Adaptive Bayesian Joint Latent Space Modeling via Cumulative Shrinkage

Network models are increasingly vital in psychometrics for analyzing relational data, which are often accompanied by high-dimensional node attributes. Joint latent space models (JLSM) provide an elegant framework for integrating these data sources by assuming a shared underlying latent representation; however, a persistent methodological challenge is determining the dimension of the latent space, as existing methods typically require pre-specification or rely on computationally intensive post-hoc procedures. We develop a novel Bayesian joint latent space model that incorporates a cumulative ordered spike-and-slab (COSS) prior. This approach enables the latent dimension to be inferred automatically and simultaneously with all model parameters. We develop an efficient Markov Chain Monte Carlo (MCMC) algorithm for posterior computation. Theoretically, we establish that the posterior distribution concentrates on the true latent dimension and that parameter estimates achieve Hellinger consistency at a near-optimal rate that adapts to the unknown dimensionality. Through extensive simulations and two real-data applications, we demonstrate the method's superior performance in both dimension recovery and parameter estimation. Our work offers a principled, computationally efficient, and theoretically grounded solution for adaptive dimension selection in psychometric network models.