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We present a comprehensive multiband photometric and spectroscopic study of the G-type binary CW Cas whose parameters have not been well determined. Our double-lined spectroscopic radial velocity curve of this system yields a reliable mass ratio of $q = 1.88(9)$. By combining $BVR_{c}I_{c}$ bands, TESS light curves and radial velocity curves, we found that CW Cas is a W-subtype shallow contact binary with a fill-out factor of 15\%. The components have masses of $0.98(6)M_{\odot}$ and $0.52(4)M_{\odot}$, separated by $2.25(5)R_{\odot}$. A notable asymmetry in the maxima of the light curves was detected and explained by a dark spot located on the surface of at least one component. Comparison of light curves from different years revealed that these dark spot activities exhibit cyclic variations with an approximate period of 1250 days. Orbital period analysis via O-C diagram spanning 125 years shows a long-term decrease superimposed with periodic oscillation caused by the light-travel-time effect (LTTE) due to a third body. This tertiary component has an orbital period of $P_{3} = 99.4(6)$ years and a minimal mass of $M_{3}=0.91(1)M_{\odot}$. The absence of detectable signatures for this massive object in either spectroscopic or photometric datasets implies it must be a compact object such as a white dwarf or neutron star. Furthermore, a visual companion was identified based on Gaia DR3 astrometric data, suggesting that CW Cas is part of a hierarchical quadruple system. As such, CW Cas represents a valuable laboratory for probing 2+1+1 hierarchical multiple system hosting compact object.

Gravitational waves from extreme mass-ratio inspirals (EMRIs) will enable subpercent measurements of massive black hole parameters and provide access to the demographics of compact objects in galactic nuclei. During the LISA mission, multiple EMRIs are expected to be detected, allowing statistical studies of massive black hole populations and their formation pathways. We perform hierarchical Bayesian inference on simulated EMRI catalogs to assess how well LISA could constrain the astrophysical population using parameterized population models. We test our inference framework on a variety of populations, including heterogeneous and homogeneous mixtures of parameterized subpopulations, and scenarios in which the assumed model is deliberately misspecified. Our results show that population parameters governing distributions with sharp features can be tightly constrained. Mixed populations can be disentangled with as few as ∼20 detections, and even with model misspecification, the inference retains sensitivity to key population features. These results demonstrate the capabilities and limitations of EMRI population inference, providing guidance for constructing realistic astrophysical population models for LISA analysis.