Switching State Estimation: Filtering with Hidden Discrete Modes

Overview

Switching state estimation is the problem of recursively estimating a continuous state that is conditionally linear-Gaussian given a discrete latent mode that itself follows a Markov chain. Both the continuous state and the mode are hidden. The optimal Bayesian filter has a posterior that is a Gaussian mixture whose number of components grows as M^t (M = number of modes, t = horizon length), making exact filtering intractable for all but the shortest horizons. The literature splits into three families of approximations:

1. Moment-collapsing methods (GPB1, GPB2, IMM) — at each step, collapse the mixture back to a small number of Gaussians. IMM is the workhorse in target tracking; it gets GPB2 accuracy at GPB1 cost.
2. Selection / sampling methods (Switching Kalman Filters with sampling, particle filters, Rao-Blackwellized particle filters / RBPF) — sample regime sequences with weights, run a Kalman filter conditional on each sample. Variance scales with the number of particles, not the mixture-component count.
3. Variational methods — fit a structured approximation (e.g. mean-field over the mode sequence) by minimizing KL divergence. Used less in dynamical filtering than in batch state-space learning.

A complementary line of work — Gaussian Mixture Reduction (GMR) — provides principled ways to merge / prune the exponentially-growing mixture so that exact-filter-like methods remain tractable; the leading methods minimize either ISE or an upper bound on KL divergence (Runnalls), and survey treatments compare them on tracking benchmarks (Crouse et al.).

For the CRE asset-pricing project, an RBPF over regime histories — with per-particle Kalman recursions for the continuous macro state and exponential-quadratic Riccati recursions for the asset-pricing factors — is the production likelihood evaluator. RBPF is uniquely well-suited because (a) the per-mode dynamics are linear-Gaussian, so Rao-Blackwellization is exact; (b) the cap-rate observation model is non-linear in the discrete state but conditionally linear in the continuous state; and (c) the regime process has only 4 compound modes, keeping per-step branching cheap.

Timeline

• 1965 — original Kalman filter; gold standard for the no-switching case.
• 1970s — Ackerson & Fu, Tugnait, Chang & Athans: early Generalized Pseudo-Bayesian (GPB1, GPB2) approximations to the optimal switching filter.
• 1988 — Blom & Bar-Shalom: Interacting Multiple Model (IMM) algorithm; standard in radar / target tracking.
• 1998 — Murphy: unified treatment of Switching Kalman Filters (SKF), GPB1/GPB2/IMM, variational learning, and EM.
• 1998 — Mazor et al.: IMM target-tracking survey; GPB-vs-IMM benchmark.
• 2007 — Runnalls: KL upper-bound pairwise-merge GMR algorithm.
• 2011 — Crouse et al.: GMR survey (GMRC, COWA, Runnalls, brute force; ISE vs KL).
• Various — Doucet, de Freitas, Gordon, Liu and others: Rao-Blackwellized particle filters for conditionally-linear-Gaussian state-space models.

Seminal works

• murphy-1998-switching-kalman-filters
• mazor-imm-target-tracking-survey
• runnalls-kl-gaussian-mixture-reduction
• crouse-gaussian-mixture-reduction-survey

SOTA tracker

• Tracking applications: IMM is dominant for radar / tracking; particle filters used when nonlinearity is severe.
• Econometrics: RBPF and Hamilton filter are standard for regime-switching state-space estimation; Hamilton filter is exact when the conditional likelihood marginalizes the continuous state, but does not handle nonlinear observation models (e.g. exponential-quadratic CRE cap rates).
• Mixture reduction: Runnalls’ KL-bound method is the simple, robust default; more sophisticated methods exist but are rarely worth the complexity.

Open problems

• Particle degeneracy under highly persistent regimes — the regime path posterior collapses onto a small number of paths, killing effective sample size. Mitigations: resampling schedules, RBPF with informed proposals, regime-history blocked Gibbs.
• Variance reduction for RBPF in MJLS — the literature largely treats RBPF as a black box; project-specific results show common-random-numbers gives only 1.4–1.6× variance reduction due to regime-path fragility.
• Honest UQ for likelihood estimates — the RBPF likelihood is a Monte Carlo estimator with non-negligible variance; tie-breaking optima within the noise floor requires paired-seed high-R sampling, which is not standard in the econometrics literature.

My position

For the CRE project the RBPF is the production likelihood evaluator. The Hamilton filter is used as a cheap deterministic surrogate for early-stage screening (≈2 ms/eval vs ≈350 ms/eval for RBPF at N=3000), but Hamilton-optimal ≠ RBPF-optimal because the Hamilton surrogate ignores the cap-rate panel. GMR has not been needed yet but is on the shelf for future variance-reduction work.

Research gaps

• Online asymptotic onset detection for the per-period Riccati horizon was tested (project Exp 12) and failed — the locked calibration objective was dominated by an oracle-distance term, turning the detector into an oracle-matcher rather than an error minimizer. Open: a reformulated objective that directly minimizes pricing error.
• Block-relative curvature at the MLE incumbent is currently DIAGNOSTIC_ONLY because of a forward-FD diagonal stencil bug at non-critical points; central-FD rerun pending.

Key people

(populated incrementally during ingest)