
German Tank Problem - bluesmoon
http://threestandarddeviationstotheleft.wordpress.com/2006/08/22/german-tank-problem/
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lmkg
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 think your production capacity is[1].
Brings back memories of reading the Cryptonomicon.

[1] My gut tells me the expected estimate is likely to be slightly different
than your max value for small values of n. Bonus points for confirming or
denying.

~~~
run4yourlives
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.

~~~
tokenadult
_the outcome of the war was decided by a few Russian snowflakes_

Not by cracking Enigma?

<http://en.wikipedia.org/wiki/Cryptanalysis_of_the_Enigma>

And remember, "the war" was also being fought in the Pacific, also with a
spectacular imbalance in cryptology strength between the Allies and the Axis.

<http://en.wikipedia.org/wiki/Japanese_naval_codes>

<http://en.wikipedia.org/wiki/Magic_%28cryptography%29>

In general, code-breaking had a lot to do with the outcome of the war.

<http://www.cs.virginia.edu/jillcrypto/>

~~~
cabalamat
What decided the war in the Pacific was that the USA had vastly more
industrial capability than Japan. Even if Japan's leaders had made all the
right decisions, and America's leaders done everything wrong, America would
still have won any protracted war between the two countries.

~~~
tokenadult
_Even if Japan's leaders had made all the right decisions, and America's
leaders done everything wrong_

If America's leaders had done everything wrong, they would have ended the war
without big territorial gains for their side, with a negotiated peace leaving
Japan in control of Asia. (That, by the way, is exactly what Japan's leaders
got wrong from the strategic point of view. They really believed before the
war started that that was a likely outcome of the war. They didn't count on
the Allies insisting upon unconditional surrender after the sneak attacks on
Pearl Harbor and Singapore.)

~~~
run4yourlives
US leaders did do almost everything wrong until well into the war.

They were saved from their mistakes by superior equipment (especially
aircraft) and eventually better and more experience personnel. They didn't
start tactically winning battles until well into the Pacific Campaign.

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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.

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leelin
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 answer
your algorithm gives will be the correct answer, and provides the minimum
variance compared to other algorithms?

However, if we can put an upper bound on the number of tanks that could
possibly exist, say definitely no more than 100 million tanks, and if we
believe it is slightly more likely that older tanks will be on the front lines
than newer tanks, or any other prior knowledge, then the proposed solution
will be more incorrect.

One of my former co-workers proposed computing a posterior distribution for
estimates of N based on equally-weighted priors for N = max-observation to N =
maximum cap (100 million tanks). Then the estimate is the expected value of
the posterior distribution.

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

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

~~~
aristus
...probably.

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alex_h
I guess one might also want to take into account the probability of capturing
5 tanks with all odd serial numbers (3.125%) vs the probability that the
Germans had numbered their tanks in increments of 2

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brg
Is there a simple derivation or justification for this estimator?

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xel02
There is, had to learn it in a probability class.
<http://en.wikipedia.org/wiki/German_tank_problem>

~~~
brg
To sum up the derivation for those looking for crib notes; model the problem
as choosing k items u.a.r. from [1..N]. Compute P(max = i), and from this
compute the expectation of the max. After simpliciation this is given in terms
of k and N, and hence we have an estimator for N in terms of k and max.

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JadeNB
As someone who spends a lot of time working with future high-school math
teachers, I'd be really interested in reading some more by this author; but I
can't find any contact information, and the blog moved to an 'invitation-only'
Blogspot account in 2007. Does anyone know how to get in touch with the author
to request an invitation?

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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.

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

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thefool
Yeah I took it as meaning a website for the brain-dead, which was much wittier
than pretentious.

Unless he meant right and said left, which is even funnier.

