Integrate the Reward, Don't Sample It: Window Likelihood for Interruptions and Backchannels in Full-Duplex Speech Models
A full-duplex spoken dialogue model must yield when the user cuts in and hold when the user merely backchannels. Sequence-level RL with a timing reward (GRPO) is the usual recipe; we test its whole-rollout form on four open models and find two failures. A binary window reward gives zero advantage on exactly the windows that most need learning (a zero-gradient trap), and the one score per rollout is spread over listening-state decisions the reward is not written in terms of (credit leakage), so the model talks into silence or stops answering. Both come from sampling a reward that is a known function of a few binary control decisions. We propose \emph{window likelihood} (\WL): sample a continuation per window, but integrate in closed form over the stop decisions it visits and maximise the log-probability of acting correctly---a noisy-OR for yielding, a survival product for holding. On HumDial-FDBench, \WL\ gives the best yield/hold F1 on all six structurally different open models we post-train, exceeds Kyutai's released GRPO checkpoints (trained on different data), and adds no re-opening of the floor on any model.