Cluster MCMC for rare neural trajectories

1. Exact integration point: use this as a sampler or trainer for a neural continuous-time state-space or world model. Let the latent state be a binary or discretized vector $z_t$, and let an MLP produce positive local rates $r_\theta(z\to z')=\operatorname{softplus}(f_\theta(z,z'))+\epsilon$. Condition on a fixed start state and a rare terminal predicate, such as reaching a target class or exceeding a threshold during a long rollout. 2. Pseudocode: initialize one valid trajectory; compute each escape rate $R_\theta$; choose $\Gamma=(1+\delta)\max R_\theta$ with $\delta=0.05$; insert virtual self-events; build a space-time graph from adjacent time slices and local event factors; seed random graph vertices; grow clusters using conditional factor weights while freezing boundary vertices; resample each cluster from its conditional distribution; remove virtual events; repeat. 3. Computed from the paper: uniformization, local factorization, detailed-balance cluster construction, and rejection-free conditional updates. Estimated empirically: neural rates, cluster bond probabilities, the safe upper bound for $\Gamma$, and integrated autocorrelation time. 4. First cheap experiment: use a 20-node SIS-like binary latent process or a two-dimensional neural jump process with horizon $T=100$, conditioning on an outbreak or target event whose forward probability is below $10^{-4}$. Compare independent forward rollouts, single-site Metropolis path updates, and cluster CPMC. Measure valid-trajectory rate, effective samples per network evaluation, and autocorrelation of event size. The prediction is that cluster updates retain nonzero valid-sample production as event probability decreases, while single-site updates exhibit rapidly increasing $\tau_{\mathrm{int}}$; at the smallest event probability, cluster $\tau_{\mathrm{int}}$ should be at least five times smaller and cluster acceptance should exceed 0.5.