77. Comparing a model with a process outside its representation
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The earlier operator-norm law bounds compare two models on one common probability cone. A misspecified generating process may not have a representation in the fitted cone. It is useful to state the more basic comparison at the level of conditional observation laws.
Let \(P\) be any causal process on a standard Borel observation space with finite actions. Let \(Q\) be a proposed process. Both use the same measurable policy. Write \(P_t(\cdot\mid h,a)\) and \(Q_t(\cdot\mid h,a)\) for their next-observation kernels, and define
The proposed kernel must be specified on every history needed for the comparison, including histories impossible under \(Q\) but possible under \(P\). A declared fallback on a model-null history completes a causal kernel without changing \(Q\)'s own law. An implementation that rejects such a history must report that rejection rather than silently claim to have filtered it.
Proposition 69: Observation-law error without a shared state space
With the same initial history, the length-\(T\) joint action-and-observation laws satisfy
An initial-history discrepancy adds its total variation. In particular, uniform bounds \(d_t(h,a)\leq\epsilon_t\) give the upper bound \(\min(1,\sum_{t<T}\epsilon_t)\).
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Proof
Introduce hybrid laws that follow \(P\) for the first \(t\) transitions and \(Q\) for the rest, with the common policy throughout. Two successive hybrids have the same \(P\) prefix distribution up to time \(t\) and the same future \(Q\) kernel after the next transition. Conditional on that prefix and its policy-chosen action, changing the next kernel costs \(d_t(h,a)\) in total variation. A common continuation kernel cannot increase total variation, because integration of a test bounded between zero and one produces another such test. Average over the shared prefix, sum the adjacent-hybrid bounds, and cap by one. The same contraction argument handles a different initial law. \(\square\)
This theorem does not make the right-hand side observable without further assumptions. Proposition 68 supplies lower evidence about visited conditional discrepancies, not an upper estimate for every future history. Turning a finite collection of residuals into an upper law guarantee requires additional regularity, coverage or a restricted class with a separating test family.
The distinction prevents a common logical jump: a model can pass several tests and still have a large untested discrepancy. Conversely, a statistically significant residual need not mean that its internal probability algebra is invalid. The model can be a valid distribution and an inadequate approximation at the same time.