47. Whole observation laws can be stable without rare-event margins
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The posterior after one rare observation can be sensitive, but that does not force the entire distribution of observation records to be equally sensitive. Work before normalization and retain the probability of each branch.
Let two models share \((C,\lambda)\) and have initial states \(z_0,\widetilde z_0\in K\). At each time use the same policy, a specified function or randomization depending only on the observed history. Thus on any common history the action probabilities are identical in the two comparisons. This is not a comparison of two policies optimized separately for the two models.
Proposition 43: Observation-tree total-variation bound
If both instruments preserve \(C\) and conserve mass, and if every action at time \(t\) has the discrepancy in Proposition 42 bounded by \(\epsilon_t\), then the distributions \(P_T,\widetilde P_T\) of length-\(T\) observation records under the common policy satisfy
The same bound holds for joint action and observation records when the common policy is randomized. No positive lower bound on individual branch probabilities is required.
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Proof
Let \(u_h,\widetilde u_h\in C\) be the unnormalized states on each history at depth \(t\), including the common policy factors. Their masses sum to one in each model. Define \(D_t=\sum_h\|u_h-\widetilde u_h\|_C\). At a history with selected action \(a\), write
Sum the first term over observations and use Proposition 42. When \(\lambda\widetilde u_h>0\), normalize only to apply the definition of \(\epsilon_t\) to the second term. Its sum is bounded by \(\epsilon_t\lambda\widetilde u_h\). If the mass is zero, strict positivity implies \(\widetilde u_h=0\). Summing histories yields \(D_{t+1}\leq D_t+\epsilon_t\). Averaging over common policy probabilities gives the same inequality for randomized actions. Finally each word-probability difference is bounded by its state difference because \(|\lambda x|\leq\|x\|_C\). Total variation is half the sum of those absolute differences. \(\square\)
This bound separates total law error from conditional-state error. It can grow linearly with the horizon and eventually become uninformative. It is not a uniform-in-time approximation theorem. It also does not control the regret of a changed planning policy without additional decision-theoretic assumptions.