Here we review the loudest event statistic, described in Refs. [77,78,83], which we will use in Sec. 4.6 to compute Bayesian 90% confidence level upper limits on the coalescence rate from our S6/VSR2-3 observations. One of the advantages to using a Bayesian formalism for estimating event rates is that it provides a natural prescription for combining the results of independent experiments. As we mentioned above, these searches each consisted of 24 independent sub-experiments. Additionally, we have upper limit estimates from the S5/VSR1 searches [47, 84, 85], which we would like to fold into our present observations. After describing the loudest event statistic, we show how we combine results from multiple independent experiments, marginalize over the various sources of uncertainty, and compute upper limits.
We formalize the problem by considering a set~µof model parameters we wish to constrain and a set ~x of outcomes for N independent experiments. In our case, the model parameters ~µ will consist of the rate densityµ plus a number or “error” parameters which we will marginalize over.
The outcomes ~x of the experiments are the loudest event observed in each experiment. To make
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inferences on the parameters ~µ after having observed the outcomes ~x, we apply Bayes’ theorem, which states
p(~µ|~x) =p(~x|~µ)p(~µ)
p(~x). (4.9)
If the observations~xare independent, as for the outcomes of our 24 sub-periods, then the expression above factorizes as
p(~µ|~x) =Y
i
p(xi|~µ) p(~µ) Q
ip(xi). (4.10)
The conditional probability densityp(~x|~µ) is called thelikelihoodof the outcome~xgiven the param- eter~µ, p(µ) is the prior distribution on the parameters~µ, and p(~x) is the prior distribution on the outcome of the experiments.
Before delving into the loudest event statistic, we take a step back and consider the simple and familiar problem of estimating event rates when the number of observed events is certain. One can view the loudest event statistic as essentially performing the same calculation but marginalizing over the unknown number of observed events. Suppose that in performing a search for gravitational waves, we observe exactly n CBC events. If the search is sensitive to a volume V and lasts for a durationT then one can show by applying Bayes theorem that the posterior distribution on the rate is
p(µ|n) = p(µ)(µV T)ne−µV T R∞
0 p(µ)(µV T)ne−µV Tdµ. (4.11) In deriving this result, we treat the foreground process as a Poisson process with mean number of eventsµV T.
For the special case in which the prior is uniform in the rate, the posterior becomes
p(µ|n) = (µV T)ne−µV T
R(µV T)ne−µV Tdµ. (4.12)
A simple way to understand this result is to compute the peak of the distribution, which turns out to be
µmax= n
V T. (4.13)
We see that given the observation of n events and no prior information about the rate (uniform prior), the most likely rate is just the number of observed events divided by the observed volume.
Another important special case are priors of the formp(µ)∝µne−µV1T1 withnan integer. This prior would arise naturally as a posterior from an initial search (using a flat prior) which identified
Table 4.2: Event rate confidence intervals for a simple counting experiment.
n 5% lower limit 90% upper limit 95% upper limit
0 - 2.302 2.996
1 0.355 3.890 4.744
2 0.818 5.322 6.296
3 1.366 6.681 7.754
4 1.970 7.994 9.154
5 2.613 9.275 10.513
nevents as in Eqn. 4.12. If in a subsequent search with sensitivityV2T2, we identifymmore events and wish to use the posterior from the previous search as a prior to this search, we find that the combined posterior is
p(µ|n, m)∝µn+me−µ(V1T1+V2T2). (4.14) This result illustrates an important property of this approach to estimating the rate: the posterior distribution one obtains starting from a flat prior depends only on the total number of observed events and the total observed volume. The loudest event statistic does not have this property – the posterior is sensitive to the way in which the data are divided.
We can use the posterior in Eqn. 4.12 to compute upper limits or confidence intervals. In the case of an experiment using a uniform prior which makes no detections, the upper limit at 90%
confidence is
ˆ
µ=2.303
V T . (4.15)
Forn= 1, the 90% upper limit is
ˆ
µ=3.890
V T . (4.16)
We will see below that in loudest event statistic formalism, these two results are limiting cases, and a parameter Λ interpolates between them to take into account uncertainty in possibly having made a false dismisal. We present upper and lower limits for other values ofn in Tbl. 4.2. This table is useful for interpreting the upper limits obtained by the loudest event method.
We now generalize these results to the case in which the identity of the foreground events (signal or noise) is unknown. The loudest event statistic provides a method to compute the consistency of the loudest event with the background, and in doing so the statistic effectively interpolates between the two extreme cases: (i) where the loudest event is definitely due to signal and (ii) where the loudest event is definitely due to noise. Beginning with the trivial observation that there were no events observed above the loudest event, one can show that Bayes’ theorem implies that the rate
90
posterior for a single search with loudest eventx, is given by
p(µ|x) = p(µ)(1 +µV(x)TΛ(x))e−µV(x)T
R p(µ)(1 +µV(x)TΛ(x))e−µV(x)Tdµ, (4.17)
whereV(x) is the average volume of the region in which events would register in the pipeline louder than x, and Λ(x) is a measure of the relative likelihood of the loudest event coming from the foreground versus coming from the background. Compare Eqn. 4.17 to Eqn. 4.12 with n= 0 and n= 1. We can think of the parameter Λ as interpolating between these two cases. We define Λ by
Λ(x) =−dlognf(x)/dx
dlogPb(x)/dx, (4.18)
where nf(x) is the expected number of signal events above the loudest event and Pb(x) is the probability of having a background event above the loudest event. For Λ<1, the loudest event is more consistent with background, while Λ>1 indicates the loudest event is inconsistent with the background. For our searches, the detection statisticxis false alarm numberx=RT, whereRis the false alarm rate, and thereforePb(x) =e−x. The mean number of foreground events is nf =µV T. Putting these into Eqn. 4.18, we obtain the expression
Λ(x) = 1 V
dV
dx, (4.19)
which is how Λ(x) is actually computed in theIHOPEpipeline.
To combine multiple experiments, we use the posterior from one analysis as the prior to the next.
We ignore marginalization over volume uncertainties for now, treating that issue in the next section.
For two experiments with sensitive volumesV1, V2and likelihoods Λ1,Λ2, the rate posterior is given by
p(µ|V1T1,Λ1, V2,Λ2)∝(1 +µV1T1Λ1)(1 +µV2T2Λ2)e−µ(V1T1+V2T2). (4.20) We note three special corner cases (see Fig. 4.4) that give a good summary of the overall behavior of this posterior. In each of these cases, the posterior in Eqn. 4.20 simply reduces to a posterior of the form in Eqn. 4.14 for a certain value ofn. We have (i) Λ1= Λ2= 0, corresponding to Eqn. 4.14 with n = 0, (ii) Λ1 = 0,Λ2 = ∞, or vice versa, corresponding to Eqn. 4.14 with n = 1 and (iii) Λ1 = Λ2=∞, corresponding to Eqn. 4.14 withn= 2. Thus, for two experiments the upper limits satisfy 2.30 ≤µ90PViTi ≤5.32. By similar reasoning, one can argue that for three experiments 2.30 ≤ µ90PViTi ≤ 6.68. Where in that range the upper limit falls depends on the computed
0 2 4 6 8 10
µP ViTi
PosteriorProbabilityDensity
Combining Two Experiments
two quiet events one loud event two loud events
Figure 4.4: Combining independent experiments with the loudest event statistic. When you combine two experiments using the loudest event statistic, the resulting 90% upper limit (solid vertical lines) is bounded by 2.302V T to 5.322V T, corresponding to the two extreme values for the counting experiment withn= 0 orn= 2 observed events, respectively. The outcome depends on whether there are large values of Λ in one or both of the experiments.
values for Λi. For an arbitrary number of independent experiments the posterior distribution on the combined analysis is given by
p(µ|V , ~~ Λ)∝p(µ) exp(−µX
k
VkT)Y
k
(1 +µΛkVkT), (4.21)
whereVk and Λk are the measured volumes and Λ values for the individual experiments.