65. Full continuous record laws inherit the stability argument
On this page · Permalink · Source line 2557
The base-norm proof from Part IV has an integral version. Its hypotheses must be supplied rather than inferred by replacing a finite sum with an integral in a program.
Proposition 59: Continuous-record total-variation bound
Let \(B_a(y)\) and \(\widetilde B_a(y)\) satisfy Proposition 52 on a common \((C,\lambda,\mu)\), and let the two models use the same measurable policy on every common observation history. Suppose their initial states satisfy \(\|z_0-\widetilde z_0\|_C\leq e_0\) and their time-\(t\) instruments satisfy
The length-\(T\) record laws, including actions when the policy is randomized, obey
No lower bound on each observation density is required. For identical initial states and the equal-bin approximation in Section 64, the example has the sufficient bound \(\min(1,2T/n)\).
Read proof · Recorded Lean scope · Visual guide: Integration retains a covariance term · Permalink · Source line 2561
Proof
For any positive decomposition \(x=u-v\), cone preservation and mass conservation give
Taking the infimum proves integrated instrument contraction. At each continuous history let \(v_h,\widetilde v_h\) be the unnormalized state densities, including the policy factors. Split the next difference as
Integrate the first term using contraction. The second is bounded by \(\epsilon_t\lambda\widetilde v_h\) after normalizing only its nonzero positive mass. Its integral over histories is at most \(\epsilon_t\). Tonelli's theorem applies to these nonnegative norm bounds, and operator integrability supplies well-defined finite-dimensional integrals. The integrated state discrepancy therefore satisfies \(D_{t+1}\leq D_t+\epsilon_t\). Finally \(|\lambda x|\leq\|x\|_C\) bounds the record-density difference, and total variation is half its absolute integral. \(\square\)
This result compares one common policy, not two policies optimized separately. It concerns the whole law and can become uninformative at long horizons. It is not a uniform posterior contraction theorem.
For a measurable event \(H\) of probability \(p=P_T(H)>0\), the restriction-and-normalization argument of Proposition 44 works for measures as well as finite vectors. When both conditional laws exist, their total-variation distance is at most \(\min(1,\delta/p)\) if the full-law distance is at most \(\delta\). The event probability is invariant under a reference-measure change. A density value at one point is not a substitute for that event probability.
The full-law comparison between \(B\) and \(B^\Pi\) permits any common measurable policy because both are continuous-output models on the same space. The exact equivalence to the finite bin model in Proposition 56 instead requires a bin-measurable policy. These are different conclusions with different observation information.