Bayesian model selection for LISA pathfinder

The main goal of the LISA Pathfinder (LPF) mission is to fully characterize the acceleration noise models and to test key technologies for future space-based gravitational-wave observatories similar to the eLISA concept. The data analysis team has developed complex three-dimensional models of the LISA Technology Package (LTP) experiment onboard the LPF. These models are used for simulations, but, more importantly, they will be used for parameter estimation purposes during flight operations. One of the tasks of the data analysis team is to identify the physical effects that contribute significantly to the properties of the instrument noise. A way of approaching this problem is to recover the essential parameters of a LTP model fitting the data. Thus, we want to define the simplest model that efficiently explains the observations. To do so, adopting a Bayesian framework, one has to estimate the so-called Bayes factor between two competing models. In our analysis, we use three main different methods to estimate it: the reversible jump Markov chain Monte Carlo method, the Schwarz criterion, and the Laplace approximation. They are applied to simulated LPF experiments in which the most probable LTP model that explains the observations is recovered. The same type of analysis presented in this paper is expected to be followed during flight operations. Moreover, the correlation of the output of the aforementioned methods with the design of the experiment is explored.

Figure 1

Schematics of a simplified LTP SSM. It is composed by smaller SSMs of the various subsystems, each one a standalone SSM, here represented with white boxes. The rhombi represent the main injection inputs to the system. We use mainly the first injection port, which represents the interferometric inputs to the system, while the second rhombus stands for the capacitance actuators injection ports. The symbol $\mathrm{n⃗}$ represents noise contributions from various sources, and finally the parameters to fit, for the sake of convenience, are located in the gray boxes inside of the respective SSMs. The interferometer signals $\mathrm{i⃗}$ are injected to the controllers (DFACS), in which the commanded forces are generated and applied through the capacitance actuators and the thrusters of the spacecraft. In the last two of experiments of Ref. [29] “out-of-loop” forces $\mathrm{g⃗}$ are applied to the three bodies (TMs and SC) of the system. The resulting movement of the bodies is monitored by the inertial sensor, the star tracker, and the interferometer. Here, the interferometer output is denoted as $\mathrm{o⃗}$.

Figure 2

First $3\times 10^4$ iteration of the RJMCMC output when comparing a seven- and a five-dimensional LTP model (models X and Y, respectively). Since the models are not competitive when the $\text{SNR}=60$, the blue line tends asymptotically to zero.

Figure 3

Power spectra of simulated differential acceleration between the two test masses. The gray curve represents the reference noise measurement, while the light blue curve is the differential interferometer read out. The value of the computed Bayes factor can be confirmed with the comparison of the equivalent estimated residual acceleration for each model. The differential interferometer read out is also plotted for comparison.

Figure 4

A RJMCMC run on a set of nested LTP models. There is a clear preference for the five-dimensional model for the given data set. The data were produced with a “perfect” model in which the two respective actuators were identical.

Figure 5

The Bayes factor as a function of SNR computed using different methods.

Figure 6

The Bayes factor as a function of the injection frequencies in the first interferometric channel. The computed evidence of model X is stronger when the sinusoidal injection signals have frequencies around $f_{i_{x_1}}\simeq 0.0006\mathrm{Hz}$ and $f_{i_{x_1}}\simeq 0.05\mathrm{Hz}$, respectively.