Grounded World Models: Efficient and Verifiable Structural Causal Prediction C Explanation-Quality Components and Heuristic Parameter Derivations E A Worked Example: GWM vs NWM on One Case

Appendix D The Sampling Budget: Latency vs Residual Monte-Carlo Error

Every GWM headline (target price, direction probabilities, scenario mix) is a functional of $n_{\mathrm{samples}}$ Monte-Carlo draws, so the sample budget trades latency against the residual noise that enters the residual-noise control $R(q)=1-\min(1,\mathrm{se}(q)/\tau)$ of Definition 11. We fix $n_{\mathrm{samples}}=10{,}000$ throughout; this appendix justifies that choice. Sweeping by powers of ten over the cost-benchmark cases of Appendix F (warm median wall-clock; standard error of $T(q)=\mathbb{E}[\text{upside}]$ estimated as $\mathrm{se}(q)=\hat{\sigma}_{\text{upside}}/\sqrt{n_{\mathrm{samples}}}$ ), Figure 7 plots the tradeoff.

Refer to caption — Figure 7: Choosing the sampling budget: latency vs residual noise. Latency per predictive distribution (log, vertical) against the residual Monte-Carlo standard error on the return summary (log, horizontal; right $=$ lower error $=$ better), swept over powers of ten and aggregated as the median over recent deployed v1 cases. The residual falls as the textbook $n^{-1/2}$ (each decade cuts it $\approx 3.2\times$ ), while latency is flat below a per-call overhead floor and then rises roughly linearly—so the curve’s *geometric* elbow is near $n{=}1\text{k}$ , not at the deployed budget. We nonetheless deploy $n{=}10\text{k}$ (circled) for *sufficiency*, not because it is the elbow: it is the smallest budget whose residual clears the decision-relevant $\mathrm{se}\!\approx\!1\%$ band (dotted; $R\!\approx\!0.83$ at $\tau{=}5\%$ ) while latency is still sub- $100$ ms. The elbow at $1\text{k}$ is too noisy ( $\mathrm{se}\!\approx\!2.7\%$ , $R\!\approx\!0.47$ —enough to flip a direction call), and the next decade to $100\text{k}$ pays $\approx 9\times$ the latency for a further $\sqrt{10}$ residual reduction that no longer moves a decision. The top axis reads the same residual as the bounded residual-noise control $R=1-\min(1,\mathrm{se}/\tau)$ (Def. 11).

Two regimes bracket the choice: below $\sim\!10^{3}$ samples a fixed per-call overhead floors the latency while the residual balloons, and above $\sim\!10^{4}$ the sampling loop dominates, so each further decade costs a near-full decade of latency for only a $\sqrt{10}$ residual gain. The $10{,}000$ -sample budget clears the floor and drives $R$ into saturation at sub-second latency; because the residual scales as $n^{-1/2}$ , only the absolute latency floor (hardware) moves with the case mix, not the choice itself.