Likelihood Methods and Classical Burster Repetition

Abstract

We develop a likelihood methodology which can be used to search for evidence of burst repetition in the BATSE catalog, and to study the properties of the repetition signal. We use a simplified model of burst repetition in which a number $N_{\rm r}$ of sources which repeat a fixed number of times $N_{\rm rep}$ are superposed upon a number $N_{\rm nr}$ of non-repeating sources. The instrument exposure is explicitly taken into account. By computing the likelihood for the data, we construct a probability distribution in parameter space that may be used to infer the probability that a repetition signal is present, and to estimate the values of the repetition parameters. The likelihood function contains contributions from all the bursts, irrespective of the size of their positional errors --- the more uncertain a burst's position is, the less constraining is its contribution. Thus this approach makes maximal use of the data, and avoids the ambiguities of sample selection associated with data cuts on error circle size. We present the results of tests of the technique using synthetic data sets. Comment: 5 pages, Revtex (aipbook.sty included), 2 PostScript figures included using psfig. To appear in the Proceedings of the 1995 La Jolla Workshop "High Velocity Neutron Stars and Gamma-Ray Bursts," eds. R. Rothschild and R. Lingenfelter, AIP, New York

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arXiv:astro-ph/9511062v1 14 Nov 1995
Likelihood Methods and Classical
Burster Repetition
Carlo Graziani
NRC/NASA Goddard Space Flight Center
Donald Q. Lamb
Department of Astronomy and Astrophysics, University of Chicago
We develop a likelihood methodology which can be used to search
for evidence of burst repetition in the BATSE catalog, and to study the
properties of the repetition signal. We use a simplified model of burst
repetition in which a number Nr of sources which repeat a fixed num-
ber of times Nrep are superposed upon a number Nnr of non-repeating
sources. The instrument exposure is explicitly taken into account. By
computing the likelihood for the data, we construct a probability dis-
tribution in parameter space that may be used to infer the probability
that a repetition signal is present, and to estimate the values of the
repetition parameters. The likelihood function contains contributions
from all the bursts, irrespective of the size of their positional errors
— the more uncertain a burst’s position is, the less constraining is its
contribution. Thus this approach makes maximal use of the data, and
avoids the ambiguities of sample selection associated with data cuts on
error circle size. We present the results of tests of the technique using
synthetic data sets.
INTRODUCTION
Classical gamma-ray burst repetition is one of the most startling possibil-
ities suggested by analyses the BATSE 1B catalog. Quashnock & Lamb (1)
searched for evidence of strong clustering of burst positions by studying the
distribution of nearest-neighbor separations, and reported an excess of bursts
clustered on 5◦scales with a significance of 2.5× 10−4. Wang & Lingenfelter
(2,3) have studied clustering of bursts in the 1B catalog in both angle and
time, and found clustering on scales of 4 days and 4◦with a significance of
2×10−5. Quashnock (4) finds the odds favoring repetition over no repetition
to be about 4:1, based on a counts-in-cells likelihood technique.
On the other hand, the data in the BATSE 2B − 1B catalog (that is, those
bursts seen in the 2B catalog after the end of the 1B epoch) show no such
significant repetition signal (5). This is not unexpected, given the lower sky
exposure due to the failure of the tape recorders aboard CGRO. Neverthe-
c ? 1993 American Institute of Physics1

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less, the absence of confirmation of the 1B signals, combined with the only
moderate significance of those signals, have fueled a controversy over whether
the evidence for burst repetition in the 1B catalog is merely a statistical
fluctuation, perhaps amplified by choice of bin intervals (in the case of the
time-angle analysis) or of cuts in error circle size (in the case of the nearest-
neighbor analysis). These criticisms have been answered convincingly through
analysis of the effect of data cuts (1) and of choice of binsize (3). Nevertheless
the evidence for repetition is not universally regarded as conclusive.
Under the circumstances, there is a premium on the development of tech-
niques to probe for burst repetition that are more sensitive than existing ones
— to achieve higher significance detections — and that do not rely on either
binning or data cuts. In this work we present such a technique, founded upon
the likelihood function for a simple model of burst repetition. We also exhibit
the results of tests of the method on synthetic data sets, which allow us to
gauge its sensitivity and to understand potential systematic effects.
THE LIKELIHOOD FUNCTION
In order to calculate the likelihood function, we require a model of burst
repetition. The model we have adopted is one in which Nrrepeating sources
are superposed upon Nnrnon-repeating sources. Each repeating source bursts
exactly Nreptimes, although not all these bursts are observed because of the
limited sky exposure. This simplified model captures the essential features of
the phenomenon — number of repeating sources and repetition rate — while
permitting practical calculation. The model parameters are Nr and Nrep.
Nnris set by the data when Nrand Nrepare chosen, and the sky exposure E
is fixed at the value appropriate for the epoch of the BATSE catalog under
study.
The likelihood function calculation requires that we identify likely “parti-
tions” π of the data. A partition is an assignment of some bursts to each of
the repeating sources. The likelihood function L(Nr,Nrep) is then given by
L(Nr,Nrep) =
?
π
(Nr+ Nnr)!
×



Nr
?
j=1
?
Nrep!
nj!(Nrep− nj)!Enj(1 − E)Nrep−nj× Lj
?

,(1)
where njis the observed number of bursts from the jth source (according to
the partition π), E is the sky exposure, and Ljis the “cluster likelihood” for
the jth source of π. Ljacts as a measure of the plausibility of the assignment
of the nj bursts to the source, and is in fact the expression for the Bayesian
odds favoring the hypothesis that the nj bursts were produced by the same
source over the hypothesis that they each had a separate source. Assuming

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that the scatter of the observed positions about the true source positions is
described by the Fisher distribution, Ljis given by the following expression:
Lj=
Σj2sinhΣj−2
k=1[σjk2sinhσjk−2]
?nj
;Σj≡
?????
nj
?
k=1
? xjk
σjk2
?????
−1/2
.(2)
Here, ? xjkis the unit vector describing the position on the sky of the kth burst
of the jth source, and σjk is the error in that position. Σj is the error in
the maximum likelihood position of the jth source. The maximum likelihood
position itself is ? zj= Σj2?nj
in which all the σ are much less than one. In this limit the Fisher distribu-
tion becomes the two-dimensional, symmetric Gaussian distribution, and the
expression for the jth cluster likelihood becomes
k=1? xjk/σjk2.
The meaning of Eq. (2) may be clarified by taking the small-angle limit,
Lj≈
1
2Σj2
?σjk2
?nj
k=1
2
? × exp
?
−1
2
nj
?
k=1
(? xjk−? zj)2
σjk2
?
;Σj≈
?nj
k=1
?
1
σjk2
?−1/2
.
(3)
From Eq. (3) we see that Lj is a product of two factors: the exponential
factor, which penalizes (decreases) the likelihood if the jth cluster of π is too
dispersed, and the rational factor, which rewards (increases) the likelihood
for making nj as large as possible (recall that σjk ≪ 1). These two factors
compete with each other, since the addition of a burst to a cluster may well
provoke a harsher penalty from the exponential factor (if the burst is implau-
sibly distant from the rest of the cluster) than the reward it reaps from the
rational factor. This fact allows Lj to be used as a tool in identifying “opti-
mum” clusters of bursts. This is an important observation, since the sum over
partitions π in the likelihood clearly ranges over an impossibly large number
of configurations. Fortunately, most partitions contribute negligibly to the
likelihood, as they assign bursts to improbably dispersed clusters. We may
thus restrict our attention to partitions that make appreciable contributions
to the likelihood by stringing together into partitions only “optimum” clusters
identified using Lj. This is a key ingredient of our technique.
Note that the dependence of Ljon the burst positions is strongest for bursts
with small error circles and weakest for bursts with large error circles. This
is the reason that the inclusion of bursts with poorly determined positions
does not weaken the repetition signal — such bursts are individually unable
to seriously constrain inferences based on the likelihood. On the other hand,
even bursts with large error circles bear some information, particularly if there
are many of them. Our approach allows that information to be exploited to
the fullest extent possible.
We use Bayes theorem to interpret the likelihood function as a probability
distribution in (Nr,Nrep) parameter space. This distribution is the basis for

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FIG. 1. Sky map resulting from a simulation with Nr = 5, Nrep = 15, and a sky
exposure E = 0.33. Note that more than three bursts are observed from only two
of the five repeating sources, even though Nrep = 15, because the sky exposure is
limited.
all our statistical inferences. By comparing the contribution of the likelihood
function to the entire parameter space to the value of the likelihood for the
non-repeating model (Nr= 0), we may compute the odds favoring the repe-
tition hypothesis over the no-repetition hypothesis, thus obtaining a sensitive
test for the presence of a repetition signal. Furthermore, the detailed shape
of the distribution allows us to estimate the maximum likelihood parameter
values, and to infer credible regions for those parameters.
TESTING THE TECHNIQUE
We have tested the method on synthetic data, simulating both repeating
and non-repeating (Nr= 0) models, in order to test its sensitivity and power,
as well as to gauge the reliability of the parameter estimates that it produces.
Fig. 1 shows a sky map resulting from a simulation assuming Nr = 5,
Nrep = 15, and sky exposure E = 0.33. There are 25 bursts from the 5
repeating sources, and 50 bursts from all sources. The assumed total error
σcirc— that is, the radius of the 68% circular confidence region including both
systematic and statistical error — is 5◦for all bursts. σcircis related to the
Fisher distribution parameters σjkabove by σjk2= 0.871(1 − cosσcirc).
Fig. 2 shows the probability distribution in the (Nr,Nrep)-plane that results
from the analysis of this simulated data set. The distribution is sharply peaked
near the true parameter values. Also shown are the 1-, 2-, and 3-σ credible
regions for the model parameters.
An interesting question that arises is whether the credible regions shown in
Fig. 2 are well-calibrated, in the sense that they bracket the true parameters
with frequency equal to the stated probability contained within the contour.

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FIG. 2. Left panel: Probability distribution in the (Nr,Nrep)-plane that results
from the analysis of the simulated data shown in Fig. 1. Right panel: 1-, 2-, and 3-σ
credible regions in the (Nr,Nrep)-plane. The cross shows the maximum likelihood
parameter estimates, while the X marks the true parameter values.
To begin addressing this question, we analyzed 20 simulated data sets with
the same parameter values as those given above. We found that the 68%
contour included the true parameters in 18 out of 20 simulations, which by
the binomial distribution is quite consistent with the 68% interpretation of
the contour. We are carrying out analyses on many more simulations in order
to refine the contour calibration.
The odds may be computed by ascribing equal a priori probability to the
repeating and the non-repeating hypothesis, and, within the repeating hy-
pothesis, equal a priori probability to each of the discrete parameter values
(Nr,Nrep) at which the likelihood is evaluated. This last assignment dilutes
the value of the odds if the parameter space is very large, and accounts nat-
urally for the “number of attempts” correction to the statistical significance
of the detection.
We calculated the odds for each of the 20 simulated data sets discussed
above, and found that the mean and standard deviation log odds were
logO = 22.5 ± 7.9. We also simulated 20 data sets assuming no repetition,
and calculated the odds for each of those. The result was logO = 0.5 ± 2.0.
Clearly, then, the likelihood function approach is quite capable of distinguish-
ing between no repetition and the kind of repetition simulated here, and thus
offers a promise of a sensitive and powerful statistic for the detection of rep-
etition signals in burst data.
REFERENCES
1. J. M. Quashnock, and D. Q. Lamb, Mon. Not. R. Astron. Soc., 265, L59 (1993).
2. V. C. Wang, and R. E. Lingenfelter, in Gamma-Ray Bursts, Proceedings of the