Tests of Bayesian Model Selection Techniques for Gravitational Wave Astronomy
Neil J. Cornish and Tyson B. Littenberg
Department of Physics, Montana State University, Bozeman, MT 59717
The analysis of gravitational wave data involves many model selection problems. The most
important example is the detection problem of selecting between the data being consistent with
instrument noise alone, or instrument noise and a gravitational wave signal. The analysis of data
from ground based gravitational wave detectors is mostly conducted using classical statistics, and
methods such as the Neyman-Pearson criteria are used for model selection. Future space based
detectors, such as the Laser Interferometer Space Antenna (LISA), are expected to produced rich
data streams containing the signals from many millions of sources. Determining the number of
sources that are resolvable, and the most appropriate description of each source poses a challenging
model selection problem that may best be addressed in a Bayesian framework. An important class of
LISA sources are the millions of low-mass binary systems within our own galaxy, tens of thousands
of which will be detectable. Not only are the number of sources unknown, but so are the number
of parameters required to model the waveforms. For example, a significant subset of the resolvable
galactic binaries will exhibit orbital frequency evolution, while a smaller number will have measurable
eccentricity. In the Bayesian approach to model selection one needs to compute the Bayes factor
between competing models. Here we explore various methods for computing Bayes factors in the
context of determining which galactic binaries have measurable frequency evolution. The methods
explored include a Reverse Jump Markov Chain Monte Carlo (RJMCMC) algorithm, Savage-Dickie
density ratios, the Schwarz-Bayes Information Criterion (BIC), and the Laplace approximation to
the model evidence. We find good agreement between all of the approaches.
I. BACKGROUND
Bayesian statistical techniques are becoming increas-
ingly popular in gravitational wave data analysis, and
have shown great promise in tackling the various difficul-
ties of gravitational wave (GW) source extraction from
modeled data for the Laser Interferometer Space Antenna
(LISA). A powerful tool in the suite of Bayesian methods
is that of quantitative model selection [1, 2]. To under-
stand why this is a valuable feature consider a scenario
where one is attempting to fit data with two competing
models of differing dimension. In general, a higher di-
mensional model will produce a better fit to a given set
of data. This can be taken to the limit where there are as
many model parameters as there are data points allowing
one to perfectly match the data. The problem then is to
decide how many parameters are physically meaningful
and to select the model containing only those parameters.
In the context of GW detection these extra parameters
could be additional physical parameters used to model
the source or additional sources in the data. If a model
is over-parameterized it will over-fit the data and produce
spurious results.
Many of the model selection problems associated with
LISA astronomy involve nested models, where the sim-
pler model forms a subset of the more complicated model.
The problem of determining the number of resolvable
galactic binaries, and the problem of determining the
number of measurable source parameters, are both ex-
amples of nested model selection. One could argue that
the later is better described as “approximation selection”
since we are selecting between different parameterizations
of the full 17 dimensional physical model that describes
the signals from binary systems of point masses in general
relativity. However, many similar modeling problems in
astrophysics and cosmology [2], as well as in other fields
such as geophysics [3], are considered to be examples of
model selection, and we will adopt that viewpoint here.
The LISA observatory [4] is designed to explore the
low frequency portion of the gravitational wave spectrum
between ∼ 0.1 → 100 mHz. This frequency region will be
heavily populated by signals from galactic binary systems
composed of stellar mass compact objects (e.g. white
dwarfs and neutron stars), of which millions are theorized
to exist. Tens of thousands of these GW sources will
be resolvable by LISA and the remaining sources will
contribute to a confusion-limited background [5]. This
is expected to be the dominant source of low frequency
noise for LISA.
Detection and subsequent regression of the galactic
foreground is of vital importance in order to then pur-
sue dimmer sources that would otherwise be buried by
the foreground. Because of the great number of galac-
tic sources, and the ensuing overlap between individ-
ual sources, a one-by-one detection/regression is inaccu-
rate [6]. Therefore a global fit to all of the galactic sources
is required. Because of the uncertainty in the number of
resolvable sources one can not fix the model dimension a
priori which presents a crucial model selection problem.
Over-fitting the data will result in an inaccurate regres-
sion which would then remove power from other sources
in the data-stream, negatively impacting their detection
and characterization. The Reverse Jump Markov Chain
Monte Carlo approach to Bayesian model selection has
been used to determine the number of resolvable sources
in the context of a toy problem [7, 8] which shares some
of the features of the LISA foreground removal prob-
lem. Meanwhile the Laplace approximation to Bayesian
http://arxiv.org/abs/0704.1808v3
model selection has been employed to estimate the num-
ber of resolvable sources as part of a MCMC based al-
gorithm to extracting signals from simulated LISA data
streams [6, 9].
Another important problem for GW astronomy is the
determination of which parameters need to be included in
the description of the waveforms. For example, the GW
signal from a binary inspiral, as detected by LISA, may
involve as many as 17 parameters. However, for massive
black hole binaries of comparable mass we expect the ec-
centricity to be negligible, reducing the model dimension
to 15, while for extreme mass ratio systems we expect
the spin of the smaller body to have little impact on
the waveforms, reducing the model dimension to 14. In
many cases the inspiral signals may be described by even
fewer parameters. For low mass galactic binaries spin
effects will be negligible (removing six parameters), and
various astrophysical processes will have circularized the
orbits of the majority of systems (removing two param-
eters). Of the remaining nine parameters, two describe
the frequency evolution - e.g. the first and second time
derivatives of the GW frequency, which may or may not
be detectable [27].
Here we investigate the application of Bayesian model
selection to LISA data analysis in the context of deter-
mining the conditions under which the first time deriva-
tive of the GW frequency, ḟ , can be inferred from the
data. We parameterize the signals using the eight pa-
rameters
~λ→ (A, f, θ, φ, ψ, ι, ϕ0, ḟ) (1)
where A is the amplitude, f is the initial gravitational
wave frequency, θ and φ are the ecliptic co-latitude and
longitude, ψ is the polarization angle, ι is the orbital
inclination of the binary and ϕ0 is the GW phase. The
parameters f , ḟ and ϕ0 are evaluated at some fiducial
time (e.g. at the time the first data is taken). For our
analysis only a single source is injected into the simulated
data streams. In the frequency domain the output s(f)
in channel α can be written as
s̃α(f) = h̃α(~λ) + ñα(f) (2)
where h̃α(~λ) is the response of the detector to the incident
GW and ñα(f) is the instrument noise. For our work we
will assume stationary Gaussian instrument noise with no
contribution from a confusion background. In our anal-
ysis we use the noise orthogonal A,E, T data streams,
which can be constructed from linear combinations of
the Michelson type X,Y, Z signals:
(2X − Y − Z) ,
(Z − Y ) ,
(X + Y + Z) . (3)
This set of A,E, T variables differ slightly from those
constructed from the Sagnac signals [10]. We do not use
the T channel in our analysis as it is insensitive to GWs
at the frequencies we are considering. The noise spectral
density in the A,E channels has the form
Sn(f) =
(1− cos(2f/f∗))
(2 + cos(f/f∗))Ss
+2(3 + 2 cos(f/f∗) + cos(2f/f∗))Sa
Hz−1 (4)
where f∗ = 1/2πL, and the acceleration noise Sa and
shot noise Ss are simulated at the levels
10−22
9× 10−30
(2πf)4L2
Hz−1 .
Here L is the LISA arm length (≈ 5× 109 m).
Of central importance to Bayesian analysis is the pos-
terior distribution function (PDF) of the model parame-
ters. The PDF p(~λ|s) describes the probability that the
source is described by parameters ~λ given the signal s.
According to Bayes’ Theorem,
p(~λ|s) = p(
~λ)p(s|~λ)
d~λ p(~λ)p(s|~λ)
where p(~λ) is the a priori, or prior, distribution of the
parameters ~λ and p(s|~λ) is the likelihood that we measure
the signal s if the source is described by the parameters ~λ.
We define the likelihood using the noise weighted inner
product
(A|B) ≡ 2
Ã∗α(f)B̃α(f) + Ãα(f)B̃
Sαn (f)
p(s|~λ) = C exp
s− h(~λ)
s− h(~λ)
where the normalization constant C depends on the
noise, but not the GW signal. One goal of the data anal-
ysis method is to find the parameters ~λ which maximizes
the posterior. Markov Chain Monte Carlo (MCMC)
methods are ideal for this type of application [11]. The
MCMC algorithm will simultaneously find the param-
eters which maximize the posterior and accurately map
out the PDF of the parameters. This is achieved through
the use of a Metropolis-Hastings [12, 13] exploration of
the parameter space. A brief description of this process
is as follows: The chain begins at some random position
~x in the parameter space and subsequent steps are made
by randomly proposing a new position in the parame-
ter space ~y. This new position is determined by drawing
from some proposal distribution q(~x|~y). The choice of
whether or not adopt the new position ~y is made by cal-
culating the Hastings ratio (transition probability)
α = min
p(~y)p(s|~y)q(~y|~x)
p(~x)p(s|~x)q(~x|~y)
and comparing α to a random number β taken from a
uniform draw in the interval [0:1]. If α exceeds β then
the chain adopts ~y as the new position. This process
is repeated until some convergence criterion is satisfied.
The MCMC differs from a Metropolis extremization by
forbidding proposal distributions that depend on the past
history of the chain. This ensures that the progress of the
chain is Markovian and therefore statistically unbiased.
Once the chain has stabilized in the neighborhood of the
best fit parameters all previous steps of the chain are
excluded from the statistical analysis (these early steps
are referred to as the “burn in” phase of the chain) and
henceforth the number of iterations the chain spends at
different parameter values can be used to infer the PDF.
The power of the MCMC is two-fold: Because the al-
gorithm has a finite probability of moving away from a
favorable location in the parameter space it can avoid
getting trapped by local features of the likelihood sur-
face. Meanwhile, the absence of any “memory” within
the chain of past parameter values allows the algorithm
to blindly, statistically, explore the region in the neigh-
borhood of the global maximum. It is then rigorously
proven that an MCMC will (eventually) perfectly map
out the PDF, completely removing the need for user in-
put to determine parameter uncertainties or thresholds.
The parameter vector that maximizes the posterior dis-
tribution is stored as the maximum a posteriori (MAP)
value and is considered to be the best estimate of the
source parameters. Note that because of the prior weight-
ing in the definition of the PDF this is not necessarily
the ~λ that yields the greatest likelihood. Upon obtaining
the MAP value for a particular model X the PDF, now
written as p( ~λ,X |s), can be employed to solve the model
selection problem.
II. BAYES FACTOR ESTIMATES
The Bayes Factor BXY [14] is a comparison of the ev-
idence for two competing models, X and Y , where
pX(s) =
d~λ p(~λ,X |s) (10)
is the marginal likelihood, or evidence, for model X. The
Bayes Factor can then be calculated by
BXY (s) =
pX(s)
pY (s)
. (11)
The Bayes Factor has been described as the Holy Grail
of model selection: It is a powerful entity that is very
difficult to find. The quantity BXY can be thought of as
the odds ratio for a preference of model X over model Y .
Apart from carefully concocted toy problems, direct cal-
culation of the evidence, and therefore BXY , is imprac-
tical. To determine BXY the integral required to com-
pute pX can not generally be solved analytically and for
high dimension models Monte-Carlo integration proves
BXY 2 logBXY Evidence for model X
< 1 < 0 Negative (supports model Y )
1 to 3 0 to 2 Not worth more than a bare mention
3 to 12 2 to 5 Positive
12 to 150 5 to 10 Strong
> 150 > 10 Very Strong
TABLE I: BXY ‘confidence’ levels taken from [1]
to be inefficient. To employ this powerful statistical tool
various estimates for the Bayes Factor have been devel-
oped that have different levels of accuracy and computa-
tional cost [1, 2]. We have chosen to focus on four such
methods: Reverse Jump Markov Chain Monte Carlo and
Savage-Dickie density ratios, which directly estimate the
Bayes factor, and the Schwarz-Bayes Information Cri-
terion (BIC) and Laplace approximations of the model
evidence.
A. RJMCMC
Reverse JumpMarkov Chain Monte Carlo (RJMCMC)
algorithms are a class of MCMC algorithms which admit
“trans-dimensional” moves between models of different
dimension [3, 15, 16]. For the trans-dimensional imple-
mentation applicable to the LISA data analysis problem
the choice of model parameters becomes one of the search
parameters. The algorithm proposes parameter ‘birth’ or
‘death’ moves (proposing to include or discard the ‘extra’
parameter(s)) while holding all other parameters fixed.
The priors in the RJMCMC Hastings ratio
α = min
p(~λY )p(s|~λY )g(~uY )
p(~λX)p(s|~λX)g(~uX)
automatically penalizes the posterior density of the
higher dimensional model, which compensate for its
generically higher likelihood, serving as a built in ‘Occam
Factor.’ The g(~u) which appears in (12) is the distribu-
tion from which the random numbers ~u are drawn and
|J| is the Jacobian
|J| ≡
∂(~λY , ~uY )
∂(~λX , ~uX)
The chain will tend to spend more iterations using the
model most appropriately describing the data, making
the decision of which model to favor a trivial one. To
quantitatively determine the Bayes Factor one simply
computes the ratio of the iterations spent within each
model.
BXY ≃
# of iterations in model X
# of iterations in model Y
Like the simpler MCMC methods, the RJMCMC is guar-
anteed to converge on the correct value of BXY making
 0  10000  20000  30000  40000  50000
iterations
FIG. 1: 50 000 iteration segment of an RJMCMC chain mov-
ing between models with and without frequency evolution.
This particular run was for a source with q = 1 and SNR=10
and yielded BXY ∼ 1.
it the ‘gold standard’ of Bayes Factor estimation. And,
like regular MCMCs, the convergence can be very slow, so
that in practice the Bayes Factor estimates can be inaccu-
rate. This is especially true when the trans-dimensional
moves involve many parameters, such as the 7 or 8 di-
mensional jumps that are required to transition between
models with differing numbers of galactic binaries.
Figure 1 shows the output of a RJMCMC search of a
simulated LISA data stream containing the signal from a
galactic binary with q = ḟT 2obs = 1 and and observation
time of Tobs = 2 years. The chain moved freely between
the 7-dimensional model with no frequency evolution and
the 8-dimensional model which included the frequency
evolution.
B. Laplace Approximation
A common approach to model selection is to approx-
imate the model evidence directly. Working under the
assumption that the PDF is Gaussian, the integral in
equation (10) can be estimated by use of the Laplace
approximation. This is accomplished by comparing the
volume of the models parameter space V to that of the
parameter uncertainty ellipsoid ∆V
pX(s) ≃ p(~λMAP, X |s)
. (15)
The uncertainty ellipsoid can be determined by calculat-
ing the determinant of the Hessian H of partial deriva-
tives of the posterior with respect to the model parameter
evaluated at the MAP value for the parameters.
pX(s) ≃ p(~λMAP, X |s)
(2π)D/2√
The Fisher Information Matrix (FIM) Γ with compo-
nents
Γij ≡ (h,i |h,j ) where h,i≡
can be used as a quadratic approximation to H yielding
pX(s) ≃ p(~λMAP, X |s)
(2π)D/2√
We will refer to this estimate of the evidence as the
Laplace-Fisher (LF) approximation. The LF approxima-
tion breaks down if the priors have large gradients in the
vicinity of the MAP parameter estimates. The FIM esti-
mates can also be poor if some of the source parameters
are highly correlated, or if the quadratic approximation
fails. In addition, the FIM approximation gets progres-
sively worse as the SNR of the source decreases.
A more accurate (though more costly) method for esti-
mating the evidence is the Laplace-Metropolis (LM) ap-
proximation which employs the PDF as mapped out by
the MCMC exploration of the likelihood surface to es-
timate H [16]. This can be accomplished by fitting a
minimum volume ellipsoid (MVE) to the D-dimensional
posterior distribution function. The principle axes of the
MVE lie in eigen-directions of the distribution which gen-
erally do not lie along the parameter directions. Here we
employ the MVE.jar package which utilizes a genetic al-
gorithm to determine the MVE of the distribution and
returns the covariance matrix of the PDF [17]. The de-
terminant of the covariance matrix can then be used to
determine the evidence via
pX(s) ≃ p(~λMAP, X |s)(2π)D/2
detC. (19)
In the MCMC literature the LM approximation is gen-
erally considered to be second only to the RJMCMC
method for estimating Bayes Factors.
C. Savage Dickie Density Ratio
Both RJMCMC and LM require exploration of the pos-
terior for each model under consideration. The Savage-
Dickie (SD) approximation estimates the Bayes Factor
directly while only requiring exploration of the highest di-
mensional space [2, 18]. This approximation requires that
two conditions are met: Model X must be nested within
Model Y (adding and subtracting parameters clearly sat-
isfies this condition) and the priors for any given model
must be separable (i.e. p(~λ) = p(λ1)×p(λ2)×. . .×p(λD))
which is, to a good approximation, satisfied in our exam-
ple. The Bayes Factor BXY is then calculated by com-
paring the weight of the marginalized posterior to the
weight of the prior distribution for the ‘extra’ parameter
at the default, lower-dimensional, value for the parameter
in question.
BXY (s) ≃
p(λ0|s)
p(λ0)
It is interesting to note that if the above conditions are
precisely satisfied it can then be shown that this is an
exact calculation of BXY (assuming sufficient sampling
of the PDF), as opposed to an approximation.
D. Schwarz-Bayes Information Criterion
All of the approximations discussed so far depend on
the supplied priors p(~λ). The Schwarz-Bayes Informa-
tion Criterion (BIC) method is an approximation to the
model evidence which makes its own assumptions about
the priors - namely that they take the form of a multi-
variate Gaussian with covariance matrix derived from the
Hessian H [16, 19]. The BIC estimate for the evidence is
ln pX(s) ≃ ln p(~λMAP, X |s)−
lnNeff (21)
where DX is the dimension of model X and Neff is the
effective number of samples in the data. For our tests we
defined Neff to be the number of data points required to
return a (power) signal-to-noise ratio of SNR2−D, where
SNR is the signal-to-noise one gets by summing over the
entire LISA band. This choice was motivated by the
fact that the variance in SNR2 is equal to D2, so Neff
accounts for the data points that carry significant weight
in the model fit. The BIC estimate has the advantage of
being very easy to calculate, but is generally considered
less reliable than the other techniques we are using.
III. CASE STUDY
To compare the various approximations to the Bayes
Factor we simulated a ‘typical’ galactic binary. The in-
jected parameters for our test source can be found in ta-
ble II. Since ḟ ∼ f11/3, higher frequency sources are more
likely to have a measurable ḟ . On the other hand, the
number of binaries per frequency bin falls of as ∼ f−11/3,
so high frequency systems are fairly rare. As a compro-
mise, we selected a system with a GW frequency of 5
mHz. To describe the frequency evolution we introduced
the dimensionless parameter
q ≡ ḟT 2obs, (22)
which measures the change in the Barycenter GW fre-
quency in units of 1/Tobs frequency bins. For q ≫ 1 it
is reasonable to believe that a search algorithm will have
no difficulty detecting the frequency shift. Likewise, for
q ≪ 1 it is unlikely that the frequency evolution can be
detected (at least for sources with modest SNR). There-
fore we have selected q ∼ 1 to test the model selection
techniques. Achieving q = 1 for typical galactic binaries
at 5 mHz requires observation times of approximately
two years. A range of SNRs were explored by varying
the distance to the source.
TABLE II: Source parameters
f (mHz) cos θ φ (deg) ψ (deg) cos ι ϕ0 (deg) q Tobs (yr)
5.0 1.0 266.0 51.25 0.17 204.94 1 2
We can rapidly calculate accurate waveform templates
using the fast-slow decomposition described in the Ap-
pendix. Our waveform algorithm has been used in the
second round of Mock LISA Data Challenges [20] to sim-
ulate the response to a galaxy containing some 26 million
sources. The simulation takes just a few hours to run on
a single 2 GHz processor.
We simulated a 1024 frequency bin snippet of LISA
data around 5 mHz that included the injected signal
and stationary Gaussian instrument noise. The Markov
chains were initialized at the injected source parameters
as the focus of this study is the statistical character of
the detection, and not the initial detection (a highly ef-
ficient solution to the detection problem is described in
Ref. [9]). We used uniform priors for all of the parame-
ters, with the angular parameters taking their standard
ranges. We took the frequency to lie somewhere within
the frequency snippet, and lnA to be uniform across
the range 1
ln(Sn/(2T )) and
ln(1000Sn/(2T ), which
roughly corresponds to signal SNRs in the range 1 to
1000. We took the frequency evolution parameter q to
be uniformly distributed between -3 and 3 and adopted
q = 0 as the default value when operating under the
7-dimensional model. In reality, astrophysical considera-
tions yield a very strong prior for q (see Section V) that
will significantly impact model selection. We decided to
use a simple uniform prior to compare the various ap-
proximations to the Bayes Factor, before moving on to
consider the effects of the astrophysical prior in Section
The choice of proposal distribution q(~x|~y) from which
to draw new parameter values has no effect on the asymp-
totic form of the recovered PDFs, but the choice is cru-
cially important in determining the rate of convergence to
the stationary distribution. We took q(~x|~y) to be a mul-
tivariate Gaussian with covariance matrix given by the
inverse of the FIM. In addition to the source parameters
we included two additional parameters, kA and kE , that
describe the noise levels in the A and E data channels:
SAn (f) = kASn(f)
SEn (f) = kESn(f). (23)
In a given realization of the instrument noise kA and
kE will differ from unity by some random amount δ. The
quantity δ will have a Gaussian distribution with variance
σ2 = 1/N , whereN is the number of frequency bins being
analyzed. The likelihood p(s|~λ) can then be written as
p(s|~λ) = C′ exp
−N ln (kAkE)
where the constant C′ is independent of the signal param-
eters ~λ and the noise parameters kA and kE . We explored
the noise level estimation capabilities of our MCMC al-
gorithm by starting kA and kE far from unity. As can be
seen in Figure 2 the chain quickly identified the correct
noise level.
 0  500  1000  1500  2000  2500
iterations
FIG. 2: Demonstration of the MCMC algorithm’s rapid de-
termination of the injected noise level. The parameters kA
and kE were initialized at 10 and 0.1 respectively.
IV. COMPARISON OF TECHNIQUES
We compared the Bayes Factor estimates obtained us-
ing the various methods in two ways. First, we fixed the
frequency derivative of the source at q = 1 and varied
the SNR between 5 and 20 in unit increments. Second,
we fixed the signal to noise ratio at SNR = 12 and varied
the frequency derivative of the source.
Favor 7D model
Not worth more than a bare mention
Positive
Strong
Very StrongV
RJMCMC
 0.001
 0.01
 1000
 10000
 6  8  10  12  14  16  18  20
FIG. 3: Plot of the Bayes Factor estimates as a function of
SNR for each of the approximation schemes described in the
text.
The results of the first test are shown in Figure 3 We
see that all five methods agree very well for SNR > 7.
As expected, the Laplace-Metropolis and Savage-Dicke
methods provide the best approximation to the “Gold
Standard” RJMCMC estimate, showing good agreement
all the way down to SNR = 5. Most importantly, all five
methods agree on when the 8-dimensional source model is
favored over the 7-dimensional model, placing the tran-
sition point at SNR ≃ 12.2. To get a rough estimate
for the numerical error in the various Bayes Factor es-
timates we repeated the SNR= 15 case 10 times using
different random number seeds. We found that the nu-
merical error was enough to account for any quantitative
differences between the estimates returned by the various
approaches.
It is interesting to compare the Bayesian model se-
lection results to the frequentist “3-σ” rule for positive
detection:
|q̄| > 3σq, (25)
where q̄ is the MAP estimate for the frequency change
and σq is the standard deviation in q as determined by
the FIM. For the source under consideration we found
the “3-σ” rule to require SNR ≃ 13 for a detection, in
good agreement with the Bayesian analysis. This lends
support to Seto’s [21] earlier FIM based study of the de-
tectability of the frequency evolution of galactic bina-
ries, but we should caution that the literature is replete
with examples where the “3-σ” rule yields results in dis-
agreement with Bayesian model selection and common
sense [22].
Favor 7D model
Not worth more than a bare mention
Positive
Strong
 0.7  0.8  0.9  1  1.1  1.2  1.3
RJMCMC
FIG. 4: Plot of the Bayes Factor estimates as a function of q
for each of the approximation schemes described in the text.
The signal to noise ratio was held fixed at SNR = 12.
The results of the second test are displayed in Figure 4
In this case all five methods produced results that agree
to within numerical error.
While the results shown here are for a particular choice
of source parameters, we found similar results for other
sets of source parameters. In general all five methods
for estimating the Bayes Factor gave consistent results
for signals with SNR > 7. One exception to this general
trend were sources with inclinations close to zero, as then
the PDFs tend to be highly non-gaussian. The Laplace-
Metropolis and Laplace-Fisher approximations suffered
the most in those cases.
V. ASTROPHYSICAL PRIORS
Astrophysical considerations lead to very strong pri-
ors for the frequency evolution of galactic binaries. The
detached systems, which are expected to account for
the majority of LISA sources, will evolve under gravita-
tional radiation reaction in accord with the leading order
quadrapole formula:
3(8π)8/3
f11/3M5/3 , (26)
where M is the chirp mass. Contact binaries undergoing
stable mass transfer from the lighter to the heavier com-
ponent are driven to longer orbital periods by angular
momentum conservation. The competition between the
effects of mass transfer and gravitational wave emission
lead to a formula for ḟ with the same frequency and mass
scaling as (26), but with the opposite sign and a slightly
lower magnitude [23].
Population synthesis models, calibrated against obser-
vational data, yield predictions for the distribution of
chirp masses M as a function of orbital frequency. These
distributions can be converted into priors on q. In con-
structing such priors one should also fold in observational
selection effects, which will favor systems with larger
chirp mass (the GW amplitude scales as M5/6). To get
some sense of how such priors will affect the model selec-
tion we took the chirp mass distribution for detached sys-
tems at f ∼ 5 mHz from the population synthesis model
described in Ref. [24], (kindly provided to us by Gijs Nele-
mans), and used (26) to construct the prior on q shown
in Figures 5 and 6 (observation selection effects were ig-
nored). The prior has been modified slightly to give a
small but no-vanishing weight to sources with q = 0. The
astrophysically motivated prior has a very sharp peak at
q = 0.64, and we use this value when fixing the frequency
derivative for the 7-dimensional model.
To explore the impact on model selection when such a
strong prior has been adopted we simulated a source with
q = 1 and varied the SNR. The RJMCMC algorithm was
applied using chains of length 107 in conjunction with
a fixed 8-dimensional MCMC (also allowed to run for
107 iterations) in order to compare the RJMCMC results
with the Savage-Dickie density ratio.
The results of this first exploration are shown in Fig-
ure 5. We found that for SNR < 15 the marginalized
PDF very closely resembled the prior distribution. This
demonstrates that the information content of the data
is insufficient to change our prior belief about the value
of the frequency derivative. As the SNR increased, how-
ever, the PDF began to move away from the prior until
we reached SNR=30 when the astrophysical prior had a
 0.005
 0.01
 0.015
 0.02
 0.025
 0.03
.5  1
.5  2
 0.005
 0.01
 0.015
 0.02
 0.025
 0.03
.5  1
.5  2
 0.005
 0.01
 0.015
 0.02
 0.025
 0.03
.5  1
.5  2
 0.005
 0.01
 0.015
 0.02
 0.025
 0.03
.5  1
.5  2
 0.005
 0.01
 0.015
 0.02
 0.025
 0.03
.5  1
.5  2
 0.005
 0.01
 0.015
 0.02
 0.025
 0.03
.5  1
.5  2
SNR=5 SNR=10 SNR=15
SNR=20 SNR=25 SNR=30
q q q
FIG. 5: Comparison between astrophysically motivated prior
distribution of q for f = 5 mHz and Tobs = 2 years (dashed,
blue) to marginalized PDF (solid, red) for sources injected
with q = 1 and SNRs varying from 5 to 30.
SNR BXY (SD) BXY (RJMCMC)
5 0.926 1.015
10 0.977 0.996
15 0.749 0.742
20 0.427 0.427
25 0.176 0.177
30 0.060 0.056
TABLE III: Savage-Dickie density ratio estimates of BXY for
sources with q = 1 and SNRs varying from 5 to 30. Compar-
isons with RJMCMC explorations of the same data set show
excellent agreement between the two methods.
negligible effect on the shape of the posterior, signaling
confidence in the quoted measurement of q. This qualita-
tive assessment of model preference is strongly supported
by the Bayes factor estimation made by the RJMCMC
algorithm as can be seen in Table III. It should also be
noted that the excellent agreement between the RJM-
CMC and S-D estimates for Bayes factor BXY . Both
methods indicate that for the chosen value of q = 1,
the signal-to-noise needs to exceed SNR ∼ 25 for the
8-dimensional model to be favored. This is in contrast
to the case discussed earlier where a uniform prior was
adopted for the frequency derivative, and the model se-
lection methods began showing a preference for the 8-
dimensional model around SNR=12.
Figure 6 shows the impact of the astrophysically moti-
vated prior when the SNR was held at 15 and four differ-
ent injected values for q were adopted, corresponding to
the full width at half maximum (FWHM) and full width
at quarter maximum (FWQM) of the prior distribution.
The Bayes factors listed in Table IV indicate that for
modestly loud sources with SNR=15 the model selection
techniques do not favor updating our estimate of the fre-
 0.005
 0.01
 0.015
 0.02
 0.025
 0.03
.5  1
.5  2
 0.005
 0.01
 0.015
 0.02
 0.025
 0.03
.5  1
.5  2
 0.005
 0.01
 0.015
 0.02
 0.025
 0.03
.5  1
.5  2
 0.005
 0.01
 0.015
 0.02
 0.025
 0.03
.5  2
q=0.35
q=0.83
q=0.48
q=1.15
FIG. 6: Marginalized PDF (solid, red) for fixed SNR=15 in-
jected sources with q corresponding to FWHM and FWQM
of the astrophysical prior (dashed, blue)
q BXY (SD) BXY (RJMCMC)
0.35 1.412 1.414
0.48 1.381 1.388
0.83 1.059 1.052
1.15 0.432 0.428
TABLE IV: Savage-Dickie and RJMCMC density ratio esti-
mates of BXY for sources with SNR=15 and q at FWHM and
FWQM of astrophysical prior
quency derivative until the frequency derivative exceeds
q = 1.2.
VI. DISCUSSION
We have found that the several common methods for
estimating Bayes Factors give good agreement when ap-
plied to the the model selection problem of deciding when
the data from the LISA observatory can be used to detect
the orbital evolution of a galactic binary. The methods
studied require varying degrees of effort to implement and
calculate, and although found to be accurate in this test
case, it is clear that some of these methods would be in-
appropriate approximations for more physically relevant
examples.
If a RJMCMC algorithm is used as the sole model
selection technique, the resistance of the algorithm
to change dimension, especially when making multi-
dimensional jumps, can result in invalid model selection
unless the chains are run for a very large numbers of
steps. In the examples we studied the transdimensional
jumps only had to span one dimension, and our basic
RJMCMC algorithm performed well. However, a more
sophisticated implementation, using e.g. rejection sam-
pling or coupled chains, will be required to select the
number of sources, as this requires jumps that span seven
or more dimensions.
The Laplace-Metropolis method for approximating the
model evidence is more robust than the commonly used
Fisher Information Matrix approximation of the Hessian
of the PDF. Implementing an LM evidence estimation is
a somewhat costly because of the need to fit the posterior
to a minimum volume ellipsoid.
The Savage-Dickie approximation is more economical
than the RJMCMC or LM methods, but is limited by the
requirement that the competing models must be nested.
The Bayes Information Criterion approximation to the
evidence is by far the cheapest to implement, and is able
to produce reliable results when the SNR is high. It has
therefore shown the most promise as an ‘on the fly’ model
determination scheme. More thorough (and therefore
more costly) methods such as RJMCMC and LM could
then be used to refine the conclusions initially made by
the BIC.
Our investigation using a strong astrophysical prior
indicated that the gravitational wave signals will need
to have high signal-to-noise (SNR > 25), or moderate
signal-to-noise (SNR > 15) and frequency derivatives far
from the peak of the astrophysical distribution, in order
to update our prior belief in the value of the frequency
derivative. In other words, the frequency derivative will
only been needed as a search parameter for a small num-
ber of bright high frequency sources.
Acknowledgments
This work was supported by NASA Grant
NNG05GI69G. We are most grateful to Gijs Nele-
mans for providing us with data from his population
synthesis studies.
Appendix A
To leading order in the eccentricity, e, the Cartesian
coordinates of the ith LISA spacecraft are given by [25]
xi(t) = R cos(α) +
cos(2α− βi)− 3 cos(β)
yi(t) = R sin(α) +
sin(2α− βi)− 3 sin(β)
zi(t) = −
3eR cos(α− βi) . (27)
In the above R = 1 AU, is the radial distance of the
guiding center, α = 2πfmt+ κ is the orbital phase of the
guiding center, and βi = 2π(i − 1)/3 + λ (i = 1, 2, 3) is
the relative phase of the spacecraft within the constel-
lation. The parameters κ and λ give the initial ecliptic
longitude and orientation of the constellation. The dis-
tance between the spacecraft is L = 2
3eR. Setting
e = 0.00985 yields L = 5× 109 m.
An arbitrary gravitational wave traveling in the k̂ di-
rection can be written as the linear sum of two indepen-
dent polarization states
h(ξ) = h+(ξ)ε
+ + h×(ξ)ε
× (28)
where the wave variable ξ = t − k̂ · x gives the surfaces
of constant phase. The polarization tensors can be ex-
panded in terms of the basis tensors e+ and e× as
ε+ = cos(2ψ)e+ − sin(2ψ)e×
ε× = sin(2ψ)e+ + cos(2ψ)e× , (29)
where ψ is the polarization angle and
+ = û⊗ û− v̂ ⊗ v̂
× = û⊗ v̂ + v̂ ⊗ û . (30)
The vectors (û, v̂, k̂) form an orthonormal triad which
may be expressed as a function of the source location on
the celestial sphere
û = cos θ cosφ x̂+ cos θ sinφ ŷ − sin θ ẑ
v̂ = sinφ x̂− cosφ ŷ
k̂ = − sin θ cosφ x̂− sin θ sinφ ŷ − cos θ ẑ . (31)
For mildly chirping binary sources we have
h(ξ) = ℜ
+ + ei3π/2A×ε
eiΨ(ξ)
where
2M(πf)2/3
1 + cos2 ι
A× = −
4M(πf)2/3
cos ι . (33)
Here M is the chirp mass, DL is the luminosity dis-
tance and ι is the inclination of the binary to the line
of sight. Higher post-Newtonian corrections, eccentric-
ity of the orbit, and spin effects will introduce additional
harmonics. For chirping sources the adiabatic approxi-
mation requires that the frequency evolution ḟ occurs on
a timescale long compared to the light travel time in the
interferometer: f/ḟ ≪ L. The gravitational wave phase
can be approximated as
Ψ(ξ) = 2πf0ξ + πḟ0ξ
2 + ϕ0 , (34)
where ϕ0 is the initial phase. The instantaneous fre-
quency is given by 2πf = ∂tΨ:
f = (f0 + ḟ0ξ)(1 − k̂ · v) . (35)
The general expression for the path length variation
caused by a gravitational wave involves an integral in
ξ from ξi to ξf . Writing ξ = ξi + δξ we have
Ψ(ξ) ≃ 2π(f0 + ḟ0ξi)δξ + const . (36)
Thus, we can treat the wave as having fixed frequency
f0 + ḟ0ξi for the purposes of the integration, and then
increment the frequency forward in time in the final ex-
pression [26]. The path length variation is then given
by [25, 26]
δℓij(ξ) = Lℜ
d(f, t, k̂) : h(ξ)
, (37)
where a : b = aijbij . The one-arm detector tensor is
given by
d(f, t, k̂) =
r̂ij(t)⊗ r̂ij(t)
T (f, t, k̂) , (38)
and the transfer function is
T (f, t, k̂) = sinc
1− k̂ · r̂ij(t)
× exp
1− k̂ · r̂ij(t)
, (39)
where f∗ = 1/(2πL) is the transfer frequency and f =
f0 + ḟ0ξ. The expression can be attacked in pieces. It is
useful to define the quantities
d+ij(t) ≡ (r̂ij(t)⊗ r̂ij(t)) : e
+ (40)
d×ij(t) ≡ (r̂ij(t)⊗ r̂ij(t)) : e
× . (41)
and yij(t) = δℓij(t)/(2L). Then
yij(t) = ℜ
yslowij (t)e
2πif0t
, (42)
where
yslowij (t) =
T (f, t, k̂)
d+ij(t)(A+(t) cos(2ψ)
+e3πi/2A×(t) sin(2ψ))
+d×ij(t)(e
3πi/2A×(t) cos(2ψ)
−A+(t) sin(2ψ))) e(πiḟ0ξ
2+iϕ0−2πif0k̂·x)
It is a simple exercise to derive explicit expressions for the
antenna functions and the transfer function appearing in
yslowij (t) using (27) and (31).
In the Fourier domain the response can be written as
yij(t) = ℜ
2πint/Tobs
e2πif0t
, (44)
where the coefficients an can be found by a numerical
FFT of the slow terms yslowij (t). Note that the sum over
n should extend over both negative and positive values.
The number of time samples needed in the FFT will de-
pend on f0 and ḟ0 and Tobs, but is less than 2
9 = 512
for any galactic sources we are likely to encounter when
Tobs ≤ 2yr. The bandwidth of a source can be estimated
B = 2 (4 + 2πf0R sin(θ)) fm + ḟ0Tobs . (45)
The number of samples should exceed 2BTobs. The
Fourier transform of the fast term can be done analyti-
cally:
e2πif0t =
2πimt/Tobs (46)
where
bm = Tobs sinc(xm)e
ixm (47)
xm = f0Tobs −m. (48)
The cardinal sine function in (46) ensures that the
Fourier components bm away from resonance, xm ≈ 0, are
quite small. It is only necessary to keep ∼ 100 → 1000
terms either side of p = [f0Tobs], depending on how bright
the source is, and how far f0Tobs is from an integer. We
now have
yij(t) = ℜ
2πijt/Tobs
, (49)
where
anbj−n . (50)
The final step is to ensure that our Fourier transform
yields a real yij(t). This is done by setting the final an-
swer for the Fourier coefficients equal to dj = (cj+c
−j)/2.
But since xm never hits resonance for positive j (we
are not interested in the negative frequency components
j < 0), we can neglect the second term and simply write
dj = cj/2.
Basically what we are doing is hetrodyning the signal
to the base frequency f0, then Fourier transforming the
slowly evolving hetrodyned signal numerically. We then
convolve these Fourier coefficients with the analytically
derived Fourier coefficients of the carrier wave.
The Michelson type TDI variables are given by
X(t) = y12(t− 3L)− y13(t− 3L) + y21(t− 2L)
−y31(t− 2L) + y13(t− L)− y12(t− L)
+y31(t)− y21(t), (51)
Y (t) = y23(t− 3L)− y21(t− 3L) + y32(t− 2L)
−y12(t− 2L) + y21(t− L)− y23(t− L)
+y12(t)− y32(t), (52)
Z(t) = y31(t− 3L)− y32(t− 3L) + y13(t− 2L)
−y23(t− 2L) + y32(t− L)− y31(t− L)
+y23(t)− y13(t). (53)
Note that in the Fourier domain
X(f) = ỹ12(f)e
−3if/f∗ − ỹ13(f)e−3if/f∗ + ỹ21(f)e−2if/f∗
−ỹ31(f)e−2if/f∗ + ỹ13(f)e−if/f∗ − ỹ12(f)e−if/f∗
+ỹ31(f)− ỹ21(f) . (54)
This saves us from having to interpolate in the time do-
main. We just combine phase shifted versions of our orig-
inal Fourier transforms.
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[27] While this count is only strictly correct for point-like
masses, frequency evolution due to tides and mass trans-
fer can also be described by the same two parameters for
the majority of sources in the LISA band.

  The analysis of gravitational wave data involves many model selection
problems. The most important example is the detection problem of selecting
between the data being consistent with instrument noise alone, or instrument
noise and a gravitational wave signal. The analysis of data from ground based
gravitational wave detectors is mostly conducted using classical statistics,
and methods such as the Neyman-Pearson criteria are used for model selection.
Future space based detectors, such as the \emph{Laser Interferometer Space
Antenna} (LISA), are expected to produced rich data streams containing the
signals from many millions of sources. Determining the number of sources that
are resolvable, and the most appropriate description of each source poses a
challenging model selection problem that may best be addressed in a Bayesian
framework. An important class of LISA sources are the millions of low-mass
binary systems within our own galaxy, tens of thousands of which will be
detectable. Not only are the number of sources unknown, but so are the number
of parameters required to model the waveforms. For example, a significant
subset of the resolvable galactic binaries will exhibit orbital frequency
evolution, while a smaller number will have measurable eccentricity. In the
Bayesian approach to model selection one needs to compute the Bayes factor
between competing models. Here we explore various methods for computing Bayes
factors in the context of determining which galactic binaries have measurable
frequency evolution. The methods explored include a Reverse Jump Markov Chain
Monte Carlo (RJMCMC) algorithm, Savage-Dickie density ratios, the Schwarz-Bayes
Information Criterion (BIC), and the Laplace approximation to the model
evidence. We find good agreement between all of the approaches.

Tests of Bayesian Model Selection Techniques for Gravitational Wave Astronomy
Neil J. Cornish and Tyson B. Littenberg
Department of Physics, Montana State University, Bozeman, MT 59717
The analysis of gravitational wave data involves many model selection problems. The most
important example is the detection problem of selecting between the data being consistent with
instrument noise alone, or instrument noise and a gravitational wave signal. The analysis of data
from ground based gravitational wave detectors is mostly conducted using classical statistics, and
methods such as the Neyman-Pearson criteria are used for model selection. Future space based
detectors, such as the Laser Interferometer Space Antenna (LISA), are expected to produced rich
data streams containing the signals from many millions of sources. Determining the number of
sources that are resolvable, and the most appropriate description of each source poses a challenging
model selection problem that may best be addressed in a Bayesian framework. An important class of
LISA sources are the millions of low-mass binary systems within our own galaxy, tens of thousands
of which will be detectable. Not only are the number of sources unknown, but so are the number
of parameters required to model the waveforms. For example, a significant subset of the resolvable
galactic binaries will exhibit orbital frequency evolution, while a smaller number will have measurable
eccentricity. In the Bayesian approach to model selection one needs to compute the Bayes factor
between competing models. Here we explore various methods for computing Bayes factors in the
context of determining which galactic binaries have measurable frequency evolution. The methods
explored include a Reverse Jump Markov Chain Monte Carlo (RJMCMC) algorithm, Savage-Dickie
density ratios, the Schwarz-Bayes Information Criterion (BIC), and the Laplace approximation to
the model evidence. We find good agreement between all of the approaches.
I. BACKGROUND
Bayesian statistical techniques are becoming increas-
ingly popular in gravitational wave data analysis, and
have shown great promise in tackling the various difficul-
ties of gravitational wave (GW) source extraction from
modeled data for the Laser Interferometer Space Antenna
(LISA). A powerful tool in the suite of Bayesian methods
is that of quantitative model selection [1, 2]. To under-
stand why this is a valuable feature consider a scenario
where one is attempting to fit data with two competing
models of differing dimension. In general, a higher di-
mensional model will produce a better fit to a given set
of data. This can be taken to the limit where there are as
many model parameters as there are data points allowing
one to perfectly match the data. The problem then is to
decide how many parameters are physically meaningful
and to select the model containing only those parameters.
In the context of GW detection these extra parameters
could be additional physical parameters used to model
the source or additional sources in the data. If a model
is over-parameterized it will over-fit the data and produce
spurious results.
Many of the model selection problems associated with
LISA astronomy involve nested models, where the sim-
pler model forms a subset of the more complicated model.
The problem of determining the number of resolvable
galactic binaries, and the problem of determining the
number of measurable source parameters, are both ex-
amples of nested model selection. One could argue that
the later is better described as “approximation selection”
since we are selecting between different parameterizations
of the full 17 dimensional physical model that describes
the signals from binary systems of point masses in general
relativity. However, many similar modeling problems in
astrophysics and cosmology [2], as well as in other fields
such as geophysics [3], are considered to be examples of
model selection, and we will adopt that viewpoint here.
The LISA observatory [4] is designed to explore the
low frequency portion of the gravitational wave spectrum
between ∼ 0.1 → 100 mHz. This frequency region will be
heavily populated by signals from galactic binary systems
composed of stellar mass compact objects (e.g. white
dwarfs and neutron stars), of which millions are theorized
to exist. Tens of thousands of these GW sources will
be resolvable by LISA and the remaining sources will
contribute to a confusion-limited background [5]. This
is expected to be the dominant source of low frequency
noise for LISA.
Detection and subsequent regression of the galactic
foreground is of vital importance in order to then pur-
sue dimmer sources that would otherwise be buried by
the foreground. Because of the great number of galac-
tic sources, and the ensuing overlap between individ-
ual sources, a one-by-one detection/regression is inaccu-
rate [6]. Therefore a global fit to all of the galactic sources
is required. Because of the uncertainty in the number of
resolvable sources one can not fix the model dimension a
priori which presents a crucial model selection problem.
Over-fitting the data will result in an inaccurate regres-
sion which would then remove power from other sources
in the data-stream, negatively impacting their detection
and characterization. The Reverse Jump Markov Chain
Monte Carlo approach to Bayesian model selection has
been used to determine the number of resolvable sources
in the context of a toy problem [7, 8] which shares some
of the features of the LISA foreground removal prob-
lem. Meanwhile the Laplace approximation to Bayesian
http://arxiv.org/abs/0704.1808v3
model selection has been employed to estimate the num-
ber of resolvable sources as part of a MCMC based al-
gorithm to extracting signals from simulated LISA data
streams [6, 9].
Another important problem for GW astronomy is the
determination of which parameters need to be included in
the description of the waveforms. For example, the GW
signal from a binary inspiral, as detected by LISA, may
involve as many as 17 parameters. However, for massive
black hole binaries of comparable mass we expect the ec-
centricity to be negligible, reducing the model dimension
to 15, while for extreme mass ratio systems we expect
the spin of the smaller body to have little impact on
the waveforms, reducing the model dimension to 14. In
many cases the inspiral signals may be described by even
fewer parameters. For low mass galactic binaries spin
effects will be negligible (removing six parameters), and
various astrophysical processes will have circularized the
orbits of the majority of systems (removing two param-
eters). Of the remaining nine parameters, two describe
the frequency evolution - e.g. the first and second time
derivatives of the GW frequency, which may or may not
be detectable [27].
Here we investigate the application of Bayesian model
selection to LISA data analysis in the context of deter-
mining the conditions under which the first time deriva-
tive of the GW frequency, ḟ , can be inferred from the
data. We parameterize the signals using the eight pa-
rameters
~λ→ (A, f, θ, φ, ψ, ι, ϕ0, ḟ) (1)
where A is the amplitude, f is the initial gravitational
wave frequency, θ and φ are the ecliptic co-latitude and
longitude, ψ is the polarization angle, ι is the orbital
inclination of the binary and ϕ0 is the GW phase. The
parameters f , ḟ and ϕ0 are evaluated at some fiducial
time (e.g. at the time the first data is taken). For our
analysis only a single source is injected into the simulated
data streams. In the frequency domain the output s(f)
in channel α can be written as
s̃α(f) = h̃α(~λ) + ñα(f) (2)
where h̃α(~λ) is the response of the detector to the incident
GW and ñα(f) is the instrument noise. For our work we
will assume stationary Gaussian instrument noise with no
contribution from a confusion background. In our anal-
ysis we use the noise orthogonal A,E, T data streams,
which can be constructed from linear combinations of
the Michelson type X,Y, Z signals:
(2X − Y − Z) ,
(Z − Y ) ,
(X + Y + Z) . (3)
This set of A,E, T variables differ slightly from those
constructed from the Sagnac signals [10]. We do not use
the T channel in our analysis as it is insensitive to GWs
at the frequencies we are considering. The noise spectral
density in the A,E channels has the form
Sn(f) =
(1− cos(2f/f∗))
(2 + cos(f/f∗))Ss
+2(3 + 2 cos(f/f∗) + cos(2f/f∗))Sa
Hz−1 (4)
where f∗ = 1/2πL, and the acceleration noise Sa and
shot noise Ss are simulated at the levels
10−22
9× 10−30
(2πf)4L2
Hz−1 .
Here L is the LISA arm length (≈ 5× 109 m).
Of central importance to Bayesian analysis is the pos-
terior distribution function (PDF) of the model parame-
ters. The PDF p(~λ|s) describes the probability that the
source is described by parameters ~λ given the signal s.
According to Bayes’ Theorem,
p(~λ|s) = p(
~λ)p(s|~λ)
d~λ p(~λ)p(s|~λ)
where p(~λ) is the a priori, or prior, distribution of the
parameters ~λ and p(s|~λ) is the likelihood that we measure
the signal s if the source is described by the parameters ~λ.
We define the likelihood using the noise weighted inner
product
(A|B) ≡ 2
Ã∗α(f)B̃α(f) + Ãα(f)B̃
Sαn (f)
p(s|~λ) = C exp
s− h(~λ)
s− h(~λ)
where the normalization constant C depends on the
noise, but not the GW signal. One goal of the data anal-
ysis method is to find the parameters ~λ which maximizes
the posterior. Markov Chain Monte Carlo (MCMC)
methods are ideal for this type of application [11]. The
MCMC algorithm will simultaneously find the param-
eters which maximize the posterior and accurately map
out the PDF of the parameters. This is achieved through
the use of a Metropolis-Hastings [12, 13] exploration of
the parameter space. A brief description of this process
is as follows: The chain begins at some random position
~x in the parameter space and subsequent steps are made
by randomly proposing a new position in the parame-
ter space ~y. This new position is determined by drawing
from some proposal distribution q(~x|~y). The choice of
whether or not adopt the new position ~y is made by cal-
culating the Hastings ratio (transition probability)
α = min
p(~y)p(s|~y)q(~y|~x)
p(~x)p(s|~x)q(~x|~y)
and comparing α to a random number β taken from a
uniform draw in the interval [0:1]. If α exceeds β then
the chain adopts ~y as the new position. This process
is repeated until some convergence criterion is satisfied.
The MCMC differs from a Metropolis extremization by
forbidding proposal distributions that depend on the past
history of the chain. This ensures that the progress of the
chain is Markovian and therefore statistically unbiased.
Once the chain has stabilized in the neighborhood of the
best fit parameters all previous steps of the chain are
excluded from the statistical analysis (these early steps
are referred to as the “burn in” phase of the chain) and
henceforth the number of iterations the chain spends at
different parameter values can be used to infer the PDF.
The power of the MCMC is two-fold: Because the al-
gorithm has a finite probability of moving away from a
favorable location in the parameter space it can avoid
getting trapped by local features of the likelihood sur-
face. Meanwhile, the absence of any “memory” within
the chain of past parameter values allows the algorithm
to blindly, statistically, explore the region in the neigh-
borhood of the global maximum. It is then rigorously
proven that an MCMC will (eventually) perfectly map
out the PDF, completely removing the need for user in-
put to determine parameter uncertainties or thresholds.
The parameter vector that maximizes the posterior dis-
tribution is stored as the maximum a posteriori (MAP)
value and is considered to be the best estimate of the
source parameters. Note that because of the prior weight-
ing in the definition of the PDF this is not necessarily
the ~λ that yields the greatest likelihood. Upon obtaining
the MAP value for a particular model X the PDF, now
written as p( ~λ,X |s), can be employed to solve the model
selection problem.
II. BAYES FACTOR ESTIMATES
The Bayes Factor BXY [14] is a comparison of the ev-
idence for two competing models, X and Y , where
pX(s) =
d~λ p(~λ,X |s) (10)
is the marginal likelihood, or evidence, for model X. The
Bayes Factor can then be calculated by
BXY (s) =
pX(s)
pY (s)
. (11)
The Bayes Factor has been described as the Holy Grail
of model selection: It is a powerful entity that is very
difficult to find. The quantity BXY can be thought of as
the odds ratio for a preference of model X over model Y .
Apart from carefully concocted toy problems, direct cal-
culation of the evidence, and therefore BXY , is imprac-
tical. To determine BXY the integral required to com-
pute pX can not generally be solved analytically and for
high dimension models Monte-Carlo integration proves
BXY 2 logBXY Evidence for model X
< 1 < 0 Negative (supports model Y )
1 to 3 0 to 2 Not worth more than a bare mention
3 to 12 2 to 5 Positive
12 to 150 5 to 10 Strong
> 150 > 10 Very Strong
TABLE I: BXY ‘confidence’ levels taken from [1]
to be inefficient. To employ this powerful statistical tool
various estimates for the Bayes Factor have been devel-
oped that have different levels of accuracy and computa-
tional cost [1, 2]. We have chosen to focus on four such
methods: Reverse Jump Markov Chain Monte Carlo and
Savage-Dickie density ratios, which directly estimate the
Bayes factor, and the Schwarz-Bayes Information Cri-
terion (BIC) and Laplace approximations of the model
evidence.
A. RJMCMC
Reverse JumpMarkov Chain Monte Carlo (RJMCMC)
algorithms are a class of MCMC algorithms which admit
“trans-dimensional” moves between models of different
dimension [3, 15, 16]. For the trans-dimensional imple-
mentation applicable to the LISA data analysis problem
the choice of model parameters becomes one of the search
parameters. The algorithm proposes parameter ‘birth’ or
‘death’ moves (proposing to include or discard the ‘extra’
parameter(s)) while holding all other parameters fixed.
The priors in the RJMCMC Hastings ratio
α = min
p(~λY )p(s|~λY )g(~uY )
p(~λX)p(s|~λX)g(~uX)
automatically penalizes the posterior density of the
higher dimensional model, which compensate for its
generically higher likelihood, serving as a built in ‘Occam
Factor.’ The g(~u) which appears in (12) is the distribu-
tion from which the random numbers ~u are drawn and
|J| is the Jacobian
|J| ≡
∂(~λY , ~uY )
∂(~λX , ~uX)
The chain will tend to spend more iterations using the
model most appropriately describing the data, making
the decision of which model to favor a trivial one. To
quantitatively determine the Bayes Factor one simply
computes the ratio of the iterations spent within each
model.
BXY ≃
# of iterations in model X
# of iterations in model Y
Like the simpler MCMC methods, the RJMCMC is guar-
anteed to converge on the correct value of BXY making
 0  10000  20000  30000  40000  50000
iterations
FIG. 1: 50 000 iteration segment of an RJMCMC chain mov-
ing between models with and without frequency evolution.
This particular run was for a source with q = 1 and SNR=10
and yielded BXY ∼ 1.
it the ‘gold standard’ of Bayes Factor estimation. And,
like regular MCMCs, the convergence can be very slow, so
that in practice the Bayes Factor estimates can be inaccu-
rate. This is especially true when the trans-dimensional
moves involve many parameters, such as the 7 or 8 di-
mensional jumps that are required to transition between
models with differing numbers of galactic binaries.
Figure 1 shows the output of a RJMCMC search of a
simulated LISA data stream containing the signal from a
galactic binary with q = ḟT 2obs = 1 and and observation
time of Tobs = 2 years. The chain moved freely between
the 7-dimensional model with no frequency evolution and
the 8-dimensional model which included the frequency
evolution.
B. Laplace Approximation
A common approach to model selection is to approx-
imate the model evidence directly. Working under the
assumption that the PDF is Gaussian, the integral in
equation (10) can be estimated by use of the Laplace
approximation. This is accomplished by comparing the
volume of the models parameter space V to that of the
parameter uncertainty ellipsoid ∆V
pX(s) ≃ p(~λMAP, X |s)
. (15)
The uncertainty ellipsoid can be determined by calculat-
ing the determinant of the Hessian H of partial deriva-
tives of the posterior with respect to the model parameter
evaluated at the MAP value for the parameters.
pX(s) ≃ p(~λMAP, X |s)
(2π)D/2√
The Fisher Information Matrix (FIM) Γ with compo-
nents
Γij ≡ (h,i |h,j ) where h,i≡
can be used as a quadratic approximation to H yielding
pX(s) ≃ p(~λMAP, X |s)
(2π)D/2√
We will refer to this estimate of the evidence as the
Laplace-Fisher (LF) approximation. The LF approxima-
tion breaks down if the priors have large gradients in the
vicinity of the MAP parameter estimates. The FIM esti-
mates can also be poor if some of the source parameters
are highly correlated, or if the quadratic approximation
fails. In addition, the FIM approximation gets progres-
sively worse as the SNR of the source decreases.
A more accurate (though more costly) method for esti-
mating the evidence is the Laplace-Metropolis (LM) ap-
proximation which employs the PDF as mapped out by
the MCMC exploration of the likelihood surface to es-
timate H [16]. This can be accomplished by fitting a
minimum volume ellipsoid (MVE) to the D-dimensional
posterior distribution function. The principle axes of the
MVE lie in eigen-directions of the distribution which gen-
erally do not lie along the parameter directions. Here we
employ the MVE.jar package which utilizes a genetic al-
gorithm to determine the MVE of the distribution and
returns the covariance matrix of the PDF [17]. The de-
terminant of the covariance matrix can then be used to
determine the evidence via
pX(s) ≃ p(~λMAP, X |s)(2π)D/2
detC. (19)
In the MCMC literature the LM approximation is gen-
erally considered to be second only to the RJMCMC
method for estimating Bayes Factors.
C. Savage Dickie Density Ratio
Both RJMCMC and LM require exploration of the pos-
terior for each model under consideration. The Savage-
Dickie (SD) approximation estimates the Bayes Factor
directly while only requiring exploration of the highest di-
mensional space [2, 18]. This approximation requires that
two conditions are met: Model X must be nested within
Model Y (adding and subtracting parameters clearly sat-
isfies this condition) and the priors for any given model
must be separable (i.e. p(~λ) = p(λ1)×p(λ2)×. . .×p(λD))
which is, to a good approximation, satisfied in our exam-
ple. The Bayes Factor BXY is then calculated by com-
paring the weight of the marginalized posterior to the
weight of the prior distribution for the ‘extra’ parameter
at the default, lower-dimensional, value for the parameter
in question.
BXY (s) ≃
p(λ0|s)
p(λ0)
It is interesting to note that if the above conditions are
precisely satisfied it can then be shown that this is an
exact calculation of BXY (assuming sufficient sampling
of the PDF), as opposed to an approximation.
D. Schwarz-Bayes Information Criterion
All of the approximations discussed so far depend on
the supplied priors p(~λ). The Schwarz-Bayes Informa-
tion Criterion (BIC) method is an approximation to the
model evidence which makes its own assumptions about
the priors - namely that they take the form of a multi-
variate Gaussian with covariance matrix derived from the
Hessian H [16, 19]. The BIC estimate for the evidence is
ln pX(s) ≃ ln p(~λMAP, X |s)−
lnNeff (21)
where DX is the dimension of model X and Neff is the
effective number of samples in the data. For our tests we
defined Neff to be the number of data points required to
return a (power) signal-to-noise ratio of SNR2−D, where
SNR is the signal-to-noise one gets by summing over the
entire LISA band. This choice was motivated by the
fact that the variance in SNR2 is equal to D2, so Neff
accounts for the data points that carry significant weight
in the model fit. The BIC estimate has the advantage of
being very easy to calculate, but is generally considered
less reliable than the other techniques we are using.
III. CASE STUDY
To compare the various approximations to the Bayes
Factor we simulated a ‘typical’ galactic binary. The in-
jected parameters for our test source can be found in ta-
ble II. Since ḟ ∼ f11/3, higher frequency sources are more
likely to have a measurable ḟ . On the other hand, the
number of binaries per frequency bin falls of as ∼ f−11/3,
so high frequency systems are fairly rare. As a compro-
mise, we selected a system with a GW frequency of 5
mHz. To describe the frequency evolution we introduced
the dimensionless parameter
q ≡ ḟT 2obs, (22)
which measures the change in the Barycenter GW fre-
quency in units of 1/Tobs frequency bins. For q ≫ 1 it
is reasonable to believe that a search algorithm will have
no difficulty detecting the frequency shift. Likewise, for
q ≪ 1 it is unlikely that the frequency evolution can be
detected (at least for sources with modest SNR). There-
fore we have selected q ∼ 1 to test the model selection
techniques. Achieving q = 1 for typical galactic binaries
at 5 mHz requires observation times of approximately
two years. A range of SNRs were explored by varying
the distance to the source.
TABLE II: Source parameters
f (mHz) cos θ φ (deg) ψ (deg) cos ι ϕ0 (deg) q Tobs (yr)
5.0 1.0 266.0 51.25 0.17 204.94 1 2
We can rapidly calculate accurate waveform templates
using the fast-slow decomposition described in the Ap-
pendix. Our waveform algorithm has been used in the
second round of Mock LISA Data Challenges [20] to sim-
ulate the response to a galaxy containing some 26 million
sources. The simulation takes just a few hours to run on
a single 2 GHz processor.
We simulated a 1024 frequency bin snippet of LISA
data around 5 mHz that included the injected signal
and stationary Gaussian instrument noise. The Markov
chains were initialized at the injected source parameters
as the focus of this study is the statistical character of
the detection, and not the initial detection (a highly ef-
ficient solution to the detection problem is described in
Ref. [9]). We used uniform priors for all of the parame-
ters, with the angular parameters taking their standard
ranges. We took the frequency to lie somewhere within
the frequency snippet, and lnA to be uniform across
the range 1
ln(Sn/(2T )) and
ln(1000Sn/(2T ), which
roughly corresponds to signal SNRs in the range 1 to
1000. We took the frequency evolution parameter q to
be uniformly distributed between -3 and 3 and adopted
q = 0 as the default value when operating under the
7-dimensional model. In reality, astrophysical considera-
tions yield a very strong prior for q (see Section V) that
will significantly impact model selection. We decided to
use a simple uniform prior to compare the various ap-
proximations to the Bayes Factor, before moving on to
consider the effects of the astrophysical prior in Section
The choice of proposal distribution q(~x|~y) from which
to draw new parameter values has no effect on the asymp-
totic form of the recovered PDFs, but the choice is cru-
cially important in determining the rate of convergence to
the stationary distribution. We took q(~x|~y) to be a mul-
tivariate Gaussian with covariance matrix given by the
inverse of the FIM. In addition to the source parameters
we included two additional parameters, kA and kE , that
describe the noise levels in the A and E data channels:
SAn (f) = kASn(f)
SEn (f) = kESn(f). (23)
In a given realization of the instrument noise kA and
kE will differ from unity by some random amount δ. The
quantity δ will have a Gaussian distribution with variance
σ2 = 1/N , whereN is the number of frequency bins being
analyzed. The likelihood p(s|~λ) can then be written as
p(s|~λ) = C′ exp
−N ln (kAkE)
where the constant C′ is independent of the signal param-
eters ~λ and the noise parameters kA and kE . We explored
the noise level estimation capabilities of our MCMC al-
gorithm by starting kA and kE far from unity. As can be
seen in Figure 2 the chain quickly identified the correct
noise level.
 0  500  1000  1500  2000  2500
iterations
FIG. 2: Demonstration of the MCMC algorithm’s rapid de-
termination of the injected noise level. The parameters kA
and kE were initialized at 10 and 0.1 respectively.
IV. COMPARISON OF TECHNIQUES
We compared the Bayes Factor estimates obtained us-
ing the various methods in two ways. First, we fixed the
frequency derivative of the source at q = 1 and varied
the SNR between 5 and 20 in unit increments. Second,
we fixed the signal to noise ratio at SNR = 12 and varied
the frequency derivative of the source.
Favor 7D model
Not worth more than a bare mention
Positive
Strong
Very StrongV
RJMCMC
 0.001
 0.01
 1000
 10000
 6  8  10  12  14  16  18  20
FIG. 3: Plot of the Bayes Factor estimates as a function of
SNR for each of the approximation schemes described in the
text.
The results of the first test are shown in Figure 3 We
see that all five methods agree very well for SNR > 7.
As expected, the Laplace-Metropolis and Savage-Dicke
methods provide the best approximation to the “Gold
Standard” RJMCMC estimate, showing good agreement
all the way down to SNR = 5. Most importantly, all five
methods agree on when the 8-dimensional source model is
favored over the 7-dimensional model, placing the tran-
sition point at SNR ≃ 12.2. To get a rough estimate
for the numerical error in the various Bayes Factor es-
timates we repeated the SNR= 15 case 10 times using
different random number seeds. We found that the nu-
merical error was enough to account for any quantitative
differences between the estimates returned by the various
approaches.
It is interesting to compare the Bayesian model se-
lection results to the frequentist “3-σ” rule for positive
detection:
|q̄| > 3σq, (25)
where q̄ is the MAP estimate for the frequency change
and σq is the standard deviation in q as determined by
the FIM. For the source under consideration we found
the “3-σ” rule to require SNR ≃ 13 for a detection, in
good agreement with the Bayesian analysis. This lends
support to Seto’s [21] earlier FIM based study of the de-
tectability of the frequency evolution of galactic bina-
ries, but we should caution that the literature is replete
with examples where the “3-σ” rule yields results in dis-
agreement with Bayesian model selection and common
sense [22].
Favor 7D model
Not worth more than a bare mention
Positive
Strong
 0.7  0.8  0.9  1  1.1  1.2  1.3
RJMCMC
FIG. 4: Plot of the Bayes Factor estimates as a function of q
for each of the approximation schemes described in the text.
The signal to noise ratio was held fixed at SNR = 12.
The results of the second test are displayed in Figure 4
In this case all five methods produced results that agree
to within numerical error.
While the results shown here are for a particular choice
of source parameters, we found similar results for other
sets of source parameters. In general all five methods
for estimating the Bayes Factor gave consistent results
for signals with SNR > 7. One exception to this general
trend were sources with inclinations close to zero, as then
the PDFs tend to be highly non-gaussian. The Laplace-
Metropolis and Laplace-Fisher approximations suffered
the most in those cases.
V. ASTROPHYSICAL PRIORS
Astrophysical considerations lead to very strong pri-
ors for the frequency evolution of galactic binaries. The
detached systems, which are expected to account for
the majority of LISA sources, will evolve under gravita-
tional radiation reaction in accord with the leading order
quadrapole formula:
3(8π)8/3
f11/3M5/3 , (26)
where M is the chirp mass. Contact binaries undergoing
stable mass transfer from the lighter to the heavier com-
ponent are driven to longer orbital periods by angular
momentum conservation. The competition between the
effects of mass transfer and gravitational wave emission
lead to a formula for ḟ with the same frequency and mass
scaling as (26), but with the opposite sign and a slightly
lower magnitude [23].
Population synthesis models, calibrated against obser-
vational data, yield predictions for the distribution of
chirp masses M as a function of orbital frequency. These
distributions can be converted into priors on q. In con-
structing such priors one should also fold in observational
selection effects, which will favor systems with larger
chirp mass (the GW amplitude scales as M5/6). To get
some sense of how such priors will affect the model selec-
tion we took the chirp mass distribution for detached sys-
tems at f ∼ 5 mHz from the population synthesis model
described in Ref. [24], (kindly provided to us by Gijs Nele-
mans), and used (26) to construct the prior on q shown
in Figures 5 and 6 (observation selection effects were ig-
nored). The prior has been modified slightly to give a
small but no-vanishing weight to sources with q = 0. The
astrophysically motivated prior has a very sharp peak at
q = 0.64, and we use this value when fixing the frequency
derivative for the 7-dimensional model.
To explore the impact on model selection when such a
strong prior has been adopted we simulated a source with
q = 1 and varied the SNR. The RJMCMC algorithm was
applied using chains of length 107 in conjunction with
a fixed 8-dimensional MCMC (also allowed to run for
107 iterations) in order to compare the RJMCMC results
with the Savage-Dickie density ratio.
The results of this first exploration are shown in Fig-
ure 5. We found that for SNR < 15 the marginalized
PDF very closely resembled the prior distribution. This
demonstrates that the information content of the data
is insufficient to change our prior belief about the value
of the frequency derivative. As the SNR increased, how-
ever, the PDF began to move away from the prior until
we reached SNR=30 when the astrophysical prior had a
 0.005
 0.01
 0.015
 0.02
 0.025
 0.03
.5  1
.5  2
 0.005
 0.01
 0.015
 0.02
 0.025
 0.03
.5  1
.5  2
 0.005
 0.01
 0.015
 0.02
 0.025
 0.03
.5  1
.5  2
 0.005
 0.01
 0.015
 0.02
 0.025
 0.03
.5  1
.5  2
 0.005
 0.01
 0.015
 0.02
 0.025
 0.03
.5  1
.5  2
 0.005
 0.01
 0.015
 0.02
 0.025
 0.03
.5  1
.5  2
SNR=5 SNR=10 SNR=15
SNR=20 SNR=25 SNR=30
q q q
FIG. 5: Comparison between astrophysically motivated prior
distribution of q for f = 5 mHz and Tobs = 2 years (dashed,
blue) to marginalized PDF (solid, red) for sources injected
with q = 1 and SNRs varying from 5 to 30.
SNR BXY (SD) BXY (RJMCMC)
5 0.926 1.015
10 0.977 0.996
15 0.749 0.742
20 0.427 0.427
25 0.176 0.177
30 0.060 0.056
TABLE III: Savage-Dickie density ratio estimates of BXY for
sources with q = 1 and SNRs varying from 5 to 30. Compar-
isons with RJMCMC explorations of the same data set show
excellent agreement between the two methods.
negligible effect on the shape of the posterior, signaling
confidence in the quoted measurement of q. This qualita-
tive assessment of model preference is strongly supported
by the Bayes factor estimation made by the RJMCMC
algorithm as can be seen in Table III. It should also be
noted that the excellent agreement between the RJM-
CMC and S-D estimates for Bayes factor BXY . Both
methods indicate that for the chosen value of q = 1,
the signal-to-noise needs to exceed SNR ∼ 25 for the
8-dimensional model to be favored. This is in contrast
to the case discussed earlier where a uniform prior was
adopted for the frequency derivative, and the model se-
lection methods began showing a preference for the 8-
dimensional model around SNR=12.
Figure 6 shows the impact of the astrophysically moti-
vated prior when the SNR was held at 15 and four differ-
ent injected values for q were adopted, corresponding to
the full width at half maximum (FWHM) and full width
at quarter maximum (FWQM) of the prior distribution.
The Bayes factors listed in Table IV indicate that for
modestly loud sources with SNR=15 the model selection
techniques do not favor updating our estimate of the fre-
 0.005
 0.01
 0.015
 0.02
 0.025
 0.03
.5  1
.5  2
 0.005
 0.01
 0.015
 0.02
 0.025
 0.03
.5  1
.5  2
 0.005
 0.01
 0.015
 0.02
 0.025
 0.03
.5  1
.5  2
 0.005
 0.01
 0.015
 0.02
 0.025
 0.03
.5  2
q=0.35
q=0.83
q=0.48
q=1.15
FIG. 6: Marginalized PDF (solid, red) for fixed SNR=15 in-
jected sources with q corresponding to FWHM and FWQM
of the astrophysical prior (dashed, blue)
q BXY (SD) BXY (RJMCMC)
0.35 1.412 1.414
0.48 1.381 1.388
0.83 1.059 1.052
1.15 0.432 0.428
TABLE IV: Savage-Dickie and RJMCMC density ratio esti-
mates of BXY for sources with SNR=15 and q at FWHM and
FWQM of astrophysical prior
quency derivative until the frequency derivative exceeds
q = 1.2.
VI. DISCUSSION
We have found that the several common methods for
estimating Bayes Factors give good agreement when ap-
plied to the the model selection problem of deciding when
the data from the LISA observatory can be used to detect
the orbital evolution of a galactic binary. The methods
studied require varying degrees of effort to implement and
calculate, and although found to be accurate in this test
case, it is clear that some of these methods would be in-
appropriate approximations for more physically relevant
examples.
If a RJMCMC algorithm is used as the sole model
selection technique, the resistance of the algorithm
to change dimension, especially when making multi-
dimensional jumps, can result in invalid model selection
unless the chains are run for a very large numbers of
steps. In the examples we studied the transdimensional
jumps only had to span one dimension, and our basic
RJMCMC algorithm performed well. However, a more
sophisticated implementation, using e.g. rejection sam-
pling or coupled chains, will be required to select the
number of sources, as this requires jumps that span seven
or more dimensions.
The Laplace-Metropolis method for approximating the
model evidence is more robust than the commonly used
Fisher Information Matrix approximation of the Hessian
of the PDF. Implementing an LM evidence estimation is
a somewhat costly because of the need to fit the posterior
to a minimum volume ellipsoid.
The Savage-Dickie approximation is more economical
than the RJMCMC or LM methods, but is limited by the
requirement that the competing models must be nested.
The Bayes Information Criterion approximation to the
evidence is by far the cheapest to implement, and is able
to produce reliable results when the SNR is high. It has
therefore shown the most promise as an ‘on the fly’ model
determination scheme. More thorough (and therefore
more costly) methods such as RJMCMC and LM could
then be used to refine the conclusions initially made by
the BIC.
Our investigation using a strong astrophysical prior
indicated that the gravitational wave signals will need
to have high signal-to-noise (SNR > 25), or moderate
signal-to-noise (SNR > 15) and frequency derivatives far
from the peak of the astrophysical distribution, in order
to update our prior belief in the value of the frequency
derivative. In other words, the frequency derivative will
only been needed as a search parameter for a small num-
ber of bright high frequency sources.
Acknowledgments
This work was supported by NASA Grant
NNG05GI69G. We are most grateful to Gijs Nele-
mans for providing us with data from his population
synthesis studies.
Appendix A
To leading order in the eccentricity, e, the Cartesian
coordinates of the ith LISA spacecraft are given by [25]
xi(t) = R cos(α) +
cos(2α− βi)− 3 cos(β)
yi(t) = R sin(α) +
sin(2α− βi)− 3 sin(β)
zi(t) = −
3eR cos(α− βi) . (27)
In the above R = 1 AU, is the radial distance of the
guiding center, α = 2πfmt+ κ is the orbital phase of the
guiding center, and βi = 2π(i − 1)/3 + λ (i = 1, 2, 3) is
the relative phase of the spacecraft within the constel-
lation. The parameters κ and λ give the initial ecliptic
longitude and orientation of the constellation. The dis-
tance between the spacecraft is L = 2
3eR. Setting
e = 0.00985 yields L = 5× 109 m.
An arbitrary gravitational wave traveling in the k̂ di-
rection can be written as the linear sum of two indepen-
dent polarization states
h(ξ) = h+(ξ)ε
+ + h×(ξ)ε
× (28)
where the wave variable ξ = t − k̂ · x gives the surfaces
of constant phase. The polarization tensors can be ex-
panded in terms of the basis tensors e+ and e× as
ε+ = cos(2ψ)e+ − sin(2ψ)e×
ε× = sin(2ψ)e+ + cos(2ψ)e× , (29)
where ψ is the polarization angle and
+ = û⊗ û− v̂ ⊗ v̂
× = û⊗ v̂ + v̂ ⊗ û . (30)
The vectors (û, v̂, k̂) form an orthonormal triad which
may be expressed as a function of the source location on
the celestial sphere
û = cos θ cosφ x̂+ cos θ sinφ ŷ − sin θ ẑ
v̂ = sinφ x̂− cosφ ŷ
k̂ = − sin θ cosφ x̂− sin θ sinφ ŷ − cos θ ẑ . (31)
For mildly chirping binary sources we have
h(ξ) = ℜ
+ + ei3π/2A×ε
eiΨ(ξ)
where
2M(πf)2/3
1 + cos2 ι
A× = −
4M(πf)2/3
cos ι . (33)
Here M is the chirp mass, DL is the luminosity dis-
tance and ι is the inclination of the binary to the line
of sight. Higher post-Newtonian corrections, eccentric-
ity of the orbit, and spin effects will introduce additional
harmonics. For chirping sources the adiabatic approxi-
mation requires that the frequency evolution ḟ occurs on
a timescale long compared to the light travel time in the
interferometer: f/ḟ ≪ L. The gravitational wave phase
can be approximated as
Ψ(ξ) = 2πf0ξ + πḟ0ξ
2 + ϕ0 , (34)
where ϕ0 is the initial phase. The instantaneous fre-
quency is given by 2πf = ∂tΨ:
f = (f0 + ḟ0ξ)(1 − k̂ · v) . (35)
The general expression for the path length variation
caused by a gravitational wave involves an integral in
ξ from ξi to ξf . Writing ξ = ξi + δξ we have
Ψ(ξ) ≃ 2π(f0 + ḟ0ξi)δξ + const . (36)
Thus, we can treat the wave as having fixed frequency
f0 + ḟ0ξi for the purposes of the integration, and then
increment the frequency forward in time in the final ex-
pression [26]. The path length variation is then given
by [25, 26]
δℓij(ξ) = Lℜ
d(f, t, k̂) : h(ξ)
, (37)
where a : b = aijbij . The one-arm detector tensor is
given by
d(f, t, k̂) =
r̂ij(t)⊗ r̂ij(t)
T (f, t, k̂) , (38)
and the transfer function is
T (f, t, k̂) = sinc
1− k̂ · r̂ij(t)
× exp
1− k̂ · r̂ij(t)
, (39)
where f∗ = 1/(2πL) is the transfer frequency and f =
f0 + ḟ0ξ. The expression can be attacked in pieces. It is
useful to define the quantities
d+ij(t) ≡ (r̂ij(t)⊗ r̂ij(t)) : e
+ (40)
d×ij(t) ≡ (r̂ij(t)⊗ r̂ij(t)) : e
× . (41)
and yij(t) = δℓij(t)/(2L). Then
yij(t) = ℜ
yslowij (t)e
2πif0t
, (42)
where
yslowij (t) =
T (f, t, k̂)
d+ij(t)(A+(t) cos(2ψ)
+e3πi/2A×(t) sin(2ψ))
+d×ij(t)(e
3πi/2A×(t) cos(2ψ)
−A+(t) sin(2ψ))) e(πiḟ0ξ
2+iϕ0−2πif0k̂·x)
It is a simple exercise to derive explicit expressions for the
antenna functions and the transfer function appearing in
yslowij (t) using (27) and (31).
In the Fourier domain the response can be written as
yij(t) = ℜ
2πint/Tobs
e2πif0t
, (44)
where the coefficients an can be found by a numerical
FFT of the slow terms yslowij (t). Note that the sum over
n should extend over both negative and positive values.
The number of time samples needed in the FFT will de-
pend on f0 and ḟ0 and Tobs, but is less than 2
9 = 512
for any galactic sources we are likely to encounter when
Tobs ≤ 2yr. The bandwidth of a source can be estimated
B = 2 (4 + 2πf0R sin(θ)) fm + ḟ0Tobs . (45)
The number of samples should exceed 2BTobs. The
Fourier transform of the fast term can be done analyti-
cally:
e2πif0t =
2πimt/Tobs (46)
where
bm = Tobs sinc(xm)e
ixm (47)
xm = f0Tobs −m. (48)
The cardinal sine function in (46) ensures that the
Fourier components bm away from resonance, xm ≈ 0, are
quite small. It is only necessary to keep ∼ 100 → 1000
terms either side of p = [f0Tobs], depending on how bright
the source is, and how far f0Tobs is from an integer. We
now have
yij(t) = ℜ
2πijt/Tobs
, (49)
where
anbj−n . (50)
The final step is to ensure that our Fourier transform
yields a real yij(t). This is done by setting the final an-
swer for the Fourier coefficients equal to dj = (cj+c
−j)/2.
But since xm never hits resonance for positive j (we
are not interested in the negative frequency components
j < 0), we can neglect the second term and simply write
dj = cj/2.
Basically what we are doing is hetrodyning the signal
to the base frequency f0, then Fourier transforming the
slowly evolving hetrodyned signal numerically. We then
convolve these Fourier coefficients with the analytically
derived Fourier coefficients of the carrier wave.
The Michelson type TDI variables are given by
X(t) = y12(t− 3L)− y13(t− 3L) + y21(t− 2L)
−y31(t− 2L) + y13(t− L)− y12(t− L)
+y31(t)− y21(t), (51)
Y (t) = y23(t− 3L)− y21(t− 3L) + y32(t− 2L)
−y12(t− 2L) + y21(t− L)− y23(t− L)
+y12(t)− y32(t), (52)
Z(t) = y31(t− 3L)− y32(t− 3L) + y13(t− 2L)
−y23(t− 2L) + y32(t− L)− y31(t− L)
+y23(t)− y13(t). (53)
Note that in the Fourier domain
X(f) = ỹ12(f)e
−3if/f∗ − ỹ13(f)e−3if/f∗ + ỹ21(f)e−2if/f∗
−ỹ31(f)e−2if/f∗ + ỹ13(f)e−if/f∗ − ỹ12(f)e−if/f∗
+ỹ31(f)− ỹ21(f) . (54)
This saves us from having to interpolate in the time do-
main. We just combine phase shifted versions of our orig-
inal Fourier transforms.
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[27] While this count is only strictly correct for point-like
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the majority of sources in the LISA band.
