[extropy-chat] calling all bayesians

[Eliezer S. Yudkowsky]

spike wrote:
> Guys help me eff this real-life effing problem:
> 
> I build 150 droobs and use 131 of them in my freem.  I
> test the remaining 19 spares destructively and find that
> all are good.  From that information only, what is the
> probability that all 131 droobs are good?
> 
> I have four Monte Carlo sims chewing on this problem
> but they are giving me puzzling results.  A closed-form
> solution to this would be impressive, winning my
> undying respect.

Depends on your prior.

If your prior belief is that any number of good droobs between 0 and 150 is 
equiprobable, then this is the *classic* Bayesian problem, the one that Bayes 
himself considered.  If I recall correctly, the closed form solution *for this 
prior* is that if you observe X good cases and Y bad cases, the posterior 
expected probability of goodness is:

X + 1
-----
X+Y+2

Thus if you started out believing that any failure rate between 0 and 150 was 
equally plausible, you would expect that the probability is 20/21 that any 
given remaining droob is good.

You actually asked a more difficult question, the probability that all 
remaining droogs are good.  Intuitively I would expect the answer to be 20/21 
* 21/22 * 22/23 ... = 20/152 but I haven't checked my work.

Other priors give different answers.  I don't think the equiprobable prior is 
a reasonable one for this case; we don't think it equally likely that a 
manufacturing facility turns out 150 straight successes vs. 150 straight 
rejects.  In real life, Spike, your problem is pretty much undefined, unless 
you can give me some kind of base rate statistics on how often your 
manufacturing technique works.

How on Earth did you set up a Monte Carlo sim on this?

-- 
Eliezer S. Yudkowsky                          http://singinst.org/
Research Fellow, Singularity Institute for Artificial Intelligence