Statement
For a discrete-time Markov jump linear system in which the operation mode θ(k) is not observed and only the output y(k) is available, the optimal nonlinear (minimum-mean-square-error) state estimator is a bank of Kalman filters indexed by all possible regime paths {θ(0), θ(1), …, θ(k)}. The number of filters in this bank, and consequently the memory and computation, grows as
|paths| = N + N² + … + N^{T+1} = (N^{T+1} − N) / (N − 1)
i.e. exponentially in the time horizon T. This intractability is the formal motivation for restricting attention to (a) the linear minimum mean-square error filter (LMMSE), which is Nn-dimensional regardless of T, or (b) approximate nonlinear filters such as the Interacting Multiple Model (IMM) filter, particle filters, and Rao-Blackwellized particle filters that carry regime histories on a finite particle set.
Evidence summary
CFM Section 5.4.1 cites Anderson and Moore [17] for the bank-of-Kalman-filters construction and explicitly works out the exponential count N + N² + … + N^T in Remark 5.2. The result is presented as a well-established fact in stochastic filtering and is repeated in §5.4 as the reason CFM derives the LMMSE alternative (Theorem 5.9) and a stationary LMMSE filter (Theorem 5.12) that achieve optimality only within the linear class. CFM §8.4.1 numerically compares LMMSE against IMM as the standard practical alternatives.
Conditions and scope
- Finite-state Markov chain over modes (N < ∞).
- Mode θ(k) is unobserved at all k; only y(k) is available.
- Discrete-time MJLS with additive (typically Gaussian) noise.
- “Optimal” here means optimal unrestricted nonlinear MMSE; the claim does not say anything about restricted filter classes (linear MMSE, IMM, finite- particle approximations), all of which trade optimality for tractability.
- Continuous-time analogues exist but require separate proof.
Counter-evidence
None for the unrestricted optimal nonlinear estimator. The CFM textbook itself, the IMM filter literature, and the entire particle-filter literature take the exponential growth as the starting axiom and design suboptimal finite-memory alternatives around it. The Rao-Blackwellized particle filter in the parent CRE asset-pricing project is one such alternative: particles carry regime histories explicitly, which is a Monte-Carlo approximation to the exact bank-of-Kalman-filters posterior.
Linked ideas
(none yet)
Open questions
- Tight bounds on the gap between practical finite-memory nonlinear filters (IMM, RBPF, smoothers) and the unattainable exact nonlinear posterior.
- Information-theoretic lower bounds: how much of the exponential complexity is necessary versus how much is an artefact of the bank-of-filters construction?
- Conditions under which a finite-memory sufficient statistic for the conditional regime distribution exists (mostly negative results so far).