Dual-Ensemble Latent Transition Model

1. Integration point: insert the module between an encoder and a latent transition model. Encode each observation as soft memberships $c_t=\mathrm{softmax}(f_\theta(x_t))\in\mathbb{R}^K$. Use one transition-logit head $g_P(h_t)$ for ordinary trajectories and a second head $g_Q(h_t,a_t,b_t)$ for trajectories carrying source and sink labels. Apply a row-wise softmax to obtain $P_\theta$ and $Q_\theta$. 2. Pseudocode: collect equilibrium pairs $(c_t,c_{t+\tau})$ and accumulate $N^P_{ij}\mathrel{+}=c_{t,i}c_{t+\tau,j}$; set $P_{ij}=N^P_{ij}/(\sum_jN^P_{ij}+\epsilon)$ and optimize transition cross-entropy. Generate source-sink rollouts; whenever a rollout first enters $B$, record its transition and reset its latent state to a source sample. Accumulate $N^Q$ identically and fit $Q$. At evaluation, obtain stationary vectors by power iteration $v_{n+1}=v_nP$ or $v_{n+1}=v_nQ$, selecting the operator according to the observable. 3. Taken from the paper: two ensemble-specific transition matrices, stationary weighted averaging, and recycling semantics. Estimated empirically: memberships, counts, stationary vectors, and bootstrap uncertainty. 4. First cheap experiment: construct the paper's eight-state Markov chain, merge adjacent microscopic states into four coarse states, and compare a single-matrix latent model with this dual model at lag $\tau=1$. Repeat on a small grid-world or MuJoCo dataset with source and target regions. Measure stationary occupancy, mean first-passage time, committors, and directed flux. The falsifiable prediction is that the dual model's coarse stationary vector converges to the aggregated microscopic vector $(18,12,6,1)/37\approx(0.486,0.324,0.162,0.027)$, whereas the single-matrix model retains a fixed-lag bias. Increasing data should reduce dual-model error approximately as $N^{-1/2}$ without requiring a longer lag.