29. Recovering the operators from an observable probability table
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Select \(r\) tests \(t_1,\ldots,t_r\) and \(r\) histories \(h_1,\ldots,h_r\), with \(t_1=\epsilon\), for which the square matrix
is invertible. Such a choice exists under the rank hypothesis of Proposition 27. The nonzero empty-test row can be extended to a row basis. Select independent history columns for that basis.
For each symbol \(s\), also obtain the shifted table and initial test vector
Proposition 28: Observable table reconstruction
Assume the full behavior rank is at most \(r\) and the displayed \(r\times r\) matrix \(H\) is invertible. Then the full rank is exactly \(r\), and
is a minimal realization of every word probability. Its state after history \(h\) is
The normalized state is \(b(h)/f(h)\) when \(f(h)>0\). Its update uses the formula in Proposition 26, with no hidden-state coordinates supplied to the reconstruction algorithm.
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Proof
Invertibility gives rank at least \(r\), so the assumed upper bound is attained. Evaluation on the selected tests is an isomorphism \(V\to\mathbb R^r\), because it maps the chosen basis of history columns to the invertible matrix \(H\). The shifted images of those basis columns are precisely the columns of \(G_s\), so \(B_sH=G_s\). The first selected test evaluates the mass. Initial evaluation gives \(b_0\). Proposition 27 supplies correctness for every history, not only the selected columns. \(\square\)
There are two ways to justify coverage in this statement. One can know that the selected row and column families span the true behavior space. Alternatively, a justified upper bound \(r\) on the process rank, together with the invertible \(r\times r\) block, proves that coverage. The observed invertibility by itself only gives a lower bound. An unknown higher-rank process can agree on a finite table and differ later.
Exact experiment probabilities are population quantities. The reference example below supplies rational tables from a declared process and keeps that table generator separate from the learner. Section 36 supplies one finite sampling protocol for estimating those entries. Neither a hidden matrix nor its probability cone is secretly passed to the observable learner.