>On 28 Mar 2004 19:57:46 GMT, [EMAIL PROTECTED] (Radford Neal)
>wrote:
>
>>You don't say what led you to think that you want to do this, but I
>>might speculate that you've encountered the idea, distressingly
>>common in Bayesian papers, that the gamma(0.001,0.001) distribution
>>is "vague", and suitable as a prior distribution in situations with
>>little knowledge. As an examination of a sample from this
>>distribution will show, this idea is utterly wrong.
In article <[EMAIL PROTECTED]>,
Duncan Murdoch <[EMAIL PROTECTED]> wrote:
>Just to be provocative:
>
>What would you expect to see when you sampled from a vague
>distribution? The idea that gamma(0.001,0.001) is vague comes from
>the fact that the density approximates 1/x pretty well.
Ah! But regardless of whether you think the DENSITY approximates 1/x
"pretty well", the DISTRIBUTION doesn't.
> If it would
> make sense to call the improper prior 1/x vague, then gamma(0.001,
> 0.001) is pretty close to being vague.
Maybe 1/x, equivalent to a uniform prior for log(x), could be
described as "vague". When they can get away with it, people often
use that, even though it's improper. Presumably, what people are
thinking when they use gamma(0.001,0.001) is that it's close to 1/x
but not actually improper. I'd guess that have in mind a distribution
for log(x) that's roughly uniform over a range of several orders of
each side of zero (ie, around x=1).
Here's what gamma(0.001,0.001) actually is like:
> r <- rgamma(100000,0.001,0.001)
> summary(r)
Min. 1st Qu. Median Mean 3rd Qu. Max.
0.000e+00 0.000e+00 4.139e-298 8.433e-01 3.787e-122 2.395e+03
> summary(log(r))
Min. 1st Qu. Median Mean 3rd Qu. Max.
-Inf -Inf -684.700 -Inf -279.600 7.781
> mean(r>1)
[1] 0.00601
Far from the distribution for log(x) being uniform over a region
around zero, it's highly biased in the downwards direction, and has
less than a 1% chance of taking on a value greater than zero (ie, a
value for x greater than one). So it's far too vague in one
direction, and not vague enough in the other.
If you want a "vague" prior centered around 1, you'd do better
with gamma(0.1,0.00001):
> summary(rgamma(100000,0.1,0.00001))
Min. 1st Qu. Median Mean 3rd Qu. Max.
1.358e-48 5.960e-02 5.982e+01 9.911e+03 3.448e+03 7.583e+05
Better yet, if you want a distribution for log(x) that's uniform over
several orders of magnitude around zero, I'd recommend using the
uniform distribution extending for several orders of magnitude around
zero. Or you could use a log normal distribution.
Radford Neal
.
.
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