A typical approach to developing an analysis algorithm for analyzing
gravitational wave data is to assume a particular waveform and use its
characteristics to formulate a detection criteria. Once a detection has been
made, the algorithm uses those same characteristics to tease out parameter
estimates from a given data set. While an obvious starting point, such an
approach is initiated by assuming a single, correct model for the waveform
regardless of the signal strength, observation length, noise, etc. This paper
introduces the method of Bayesian model selection as a way to select the most
plausible waveform model from a set of models given the data and prior
information. The discussion is done in the scientific context for the proposed
Laser Interferometer Space Antenna.Comment: 7 pages, 2 figures, proceedings paper for the Sixth International
  LISA Symposiu

Rubbo, Louis J.

English

arXiv

Louis J. Rubbo

Crossref

When is Enough Good Enough in Gravitational Wave Source Modeling?

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When is Enough Good Enough
in Gravitational Wave Source Modeling?
Louis J. Rubbo
Center for Gravitational Wave Physics, 104 Davey Lab, University Park, PA 16802
Abstract. A typical approach to developing an analysis algorithm for analyzing gravitational wave
data is to assume a particular waveform and use its characteristics to formulate a detection criteria.
Once a detection has been made, the algorithm uses those samech racteristics to tease out parameter
estimates from a given data set. While an obvious starting point, such an approach is initiated by
assuming a single, correct model for the waveform regardless of the signal strength, observation
length, noise, etc. This paper introduces the method of Bayesi n model selection as a way to select
the most plausible waveform model from a set of models given th data and prior information. The
discussion is done in the scientific context for the proposedLaser Interferometer Space Antenna.
INTRODUCTION
The anticipated data from the proposed Laser Interferometer Space Antenna (LISA) in-
troduces a number of exciting and original challenges. Central in these challenges is the
development of data analysis routines capable of coaxing out and characterizing individ-
ual signals from the noisy time series LISA will return. A great deal of work has already
been invested into the development of algorithms applicable to the LISA data. While a
number of these algorithms have demonstrated favorable capabilities on simulated data,
each make an initial assumption about the functional form for the waveform under con-
sideration. This paper introduces the use of Bayesian modelsel ction as a quantitative
method to selecting the waveform model. Using Bayes’ theorem w show how the data
and prior information picks out the most plausible model from a set of proposed models.
Gravitational wave data analysis can be loosely described as a three step process
as depicted in figure 1. In the first step, a signal is detected within a set of noisy
time streams retrieved from the detector. In step two, the signal is characterized by
producing estimates for the parameterization variables. Finally, step three is to make
physical interpretations based on the estimated parametervalues. These steps are not
necessarily mutually exclusive. There are no obvious boundaries and areas of overlap
do exist. However, each step is necessary when analyzing a detected signal.
In making the transition form detection to characterization (and quite often in the
detection process itself) a particular waveform is assumedprior to the investigate. While
an obvious assumption to make in the early developmental stage for an algorithm, it can
lead to needless complications and even misidentifications. For example, if a signal is
characterized by a low signal-to-noise ratio, some of the intricate waveform features can
be lost in the noise and therefore a simpler model would have sufficed in the analysis.
In the Bayesian model selection approach presented here, thda a and prior information
Detection
Characterization
Scientific Inference
Is there a signal present
in the data?
How is the signal parameterized
and what are the estimates for
the parameters?
What new science have we
gained from the data?
FIGURE 1. Data analysis flow chart.
justify the selection of a particular waveform model by calculating the most plausible
model from a proposed library of models.
Bayesian model selection is not a new methodology, but it is one that has not been
fully adopted by the still infant gravitational wave community. The aim of this paper is to
briefly summarize the theory and to discuss possible applications for analyzing the LISA
data. To this end, the paper first introduces the rules of probability theory, including a
derivation of Bayes’ theorem. It then outlines the necessary c lculations for performing
a model selection procedure. From here we give a simple, qualitative example of its use
for the LISA data. We conclude by suggesting a few other applications associated with
LISA.
BAYESIAN STATISTICS
Rules of Probability Theory
We begin by introducing a notation first used by Jeffreys [1].We will denote the
statement “the probability that propositionA is true given propositionB” as P(A|B).
Similarly, “the joint probability that bothA and B are true givenC” is denoted by
P(A,B|C). The notation “|C)” is the conditional that propositionC is assumed to be true.
In Bayesian statistics probability statements such asP(A) are not clear because they do
not explicitly state their dependencies. Furthermore,all probabilities are conditional.
Starting with the desiderata that degrees of plausibility are represented by real num-
bers, the rules for manipulating plausibility statements should agree with common sense,
and they should be consistent, then it is possible to show that the only two rules are re-
quired for manipulating probabilities [2]: the Sum Rule,
P(A+B|C) = P(A|C)+P(B|C)−P(A,B|C) (1)
where the plus sign inside the probability argument means “or”, and the Product Rule,
P(A,B|C) = P(A|C)P(B|A,C) . (2)
By standard Aristotelian logic it must be the case thatP(A,B|C) = P(B,A|C). Conse-
quently, the Product Rule may be re-expressed as
P(B,A|C) = P(B|C)P(A|B,C) . (3)
Equating the last two expressions results in Bayes’ theorem,
P(A|B,C) = P(A|C)
P(B|A,C)
P(B|C)
. (4)
Although Bayes’ theorem receives the accolades, it is simply a consistency statement
for the Product Rule.
In words, Bayes’ theorem is often stated as
Posterior= Prior
Marginal Likelihood
Global Likelihood
.
In this form it is evident that Bayes’ theorem quantitatively describes a learning process.
We start with a prior state of knowledge about propositionA whenC is assumed true,
P(A|C). We then gain new informationB, which in return updates our final state of
knowledge as given by the posterior probability,P(A|B,C). The proportionality factor
between our prior and posterior states of knowledge is a normalized statement about
how likely the propositionB will occur given that bothA andC are true.
While Bayes’ theorem is a useful byproduct of the Product Rule, the use of the Sum
Rule is equally important. It is through the Sum Rule that we are able to take a joint
probability of multiple propositions, and reduce it to a distribution of a smaller subset of
the larger joint distribution. For example, consider the joint distribution betweenA and
a set ofn exhaustiveBi’s, given prior informationI. From the Sum Rule we have
P(A,
n
∑
i=1
Bi|I) = P(A|I)
= P(A,B1|I)+P(A,
n
∑
i=2
Bi|I)−P(A,B1,
n
∑
i=2
Bi|I) , (5)
where the first equality follows from the Product Rule and thefact that theBi’s are
exhaustive. If theBi’s are mutually exclusive, that is only one value can be realiz d at a
time, then the last term is zero. Repeated applications of the Sum Rule leads to
P(A|I) =
n
∑
i=1
P(A,Bi|I) . (6)
When theBi’s take on continuous values the above goes over to an integral,
P(A|I) =
∫
P(A,B|I)dB . (7)
The process which we have just described is referred to asmarginalization. In it we have
removed anuisance parameter, B, from a joint distribution by a repeated application of
the Sum Rule.
Model Selection
In model selection the central question that is being addressed is the following:
“Given a particular set of data, and prior information, which hypothesis from a library
L ≡ {H1, . . . ,Hℓ} of hypotheses is the most plausible?” Key to this question are the
ideas that all prior information is included and that the most plausible hypothesis is based
on the given data. The hypotheses within a library are eitherassumed to be exhaustive
or, by a careful choice in models, the space is made so [3].
A model itself consists of a functional form dependent on a vector of parameters
~λ , and two probability distributions [4]. The first distribution describes the probability
distribution for the parameter values given the model priorto the new data,P(~λ |Hα).
This is a key point; two models are distinct even if they have the same parameterization
but different priors about how those parameters are believed to be distributed. The
second distribution is the probability of a data set given the model and a particular set of
parameter values,P(D|~λ ,Hα).
From Bayes’ theorem (4), the posterior probability for a particular model is given by
P(Hα |D, I) = P(Hα |I)
P(D|Hα , I)
P(D|I)
, (8)
where I symbolizes our unenumerated prior information. The denomiator can be
viewed as a normalization constant,
P(D|I) =
ℓ
∑
α=1
P(Hα |I)P(D|Hα, I) . (9)
By investigating theodds ratio between two competing models, we can eliminate the
need to calculate the normalization constant,
O12 =
P(H1|D, I)
P(H2|D, I)
=
P(H1|I)P(D|H1, I)
P(H2|I)P(D|H2, I)
=
P(D|H1, I)
P(D|H2, I)
. (10)
The second line arises by assuming that our prior information does not favor one model
over the other. The odds ratio gives us a means to directly compare competing models.
If our library contains more than two models, one model may beused as a reference. For
example, the reference model may be a constant (i.e. a no signal present model), while
the remaining library contains a spectrum of waveform models.
From the odds ratio it is apparent that to compare models in a libr ry only their
marginal likelihoods need to be calculated. The likelihoods are found by marginalizing,
Δλ
δλ
λ
ML
P(λ|I)
P(D|λ, H
α
, I)
FIGURE 2. A pictorial representation for the origins of Occam factorsin Bayesian model comparisons.
over all model parameters, the joint distribution for the data nd the model parameters,
P(D|Hα , I) =
∫
P(D,~λα |I) d~λα =
∫
P(~λα |Hα , I)P(D|~λα,Hα , I) d~λα , (11)
where the second equality follows from the Product Rule.
If the data is informative, i.e. we have learned something new, then the parameter
likelihood function,P(D|~λα ,Hα , I), will be more peaked than the parameter priors,
P(~λα |Hα , I). Figure 2 illustrates this for a one dimensional model. In this instance we
can estimate the marginal likelihood as
P(D|Hα , I)≈ P(D|λML,Hα , I) [P(λML|Hα , I) δλ ] . (12)
HereλML is the parameter value at the maximum likelihood andδλ is the characteristic
width for the parameter likelihood function. The term in square brackets is anOccam
factor; a term that naturally penalizes complicated models. To seethis consider a uni-
form prior, P(λ |I) = (∆λ )−1, where∆λ is the interval width for the range of expected
parameter values before the data is collected. The marginallikelihood is now
P(D|Hα , I)≈ P(D|λML,Hα , I)
δλ
∆λ
. (13)
For informative data the Occam factor is always less than unity. Consequently, for a
complicated model to be favored over a simpler one, the data must justify it by having a
corresponding larger value for the parameter likelihood function.
The proceeding argument is quickly extended to multiple dimensions. If the model has
more than one parameter, then there is a corresponding Occamfa tor for each parameter,
P(D|Hα , I)≈ P(D|~λML,Hα , I)
δλ1
∆λ1
· · ·
δλi
∆λi
, (14)
wherei is the number of parameters.
As a last point of emphasis, it is not enough to perform a parameter estimation analysis
and find thatλi = 0, therefore ruling out the model that includesλi. Doing so would
neglect the Occam factors that arise in Bayesian model selection and are not present in
a parameter estimation analysis, even a Bayesian analysis.
WHITE DWARF TRANSFORM
As a conceptually trivial but applicable example of Bayesian model selection for the
LISA mission, consider the detection of a supermassive black hole binary inspiral. For
black hole binaries with component masses in the range of 104−7 M⊙, LISA will observe
the binary evolution as the binary sweeps through frequencies from∼0.01 mHz up to
a few milliHertz (depending on the actual masses). In this same range of frequencies
is the gravitational wave background formed from the∼ 108 solar mass binaries in
our own galaxy. As the black holes inspiral, their detected signal will overlap with the
collective galactic background signal. Moreover, at any insta t of time the black hole
binary looks like a monochromatic binary. That is, as a supermassive black hole binary
with a time to coalescence oftc sweeps past a galactic binary of periodT , the two signals
have a significant overlap for an interval equal to the geometric mean oftc andT [5].
Consequently the black hole inspiral signal may be decomposed into a population of
monochromatic galactic binaries. Such a process is often ref rred to as awhite dwarf
transform.
For a gravitational wave data analyst the task is to select which of two models is more
plausible. The models under consideration are
HWD =
(
the detected signal is from a population
of monochromatic galactic binaries
)
HBH =
(
the detected signal is from a single
supermassive black hole binary
)
.
Model HWD is parameterized by 7N variables, whereN is the number of binaries
required to describe the apparent inspiral signal. For an inspiral signal between 0. 1 and
1 mHz,N is on the order of 104 assuming a binary per frequency bin and for a one year
observation1. Conversely, modelHBH is characterized by only seventeen parameters.
Estimating the posterior probabilities using equation (14) quickly leads to the conclu-
sion that the large parameter space associated with the whitdwarf population model has
associated with it an overwhelming number of Occam factors.These Occam factors pe-
nalize the white dwarf population model and in turn make the plausibility for the model
extremely low. The black hole model, on the other hand, only has seventeen Occam fac-
tors and therefore is not as severely penalized. Consequently, although an ensemble of
galactic binaries could conspire to look like a supermassive black hole binary inspiral,
1 A frequency bin∆ f is equal to one on the observation time,∆ f = T−1. For a one year observation,
which is used here,∆ f = 3.2×10−8 Hz.
the relative probability for such a model is many orders of magnitude less than a model
that contains a single black hole binary.
CONCLUDING REMARKS
The white dwarf transform is an obvious application of Bayesian model selection. More
informative and interesting examples include using Bayesian model selection as a cri-
teria for deciding when a signal is present in the data; characte izing complicated but
detected signals that have low signal-to-noise ratios; andcounting the number of de-
tectable galactic binaries within the larger population. The first application is simply
answering the question, when does the data justify declaring detection for a particular
waveform? The second application is concerned with deciding the information content
from a weak signal. That is, what features of an emitting system are actually measur-
able and what features are lost to the noise. Counting the number of detectable galactic
binaries is one of the few Bayesian model selection applications used in the LISA liter-
ature [6, 7]. Embedded within Reversible Jump Markov Chain Monte Carlo techniques
is the use of odds ratios in deciding the number of galactic binaries that are detectable.
In general, Bayesian model selection gives a logical and quantitative approach to
directly comparing competing models. By using a model selection procedure we are
able to maximize the amount of information we can extract from LISA’s data. The
most plausible model is the one that is most justified by the data and our prior state
of knowledge prior to the experiment. As progress is made in the development of LISA
analysis routines it is conceivable that Bayesian approaches will be a central tool.
ACKNOWLEDGMENTS
The author would like to thank Edward Cazalas, Matthew Francis, and Deirdre Shoe-
maker for a number of helpful discussions. Also, Lee Samuel Finn for introducing the
author to the Bayesian approach and for guidance on a number of its subtler points. This
work was supported by the Center for Gravitational Wave Physics. The Center for Grav-
itational Wave Physics is funded by the National Science Foundation under cooperative
agreement PHY 01-14375.
REFERENCES
1. H. Jeffreys,Theory of Probability, The International Series of Monographs on Physics, Clarendon
Press, Oxford, 1961, 3 edn.
2. R. T. Cox,American Journal of Physics 14, 1–13 (1946).
3. G. L. Bretthorst, “An Introduction to Model Selection using Probability Theory as Logic,” inMaximum
Entropy and Bayesian Methods, edited by G. R. Heidbreder, Kluwer Academic Publishers, Dordrecht,
1996, pp. 1–42.
4. D. J. C. MacKay,Neural Systems 4, 415–447 (1992).
5. N. J. Cornish, and J. Crowder,Physical Review D 72, 043005 (2005).
6. R. Umstätter, et al.,Physical Review D 72, 022001 (2005).
7. A. Stroeer, and A. Vecchio, Automatic Bayesian Inferencefor LISA Data Analysis (2006), presenta-
tion at the Sixth International LISA Symposium.


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