From: Greg Aldering (aldering@panisse.lbl.gov)
Date: Thu May 22 2003 - 21:48:28 PDT
I caution, this is preliminary:
I generated A_B values according to the Hatano et al distribution
function. I then generated random values of A_B in accord with this
distribution function. I then "observed" those values of A_B by
generating a new A_B using our error bars and a Gaussian random number
generator with the observed A_B as the central value. Next, I generated
a Gaussian with the sigma of our error bars, centered at the "observed"
A_B, then multiplied that Gaussian by the Hatano et al prior. I then
calculated the mean A_B from the resulting PDF. This mean corresponds
to the central value that Riess would use. I did this 10000 times
and then took the mean of all the realizations. Since I did not
try to mimic Riess' style of creating error bars, the means are not
weighted (I don't know if this is a significant problem or not).
Here is what I found:
High-z:
True mean AB = 0.216 mag
Bayesian mean AB = 0.195 mag
Low-z:
True mean AB = 0.216 mag (this should be the same at both z's)
Bayesian mean AB = 0.205 mag
You can see that the bias between high- and low-redshift is only
0.01 magnitudes in A_B.
So, *if* the Hatano et al distribution is correct, *and* if Riess
et al really used it, *and* their uncertainties are like ours, then
the bias is very small.
Relative to formula A4 in P99, I think the bias is small because
sigma_G -- the sigma of the Gaussian prior -- is very small for the
Hatano et al extinction distribution. I have not examined what
happens if the wrong prior is used as the Bayesian in Riess'
method.
Note that as the true dust extinction distribution widens, not only
will Riess' bias increase, but, the bias in our technique will increase
as well. So, a wider extinction distribution is bad for everyone.
- Greg
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