Bayesian parameter estimation for targeted anisotropic gravitational-wave background

Extended sources of the stochastic gravitational backgrounds have been conventionally searched on the spherical harmonics bases. The analysis during the previous observing runs by the ground-based gravitational wave detectors, such LIGO and Virgo, have yielded the constraints on the angular power spectrum $C_\ell$, yet it lacks the capability of estimating model parameters. In this paper, we introduce an alternative Bayesian formalism to search for such stochastic signals with a particular distribution of anisotropies on the sky. This approach provides a Bayesian posterior of model parameters and also enables selection tests among different signal models. While the conventional analysis fixes the highest angular scale \textit{a priori}, here we show a more systematic and quantitative way to determine the cut-off scale based on a Bayes factor, which depends on the amplitude and the angular scale of observed signals. Also, we analyze the third observing runs of LIGO and Virgo for the population of milli-second pulsars and obtain the 95 % constrains of the signal amplitude, $\epsilon < 2.7\times 10^{-8}$.