Preferential dynamic modeling with forward-backward smoothing

Estimating a secondary signal (e.g., behavior) from neural activity over time is central to both causal online decoding and non-causal offline inference in neuroscience, yet existing two-signal latent state-space models rarely support both. In this work, we provide an analytical extension of a linear method (PSID) beyond causal prediction to also support non-causal inference. We provide theoretical derivations extending PSID to enable optimal filtering and optimal smoothing of the secondary signal. We show that, in the PSID setting, the presence of a secondary signal increases identifiability. This allows us to uniquely learn the quantities needed for the optimal Kalman update via a reduced-rank regression step, yielding our first contribution, PSID with filtering. We next design a forward-backward construction for smoothing, yielding our second contribution, PSID with smoothing. In simulations, we validate that both PSID with filtering and smoothing reach ideal performance. In non-human primate motor cortex data, PSID with smoothing consistently improves over PSID with filtering, which improves over one-step-ahead prediction with standard PSID. Finally, we show that the connection between PSID and an existing nonlinear two-signal model, DPAD, extends naturally to the smoothing setting: applying our forward-backward formulation to DPAD yields DPAD with smoothing, which achieves competitive performance with leading methods on the Neural Latents Benchmark (NLB) in behavior decoding and held-out neural prediction across three datasets. Together, this work provides a theoretical foundation for prediction, filtering, and smoothing in the two-signal setting, spanning causal online decoding to offline inference, in both linear and nonlinear settings.