Multivariate Bayesian P-spline estimation of spectral density matrices, with application to LISA TDI noise

We present a Bayesian P-spline method for estimating the frequency-dependent cross-spectral density matrix of stationary multivariate time series. The inverse spectral matrix is parametrised through its frequency-varying Cholesky decomposition, which guarantees Hermitian positive definiteness at every frequency. Each real log-diagonal entry and each real and imaginary off-diagonal entry is given an independent penalised B-spline prior that controls smoothness. Inference uses a blocked, coarse-grained Whittle likelihood with safe-Bayes $\eta$-tempering to stabilise posterior calibration, sampled by the No-U-Turn Sampler from a variational initialisation. On synthetic VAR(2) benchmarks with known ground truth, the method recovers both diagonal and cross-spectral structure, attains near-nominal credible-interval coverage, and achieves a relative integrated squared (Frobenius) error (RISE) that decreases with sample size. We then apply the method to publicly released simulated LISA time-delay interferometry (TDI) data in two noise configurations. In the idealised symmetric case, the full multivariate model and a reduced model that assumes a diagonal AET noise covariance agree to within $\sim10^{-3}$ in RISE. Under realistic noise that is asymmetric across the six Movable Optical Sub-Assemblies (MOSAs), the AET-diagonal assumption fails by more than an order of magnitude in RISE ($\sim\!3.3\!\times\!10^{-2}$ versus $\sim\!10^{-3}$), whereas the full multivariate model recovers the cross-spectral structure.