http://justaddnoise.blogspot.com/2009/10/here-is-probability-that-true.html

but the short answer is just that the median estimate is greater than the mean and maximum a posterori estimates of small quantities and so it’s actually likely that small quantities are underestimated.

]]>However, one could take a proper Bayesian approach and ask, what is the probability that the true mean, lambda, is greater than this estimate? It turns out that the smaller N is the more likely it is that the true rate is greater.

To see this, place the conjugate Gamma prior on lambda so that the posterior distribution on lambda is also Gamma distributed to obtain,

p(lambda|n)

Now we can ask, what is the probability that lambda is actually greater than the mean of this distribution or even greater than the maximum aposteriori estimate of lambda? It turns out the answer is more or less independent of the parameters of the prior in this case. And its quite high. See figure here:

This result also holds for binomial distributions with uniform prior. x=lambda in the figure. For the Poisson case, the observed rate has been normalized by 1000 to fit on the same plot.

]]>Results 1 – 10 of about 284 for “the true number may be higher” -“the true number may be higher or lower”. (0.19 seconds)

Ah, search engines. And english. ]]>

Say you’re an academic who researches GID (or anything else). If you can say “GID affects maybe 1 in 100 people” that looks a lot better when you’re asking for grant money than if you say “GID affects 1 in 100,000 people”.

Or say you’re an activist for the interests of people with GID, you will naturally want to make it seem as common as possible because this is, unfortunately, equated with “normality” in most people’s minds. ]]>

Yes, but the true number may be higher. ]]>