96. Structured uncertainty has an observable error bound
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Consider two parameters in a certified family. For normalized \(z=(1,x,y)\), their density difference is
The state-update vector is the same \(v(u)\) in both models after observing \(u\). This does not make the models equivalent. Their likelihoods and hence their predicted observation laws can differ.
Proposition 85: Parameter-to-law error without a state-estimation term
Let
If \(|\Delta a|\leq d_a\) and \(|\Delta b|\leq d_b\), the conditional observation laws on any common normalized state obey
The same bound holds on every realized common observation history. With the same initial observation, complete future records of length \(T\) have total variation at most \(\min(1,T\Delta)\). For the five-node measure,
For \(du/2\), the respective constants are \(1/2\) and \(2/(3\sqrt3)\).
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Proof
The first claim follows by integrating \(|\Delta p|/2\), using \(|x|\leq1\) and \(y\leq1\). After the same observed value both models have exactly \(v(u)\), so the conditional bound applies again without accumulated coordinate-state error. Telescoping conditional kernels gives the record bound, equivalently the common-history coupling argument of Proposition 69. Direct summation gives the discrete constants. For the continuous constant \(c_2\), split the integral at \(u=\pm1/\sqrt3\) and integrate the quadratic polynomial with its respective signs. \(\square\)
The use of \(\Delta\) here is an upper bound proved from a confidence set and a model class. It is not the lower discrepancy bound from a rejected residual test in Part VI. A test that does not reject cannot supply the displayed upper bound by itself.
For the final reference rectangle and fitted center,
These bounds compare the fitted model with every member of the rectangle. They apply to the true process on the conditional coverage event. Even when that event fails, the rectangle's internal probability certificate remains an exact statement about its members.