4 From corrected records to decisions and explanations
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The same record, model, and continuation must support the decision being certified.
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Correctness of a probability calculation is not a guarantee that a real process follows its likelihood model. Within the declared model, however, a small error in the retained distribution can support a precise decision bound. We state that bridge before introducing communication, so that an explanation has a definite object to preserve.
4.1 One quantitative decision contract
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Let \(p\) and \(\widehat p\) be distributions on the same finite world set. For each world let \(v_\omega\) be a response and \(\widehat v_\omega\) its approximation in one metric space. Assume
For each action \(a\) in a finite nonempty set, let the loss \(\ell(a,v)\) lie in \([0,1]\) and be \(K\)-Lipschitz in \(v\). Define the model risks
Theorem 4.1: A decision bound for one actual approximation chain
Put \(\xi=\min\{1,e+Kd_0\}\). For every action, \(|R(a)-\widehat R(a)|\le\xi\). If \(\widehat a\) minimizes the approximate risk, then
If the approximate gap between its unique minimizer and every alternative exceeds \(2\xi\), that action is also the unique true model-risk minimizer.
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Proof
Add and subtract \(\sum_\omega p(\omega)\ell(a,\widehat v_\omega)\). The response term has absolute value at most \(Kd_0\). The expectation of a function in \([0,1]\) changes by at most total variation, giving the other term at most \(e\). Risks lie in \([0,1]\), so their difference is also at most one.
For a true minimizer \(a_*\),
The loss range supplies the cap one. Finally, for any other \(a\), subtract the two error bounds to obtain \(R(a)-R(\widehat a)\ge\widehat R(a)-\widehat R(\widehat a)-2\xi>0\).
Source relationship. [GA], “A common quantitative grounding certificate”; [MU], “A joint posterior and recipient decision certificate.” This is a specialization to a common finite candidate set and bounded loss.
The phrase “model risk” matters. The theorem cannot repair an omitted candidate, a false sensor convention, or an action outside the stated loss function. It also cannot combine a posterior bound from one history with a response bound from another. The record must bind the actual calculation and its premises.
For the noisy calibration example, use zero-one loss for selecting the target bit. Under \(P\) and report one, choosing one has risk \(1/4\), while choosing zero has risk \(3/4\). An approximation with total variation at most \(1/10\) and no response error preserves the unique decision because the gap \(1/2\) exceeds \(2/10\). Under \(P'\) the correct decision reverses. A valid error bound around the wrong prior is not a substitute for retaining the correct association.
4.2 An explanation belongs to a recipient
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An artifact is not an explanation merely because its producer can interpret it. Fix a finite case set \(X\), a sender-access map \(\alpha:X\to H\), and a finite artifact catalogue \(\mathcal C\). Recipient \(i\) has a background record \(\beta_i(x)\), a demanded output \(t_i(x)\), and a fixed bounded procedure \(L_i\). A timeout, malformed artifact, wrong output, or budget overrun counts as failure.
Let \(\mathcal C(x)\) be the artifacts that succeed for every recipient in case \(x\). For an attained sender value \(h\), put
Theorem 4.2: A common artifact for fixed recipients
A sender restricted to \(\alpha(x)\) can provide one exact artifact for all recipients if and only if every \(\mathcal C_h\) is nonempty. When feasible, the smallest possible number of distinct transmitted artifacts is
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Proof
A sender seeing \(h\) must choose the same artifact for all cases in that fibre. It must therefore select a member of \(\mathcal C_h\), proving necessity. Choosing one member of each nonempty finite intersection gives a valid encoder. The image of any encoder intersects every \(\mathcal C_h\). Conversely, any catalogue meeting all those sets supplies such an encoder by choosing from the intersections. Finite minimization gives the formula.
Source relationship. [MU], “The fixed-recipient covering criterion.” The earlier designed-decoder coloring criterion is not substituted for the installed-recipient problem.
Counterexample 4.3: Individual acceptability is not a common message
Two cases have the same sender record. In one, the sole acceptable artifact is \(c_0\); in the other it is \(c_1\ne c_0\). Each case has a valid explanation if its hidden identity is known. Their intersection is empty, so the sender cannot select a valid common explanation from its actual information.
In the sensor, a sender observing \(y\) can transmit one bit to a recipient holding \(s\). The recipient's fixed XOR procedure returns \(b=y\oplus s\). The sender need not know \(b\): its message combines with the recipient's background.
A decoder that ignores this message fails where background alone cannot distinguish the target. More strings do not fix that procedure. Installed information, interpreter work, and message size are separate resources. The next chapter asks how the sender can acquire the needed information within one budget, counting every stored branch of the controller.