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German Tank Problem

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Re: German Tank Problem

#11

Post by MarkBook »

mturmon wrote: post #8
MarkBook wrote: post #6

Earlier quoted context omitted.

continuing my foolish musings I would have thought originally that getting the mean of the sample and multiplying by 2 would have been a good estimate. But one problem with that is that there is no possibility of the maximum being lower than anything in the sample so there has to be an allowance for uncertainty upwards. Is that the reason for adding 1 to s?

You can show that the max is a comprehensively better estimator than twice the mean. This is one of the rare cases where you can outperform the mean by a large margin. The standard deviation of the mean-based estimator, which measures its accuracy, will go down like 1/sqrt(n) where n is the number of samples in the finite set. This is the rate standardly seen in lots of estimation problems. But the standard deviation…

Thanks, I think I get it on a superficial level. The British seemed to have some pretty useful statisticians during WWII. See this also : http://www.dur.ac.uk/stat.web/bomb.htm

EDIT : So there's about a fifty fifty chance that the second observation will be bigger than the first. After the second observation the range of possible future observations will be divided into 3 zones, 1 below the lowest of the previous 2 observations 1 above the higher and 1 between the two previous observations, so there is now only a 1/3 possibility of the next observation being above the higher of the 2 previous observations. Next time that drops to 1/4 then 1/5, so that's how the standard deviation decreases proportional to 1/n

Re: German Tank Problem

#12

Post by jules »

leelin wrote: post #10

There are a few "flaws" to this solution actually, although the flaws are in assumptions + bayesian vs. frequentist philosophy differences. The solution mentioned in the link solves a different problem: You get a sample of K ints uniform randomly from 1 to N, where N is unknown. Based on your sample, what is your best estimate of N, so that if we repeatedly continue to give you independent random samples, the average…

Is there for every commonly used frequentist estimation method a bayesian prior that gives the same answer?

Re: German Tank Problem

#13

Post by run4yourlives »

lmkg wrote: post #3

Key quote: > The statisticians believed that the Germans, being Germans, had logically numbered their tanks in the order in which they were produced. The moral of the story is, never do anything that could give your opponent information, unless you're controlling what information they receive. It would be better to produce tanks with pseudo-random serial numbers, up to a maximum of whatever you want your enemy to thi…

Sometimes though, information denial is more of a pain in the ass than a gain.

For all the intelligence directed at Nazi Germany during WWII, the outcome of the war was decided by a few Russian snowflakes, really.

Re: German Tank Problem

#14

Post by bpick »

I recall reading that the same kind of thing used to happen with standing field armies, (think Napoleon sized blocks) where you could almost instantly assess an opponents forces based on simply recognizing divisions, regiments, etc.

My guess is Napoleon-era field marshals did a cost-benefit analysis and decided that managing their forces simplistically (i.e. no irregular sized units) outweighed whatever inconvenience enemy intelligence posed.

Relating this to the OP my guess is the Germans made a mistake and overlooked the serial numbers, rather than simply not caring.

Re: German Tank Problem

#15

Post by mturmon »

MarkBook wrote: post #11
mturmon wrote: post #8

Earlier quoted context omitted.

You can show that the max is a comprehensively better estimator than twice the mean. This is one of the rare cases where you can outperform the mean by a large margin. The standard deviation of the mean-based estimator, which measures its accuracy, will go down like 1/sqrt(n) where n is the number of samples in the finite set. This is the rate standardly seen in lots of estimation problems. But the standard deviation…

Thanks, I think I get it on a superficial level. The British seemed to have some pretty useful statisticians during WWII. See this also : http://www.dur.ac.uk/stat.web/bomb.htm EDIT : So there's about a fifty fifty chance that the second observation will be bigger than the first. After the second observation the range of possible future observations will be divided into 3 zones, 1 below the lowest of the previous 2 o…

I like that explanation.

Re: German Tank Problem

#16

Post by mturmon »

jules wrote: post #12
leelin wrote: post #10

There are a few "flaws" to this solution actually, although the flaws are in assumptions + bayesian vs. frequentist philosophy differences. The solution mentioned in the link solves a different problem: You get a sample of K ints uniform randomly from 1 to N, where N is unknown. Based on your sample, what is your best estimate of N, so that if we repeatedly continue to give you independent random samples, the average…

Is there for every commonly used frequentist estimation method a bayesian prior that gives the same answer?

For every commonly used frequentist estimator, there is an uncountable infinity of intractable Bayesian estimators.

Re: German Tank Problem

#17

Post by sliverstorm »

run4yourlives wrote: post #13
lmkg wrote: post #3

Key quote: > The statisticians believed that the Germans, being Germans, had logically numbered their tanks in the order in which they were produced. The moral of the story is, never do anything that could give your opponent information, unless you're controlling what information they receive. It would be better to produce tanks with pseudo-random serial numbers, up to a maximum of whatever you want your enemy to thi…

Sometimes though, information denial is more of a pain in the ass than a gain. For all the intelligence directed at Nazi Germany during WWII, the outcome of the war was decided by a few Russian snowflakes, really.

Mother Russia teams up with Mother Nature

Re: German Tank Problem

#18

Post by hngryhppos »

I want to comment on how pretentious that website title is ("three standard deviations to the left"). I took that to imply the author (or his intended audience) is three standard deviations above average (intelligence?).

Anyway, he says he am IB teacher... not surprised, this attitude of intellectual superiority is an IB hallmark.

Re: German Tank Problem

#19

Post by Devilboy »

hngryhppos wrote: post #18

I want to comment on how pretentious that website title is ("three standard deviations to the left"). I took that to imply the author (or his intended audience) is three standard deviations above average (intelligence?). Anyway, he says he am IB teacher... not surprised, this attitude of intellectual superiority is an IB hallmark.

Wouldn't that be 3 SDs to the RIGHT instead?

Re: German Tank Problem

#20

Post by eru »

run4yourlives wrote: post #13
lmkg wrote: post #3

Key quote: > The statisticians believed that the Germans, being Germans, had logically numbered their tanks in the order in which they were produced. The moral of the story is, never do anything that could give your opponent information, unless you're controlling what information they receive. It would be better to produce tanks with pseudo-random serial numbers, up to a maximum of whatever you want your enemy to thi…

Sometimes though, information denial is more of a pain in the ass than a gain. For all the intelligence directed at Nazi Germany during WWII, the outcome of the war was decided by a few Russian snowflakes, really.

The convoy system in the North Atlantic also made a huge difference. Imagine Britain and the US shut off.
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