocean | seashell) will be controlled mostly by your low prior probability, P(near ocean). If you're extremely unlikely to be near the ocean to begin with, then it won't make a difference whether you pick up the shell or not. P(near ocean | seashell) will still be low because your prior belief of being far away is so strong.


Wait, where did P(seashell) go? You can't just drop it, it depends on P(near ocean)! I think we can agree that everyone sees about the same number of non-ocean seashells, at children's museums and stuff. However, people who live near (seashell-strewn) beaches see a lot more ocean seashells than people who don't, so their total number of seashells is much higher, and for them any given seashell is very likely to be an ocean seashell. I think this is true of Randall, since he lives in Massachusetts and seems to like beaches.

Your argument does make sense for people who mostly (or always) see non-ocean seashells. I personally encounter shells big enough to listen to so rarely that I'm not sure I can weigh in, though I think one of my elementary school teachers had one and my school wasn't on a beach, so I guess I was in the "mostly non-ocean seashells" group while I was in her class.

statismensch wrote:Doctors have a similar problem testing for rare diseases. Let's say I test positive for the Siberian Swazi-flu (a rare condition that I'm making up). My posterior probability of having the disease is

P(sick | +) ~ P(+ | sick) P(sick)

Since the disease is so rare, P(sick) is extremely low, and so the chances of my having the disease are still low even though I tested positive. Epidemiologists think about this sort of thing a lot.


You could apply similar logic here if testing positive is also extremely rare, and P(+ | sick) is reasonably sized. You can make up for even a small P(+ | sick) if P(+) is small enough, but such a test would not be very useful :). Of course, practically speaking I bet it's really hard to develop a good test for diseases that are seen rarely, just because you don't have many opportunities to study it...?

greymatters wrote:Better title for this comic: "Bays" Theorem


[sniff] God bless you, sir...god bless you.

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Re: 1236: "Seashell"

Postby tyler.daniels » Thu Jul 11, 2013 10:43 pm UTC

garaden wrote:Wait, where did P(seashell) go? You can't just drop it, it depends on P(near ocean)!


He treated P(seashell) as a constant and changed his equation to "is proportional to". It works mathematically. :)

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Re: 1236: "Seashell"

Postby garaden » Fri Jul 12, 2013 3:21 am UTC

tyler.daniels wrote:
garaden wrote:Wait, where did P(seashell) go? You can't just drop it, it depends on P(near ocean)!


He treated P(seashell) as a constant and changed his equation to "is proportional to". It works mathematically. :)


Sure, I agree that algebraically it's a perfectly fine thing to do :) But in terms of probability theory it doesn't make much sense. Now that I think about it, the problem isn't really that he decided P(S) was constant; sometimes you can do that without running into trouble. The trouble is, if the ratio P(S|O)/P(S) is at least as big as P(O) is small, then P(O) is no longer dominating and P(O|S) can become quite a bit larger than P(O) (up to 1, actually). By dropping P(S) and leaving P(S|O) to fend off P(O) all on its lonesome, it certainly looks like P(O|S) can be no bigger than P(O), which is not the case.

Thing is, probabilities aren't like physics equations. You don't need to know the exact ratio of force to electric field strength to understand the Lorentz force, because if you pick different units you'll get a different ratio, so it's the proportionality that actually matters (though why coulombs etc. are so friggin' huge is beyond me). But with probabilities, the difference between a ratio being >1 or <1 is the difference between evidence and counter-evidence. The information hiding in the proportionality squiggle is REALLY important!

If you like thinking about this stuff (I might. A lot.), I recommend http://yudkowsky.net/rational/bayes.

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Re: 1236: "Seashell"

Postby statistical purist » Fri Jul 12, 2013 11:20 am UTC

The events A: "I am near the ocean" and B: "I picked up a seashell" are likely not independants, so the real formula should be:
P(A|B) = P(A and B)/P(B).

In the case of A and B being independant, then:
P(A and B) = P(A|B)P(B) = P(B|A)P(A).


Of course in practice, all events are considered independants (otherwise it is too complicated to compute). Often it works rather well, but when it does not, this could lead to dire consequences like the sub-primes crisis.

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Re: 1236: "Seashell"

Postby orthogon » Fri Jul 12, 2013 2:39 pm UTC

statistical purist wrote:The events A: "I am near the ocean" and B: "I picked up a seashell" are likely not independants, so the real formula should be:
P(A|B) = P(A and B)/P(B).

In the case of A and B being independant, then:
P(A and B) = P(A|B)P(B) = P(B|A)P(A).


Of course in practice, all events are considered independants (otherwise it is too complicated to compute). Often it works rather well, but when it does not, this could lead to dire consequences like the sub-primes crisis.


Hang on a minute. Your first equation, which you say is generally true, is equivalent to the first equality in your second equation, P(A and B) = P(A|B) P(B), which you say is true only for independent events. You can get from one to the other simply by multiplying both sides by P(B). (The case where P(B)=0 is a bit pathological, I admit, but I don't think that concerns us here).

Perhaps what you are getting at is that if A and B are independent, then P(A|B) = P(A); in other words whether B occurs or not has no bearing on the probability of A, so the conditional probability equals the marginal probability. From this we get the famous result that P(A and B) = P(A|B)P(B) = P(A)P(B) if A and B are independent.

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Re: 1236: "Seashell"

Postby garaden » Fri Jul 12, 2013 6:08 pm UTC

statistical purist wrote:Of course in practice, all events are considered independants (otherwise it is too complicated to compute). Often it works rather well, but when it does not, this could lead to dire consequences like the sub-primes crisis.


...yeah. "Most major investment banks, strongly correlated to the housing market by accident? Preposterous!" KABOOM. "Whoops."

Statistically speaking, if you accidentally the economy but have a golden parachute, you can probably hear the ocean. :-/

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Re: 1236: "Seashell"

Postby arthurd006_5 » Sun Jul 14, 2013 9:00 am UTC

garaden wrote:... why coulombs etc. are so friggin' huge is beyond me...

Because it's set sensibly for current electricity (an ampere-second); and current electricity is very different from static electricity.

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Re: 1236: "Seashell"

Postby Kit. » Sun Jul 14, 2013 5:18 pm UTC

arthurd006_5 wrote:
garaden wrote:... why coulombs etc. are so friggin' huge is beyond me...

Because it's set sensibly for current electricity (an ampere-second); and current electricity is very different from static electricity.

Just to put the things into a "proper" perspective: my mobile phone's battery is rated @1200 mA*h, that's 4320 coulombs.

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Re: 1236: "Seashell"

Postby garaden » Sun Jul 14, 2013 9:17 pm UTC

Kit. wrote:
arthurd006_5 wrote:
garaden wrote:... why coulombs etc. are so friggin' huge is beyond me...

Because it's set sensibly for current electricity (an ampere-second); and current electricity is very different from static electricity.

Just to put the things into a "proper" perspective: my mobile phone's battery is rated @1200 mA*h, that's 4320 coulombs.


Good point; I was definitely thinking about capacitors when I felt like 1 C was ludicrously big. That said, I think I would have preferred it to be an order of magnitude or three lower, and then we wouldn't have to e.g. rate batteries with milliamp-hours. It'd be nice if household currents could be expressed in the base unit rather than needing fractions of it.

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Re: 1236: "Seashell"

Postby addams » Sun Jul 14, 2013 10:33 pm UTC

Off Topic:

Not anything to do with math here.
The Sea Shell is left behind, also.
http://www.spiritsite.com/writing/annlin/part2.shtml

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Do you think P(sea shell) are welks?

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Re: 1236: "Seashell"

Postby Kit. » Sun Jul 14, 2013 11:34 pm UTC

garaden wrote:I was definitely thinking about capacitors when I felt like 1 C was ludicrously big.

I guess it depends on how you define "ludicrously". I remember making a 12 V * 0.4 F capacitance box for some semiconductor lab testing. It was quite heavy, but still portable.

garaden wrote:That said, I think I would have preferred it to be an order of magnitude or three lower, and then we wouldn't have to e.g. rate batteries with milliamp-hours. It'd be nice if household currents could be expressed in the base unit rather than needing fractions of it.

That could backfire as well. Look what happened to [kilo]calories. The 1200 mA*h can be written as 1.2 A*h anyway.

Besides, the current system is pretty suitable for household currents, if you look at European circuit breakers' standard ratings. "16A" at a circuit breaker doesn't look as scary as "16kA" would.


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