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Re: [gnubg] Help with a new MET

From: Joseph Heled
Subject: Re: [gnubg] Help with a new MET
Date: Wed, 13 Nov 2019 06:16:35 +1300

Hi Tim,

"but for most practical purposes this is an irrelevant technicality"

Are you saying that I can treat each of the 4 estimates independently? That is, use sqrt(pq/N) as the std for each? seems problematic to me :)

Yes, a Bayesian approach would be better, but this probably involves things like contour integration or other horrors. I hoped for something simpler.


On Wed, 13 Nov 2019 at 06:04, Timothy Y. Chow <address@hidden> wrote:
On Tue, 12 Nov 2019, Joseph Heled wrote:
> Hi Timothy,
> Here is a stats question I encounter from time to time. 
> Suppose I run N BG games and collect the average win rates and gammon
> rates.
> 4 estimates which are dependent as they sum to 1.  How do I determine
> the confidence intervals for each? This is a 4d vector and it seems
> like a non trivial Q, but I assume this crops up a lot and must have a
> standard answer.  what is your take?
> Thanks, Joseph


I'm guessing that what you're really interested in is some measure of the
variation or dispersion of your sample dataset.  In that case, you can
simply compute the sample standard deviation for each parameter of
interest.  The fact that each sample consists of 4 numbers that satisfy
the equation that their sum equals 1 just means that your 4 estimated
standard deviations aren't independent estimates, but for most practical
purposes this is an irrelevant technicality.

On the other hand, if you really want to compute a confidence interval for
the purposes of hypothesis testing, then you need to be explicit about
what your null hypothesis and alternative hypotheses are.  If you're not
sure what your null and alternative hypotheses are, then to me that
confirms that what you're interested in is not hypothesis testing but some
sense of how good an estimate your averages are.

It's important to realize that a 95% confidence interval does *not* mean
that there is a 95% probability that the quantity you're trying to
estimate lies in your interval.  This is a common misconception about what
confidence intervals are.


If you really want to make statements of the form "there is a 95%
probability that the win rate is in such-and-such an interval" then you
need to adopt a Bayesian rather than a frequentist framework.  In
particular you'll need to choose some prior probability distribution and
compute the posterior probability distribution by applying Bayes's rule to
your data.


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