35. Conditioning a prediction can amplify its error
On this page · Permalink · Source line 1453
Let \(C\) be a probability cone with compact normalized base \(K\) and mass row \(\lambda\). Work in a declared norm and its dual. Suppose true normalized states have norm at most \(R\). Let a true branch have
For a predicted state \(\widehat z\), form \(\widehat u=\widehat B_s\widehat z\), \(\widehat p=\lambda\widehat u\), and define \(d=\widehat u-u\). Write \(e=\|\widehat z-z\|\), \(\|\widehat B_s\|\leq L\), and \(\|\widehat B_s-B_s\|\leq\epsilon_B\). Then
Proposition 34: Normalization-aware filtering stability
If \(\|\lambda\|_*D<\eta\), then \(\widehat p>0\) and
so
Alternatively, if each realized fitted denominator is checked to satisfy \(\widehat p_t\geq\widehat\eta_t>0\), the pathwise recursion obeys
The product-sum bound of Proposition 18 applies. A uniform bound follows if these constants have a common contraction margin and bounded forcing. The algebraic bound alone does not certify that a fitted state or every unchosen branch is probability-valid.
Read proof · Recorded Lean scope · Permalink · Source line 1467
Proof
The branch likelihood differs by \(\lambda d\), giving the lower bound on \(\widehat p\). Subtract \(u/p\) from \((u+d)/(p+\lambda d)\) and use \(u=pz^+\). The induced norm of \(I-z^+\lambda\) is at most \(1+R\|\lambda\|_*\). Applying that identity at each step gives the recursion. \(\square\)
Rare evidence can create a large posterior change from a small absolute error. For example, take a true unnormalized vector \((\eta,0)\) and a perturbed vector \((\eta,\eta)\). Their \(\ell^1\) difference is \(\eta\), while their normalized vectors differ by one in \(\ell^1\). Both arise as possible branch outputs of valid probability instruments when \(0<\eta\leq1/2\), after assigning the remaining mass to another branch. There is no uniform posterior Lipschitz constant independent of a probability margin.
When a known closed convex feasible base is available, Euclidean projection of a tentative posterior back onto that base cannot increase its distance from the true posterior. This is a declared repair to the estimator, not an identity for an unmodified update. It also does not repair negative branch probabilities of arbitrary fitted operators before normalization.