[racket] Neil: Is there a sensible way to sample from the naturals? EOM
Thanks a lot, Neil!
I see that I didn't read your earlier message correctly and I now get what
you're suggesting. It makes a lot of sense (and the followup discussion
also helped me).
Robby
On Thu, Feb 20, 2014 at 10:49 AM, Neil Toronto <neil.toronto at gmail.com>wrote:
> The connection between my final answer and sampling between 2^(X-1) and
> 2^X is... mostly my intuition, but I think it can be made rigorous. I
> noticed that the mean number of bits seemed to always be near 1/p, which is
> the mean of a geometric distribution with parameter p. So I defined a
> random variable whose log mean would be close to the same. It appears to
> have the same characteristics: it looks like an "anything can happen" sort
> of distribution, and computing the mean never seems to converge, no matter
> how many samples I put into it.
>
> Sampling between 2^(X-1) and 2^X is just to make sure the sampling
> algorithm will return any natural number. (Well, any that will fit in your
> computer.) I can prove this distribution has no expected value.
>
> The expected value of a discrete random variable is
>
> x_0 * p(x_0) + x_1 * p(x_1) + x_2 * p(x_2) + ...
>
> where the x_i are the possible outcomes, and p(x_i) are their
> probabilities. It's a generalization of computing the mean of finitely many
> uniform outcomes, in which p(x) = 1/n:
>
> (x_0 + x_1 + ... + x_{n-1}) / n
> = x_0 * 1/n + x_1 * 1/n + ... + x_{n-1} * 1/n
>
> If X ~ Geometric(1/2), its expected value is
>
> 0 * 1/2 + 1 * 1/4 + 2 * 1/8 + 3 * 1/16 + ...
> = 1/4 + 2/4 + 3/16 + ...
> = 1
>
> Decision theory and economics are dominated by the idea that making
> choices to maximize expected values is the best thing to do. But expected
> values aren't stable, in the sense that applying a common function to a
> random variable that has an expected value can yield another random
> variable that doesn't have one. If you exponentiate those geometric
> outcomes, you get this expected value:
>
> 1 * 1/2 + 2 * 1/4 + 4 * 1/8 + 8 * 1/16 + ...
> = 1/2 + 1/2 + 1/2 + 1/2 + ...
> = +inf
>
> If you use X ~ Geometric(1/3) and Y = 2^X, you get
>
> 1 * 1/3 + 2 * 2/9 + 4 * 4/27 + ...
> = 1/3 + 4/9 + 16/27 + ...
> = +inf
>
> for the expected value of Y. More generally, if X ~ Geometric(p), the
> formula for the expected value of Y is
>
> sum_{x=0}^inf 2^x * (1-p)^x * p
> = sum_{x=0}^inf (2*(1-p))^x * p
>
> By the geometric series convergence test, this fails to converge when
> |2*(1-p)| >= 1, which happens when p <= 1/2. So if p <= 1/2, Y = 2^X has no
> expected value, and we can use this fact to baffle economics students and
> impress the ladies.
>
> The expected value of the random variable I gave (roughly, sampling
> between 2^(X-1) and 2^X) is bounded below by substituting the lowest
> possible values (i.e. 2^(x-1) if x > 0, otherwise 0):
>
> 0 * p(x_0) + 1 * p(x_1) + 2 * p(x_2) + 4 * p(x_3) + ...
>
> The tail of this series starting at 1 is 1/2 * the tail of the series for
> the expected value of Y starting at 1, which diverges.
>
> A proof of divergence for the expected value of the final function I gave
> won't be as easy because the uniform numbers it chooses aren't bounded
> below by a function that grows without bound (i.e. 2^(X-1)). It'll probably
> have something to do with the linearity of expectation.
>
> Linearity, BTW, is what makes expected value popular. It often yields
> closed-form solutions, and makes expected values scale and add together
> easily. Still, medians might be a better choice because they always exist.
> In the Y = 2^X game, if X ~ Geometric(1/2), the median of Y's distribution
> is $1. If X ~ Geometric(1/4), it's $4, and if X ~ Geometric(1/16), it's
> $1024.
>
> Neil ⊥
>
>
> On 02/19/2014 08:07 PM, Robby Findler wrote:
>
>> Thanks Neil! (I think probably Max and Burke are also interested.)
>>
>> What is the connection between the economic philosophers argument
>> against maximizing expected value, the idea that we should sample
>> between 2^(X-1) and 2^X, and then the answer that you gave?
>>
>> I think that I know that the geometric distribution is something you get
>> by repeatedly playing a game much like that first one, but I'm not
>> really sure of the details and unable to piece together the rest....
>>
>> Robby
>>
>>
>>
>> On Wed, Feb 19, 2014 at 4:02 PM, Neil Toronto <neil.toronto at gmail.com
>> <mailto:neil.toronto at gmail.com>> wrote:
>> >
>> > I've CC'd the Racket users mailing list because I thought more people
>> (e.g. Robby) would be interested in the answer.
>> >
>> > You can't sample uniformly from the naturals, but you can sample
>> according to an "anything can happen" sort of distribution. You want a
>> distribution with no expected value (aka mean, average).
>> >
>> > Economic philosophers use such distributions to disabuse people of
>> the idea that maximizing expected value is always the best way to make
>> decisions. Here's a simple example.
>> >
>> > 1. Flip a coin until you get heads. Call the number of flips X.
>> > 2. You receive Y = 2^X dollars.
>> >
>> > You can flip a fair coin, or a coin biased toward tails. Which coin
>> do you choose? Obviously the biased coin. But in either case, the
>> expected value of Y is infinite. Voila! Paradox.
>> >
>> > (The problem isn't that there's no upper bound. For Y = X, the award
>> is still unbounded, but for any coin with nonzero heads probability, the
>> expected value is finite.)
>> >
>> > You want *every* natural to be possible, though, so we'd have to do
>> something like sample uniformly between 2^(X-1) and 2^X. Here's a
>> function that does it:
>> >
>> > (: random-natural/no-mean (-> Real Natural))
>> > (define (random-natural/no-mean prob-zero)
>> > (define n (exact-floor (sample (geometric-dist prob-zero))))
>> > (define m1 (assert (expt 2 n) integer?))
>> > (define m0 (quotient m1 2))
>> > (max 0 (random-integer m0 m1)))
>> >
>> > The "max 0" keeps TR from complaining that `random-integer' returns
>> an Integer. The `prob-zero' argument is the probability the coin is
>> heads, which is also the probability this function returns 0.
>> >
>> > This looks sort of like what you want:
>> >
>> > > (random-natural/no-mean 0.001)
>> > - : Integer [more precisely: Nonnegative-Integer]
>> > 56136474695225172011728291802626216994256833545894766283873703181
>> > 10364986394536406497817120521834403457182656624358136577815679690
>> > 73469994282060833573766848756431669747238563269112899118707963866
>> > 08154252648824183995287333693058951314331721341986222320438359883
>> > 50861513517737507150144340359987088543453799423969409721165923104
>> > 82128386772489312894482659745630141444108439622157113717027784284
>> > 7612786588731040573293479397701924913229558559022675650838885440
>> >
>> > > (random-natural/no-mean 0.001)
>> > - : Integer [more precisely: Nonnegative-Integer]
>> > 57
>> >
>> > If it returns small numbers too often, you can mitigate that by
>> taking the max of a few of them.
>> >
>> > If you want a feel for the "average" output, you can put it in terms
>> of average number of bits:
>> >
>> > > (mean (build-list 1000 (λ _ (integer-length (random-natural/no-mean
>> 0.01)))))
>> > - : Real
>> > 99 407/1000
>> >
>> > In fact, now that I think about it, this is probably nearly equivalent:
>> >
>> > (: random-natural/no-mean (-> Real Natural))
>> > (define (random-natural/no-mean p)
>> > (random-bits (add1 (exact-floor (sample (geometric-dist p))))))
>> >
>> > The average number of bits should be near (/ 1 p).
>> >
>> > Neil ⊥
>> >
>> > ____________________
>> > Racket Users list:
>> > http://lists.racket-lang.org/users
>>
>
>
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