93. From observable moment regression to operator sets
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There is a direct connection to the unnormalized instrument. Assume that the normalized predictive state \(Z_t\) is an exact, fixed, measurable function of the available observation history. For action \(a\), suppose
on realized positive-density branches. Crucially, \(Z_{t+1}\) must also be available from the observations using the specified state map. This is an accessibility assumption, not a consequence of successful regression.
Let \(B_a(u)=\sum_{k=0}^{D}\psi_k(u)C_{a,k}\). Choose functions \(g_j\) so that the known matrix \(W_{jk}=\int g_j(u)\psi_k(u)\,\mu(du)\) is invertible.
Proposition 82: Sequential observable-moment operator sets
Define \(M_{a,j}=\int g_j(u)B_a(u)\,\mu(du)\). Then
Under the rowwise noise and parameter-bound assumptions of Proposition 81, simultaneous confidence ellipsoids for the rows of \(M_{a,j}\) give a confidence set for every \(C_{a,k}\) by this known linear map. If row \(i\) of \(\widehat M_{a,j}-M_{a,j}\) has \(V_{a,t}\)-norm at most \(\beta_{a,j,i}\), then for any direction \(x\),
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Proof
In the conditional integral, multiply the normalized state by the observation density. The denominator cancels, giving \(\int g_j(u)B_a(u)Z_t\,\mu(du)\). The polynomial expansion gives \(M_{a,j}=\sum_kW_{jk}C_{a,k}\) and hence the inverse formula. Apply Proposition 81 to each row, using the predictable regressor \(Z_t\) on visits to action \(a\), or its action-gated version. A union bound over the predeclared finite rows and moments gives simultaneous coverage. Independence between these response coordinates is unnecessary. Weighted Cauchy–Schwarz and the triangle inequality give the displayed directional bound. \(\square\)
Exact mass constraints can be imposed by intersecting the confidence set with the known mass affine subspace. This preserves coverage because the true model is assumed to obey that law. Arbitrarily shrinking a set around a convenient corrected operator does not have the same justification. Data-selected geometry can be checked against the whole resulting set using Proposition 78.
The theorem is conditional on a correct finite observation basis, accessible predictive state and parameter bounds. It does not obtain a hidden predictive state from a single dependent path without additional assumptions. Unknown dictionary discovery, state-estimation error and polynomial misspecification remain distinct problems. The following example makes the state accessible by construction rather than concealing that requirement.