INTERFEROMETER LASER GRAVITATIONAL WAVE OBSERVATORY - LIGO CALIFORNIA INSTITUTE OF TECHNOLOGY MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Document Type
LIGO-T030248-00-Z
Is the Conservative frequentist upper limit always worse than the Optimistic Bayesian upper limit ? NoA Simple Counterexample
Yousuke Itoh Max-Planck-Institute fr Gravitationsphysik, Albert-Einstein-Institut
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Prepared for the S1 paper
Continuous Waves Upper Limit Group
California Institute of Technology LIGO Project - MS 51-33 Pasadena CA 91125 Phone [phone] Fax [phone] E-mail: [e-mail]
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A Processed with LTEX on 2003/11/20
Prepared for the S1 paper
LIGO-T030248-00
1
Model
Take a one dimensional Gaussian distribution for X 0 with a parameter A 0. P (X|A) = N exp(XA) 1 N = (1 + Erf(A)) 2
2
(1) (2)
Suppose a relation A = HD where D is not known but we know D is bounded: 0 < W D B < . W (worst) and B (best) denote arbitrary nite real constants. (In this model, A is akin to gravitational-wave strain at the detector, projected onto the detectors antenna pattern, while H is akin to the sources intrinsic strength h0 , and D is the uncertain, angle-dependent factor that relates them.) Now suppose we want to set an upper limit on H, given a data x. Given a signicance level p (and thus the condence level 1-p), the frequentist upper limit on A, which we denote by Af = Af (x, p), is given implicitly by
x
p=
0
P (X|Af )dX = (Erf(Af ) Erf(Af x))/(1 + Erf(Af )).
(3)
The conservative frequentist upper limit on H, which we call Hf w , is then Hf w = Af /W . Given the uniform priori probability P (A) =const (0 A < ) the Bayesian upper limit on A denoted by Ab = Ab (x, p) is by1
1p= given =N
0
Ab
Ab
P (A|x) dA = N
0 Ab 0
P (x|A)P (A) dA =
Ab
P (x|A) dA = N N
0
exp((x A)2 ) dA = (4)
= (Erf(Ab x) + Erf(x))/(1 + Erf(x)). where we used (N N )
1
=
(1 + Erf(x)). 2
(5)
The optimistic Bayesian upper limit on H, which we call Hbo , is then just given by Hbo = Ab /B.
2
Two examples
Suppose (W,B) = (0.5,1) and we take p = 0.05.
This is not quite the procedure we followed in the Bayesian pulsar analysis. There we assumed the uniform priori probabilities for h0 > 0, for the cosine of the inclination angle cos that is the angle between the angular momentum of the pulsar and the line of the sight, the gravitational-wave polarization angle , and the gravitational-wave initial phase 0 . Then we marginalised the probability with respect to cos , , and 0 to set an upper limit on h0 . Analogously, in this toy model, one may assume P (H) =const. (0 H < ), P (D) =const. (W D B), and then marginalise with respect to D to obtain the marginalised probability for H. Here, for simplicity, we do not marginalise the probability, but simply set D=B to yield the most optimistic limit as stated below by...