Emergency chest X-ray (CXR) triage has a structural modality gap: reports arrive after triage decisions, yet multimodal foundation models require image-text inputs.
We present Variational Risk Minimization (VRM), a distillation framework that treats LVLM-generated report variants as Monte Carlo samples of latent clinical interpretations.
Rather than distilling from a single teacher target, VRM learns from a variationally marginalized teacher distribution, enabling uncertainty-aware supervision under missing-modality constraints.
Under matched encoder families, VRM outperforms direct fine-tuning baselines and improves calibration with strong recovery from hallucinated supervision.
Marginalized supervision reduces report-selection instability.
In our compact edge-student instantiation, a confidence-gated cascade reaches AUC 0.941 at 103ms average latency with 20.3% cloud escalation, yielding an explicit reliability-latency operating point for cloud-edge clinical workflows.