Wavelet-domain hidden Markov model

⚠️ Curated Limitations & Negative Results

• Wavelet decorrelation does not imply independence. Zero-mean conditional Gaussian coefficients can be uncorrelated while their squared magnitudes remain dependent through hidden states. Marginalizing the states does not generally leave an observed Markov process.
• The two zero-mean states and chosen graph are modeling choices, not universal properties of wavelets. More states, nonzero means and alternative emissions are possible. Scaling coefficients may be modeled separately or left unchanged in a particular application. Directed dependence does not establish intervention causality.
• With K states and dense transitions, exact chain/tree message cost is O(edges*K^2). Richer or loopy graphs need separate inference and complexity analysis; linearity in graph size for chordal models also depends on bounded clique and state sizes. Parameter tying imposes equality constraints and requires pooled sufficient statistics with proper normalization.
• Exact EM under suitable model and optimization conditions does not ensure a global likelihood optimum. Empty posterior components, zero transition occupancy, singular Gaussian variances and constrained parameters need explicit handling. The record does not certify an author implementation or fit any parameters.
• Appendix A in the inspected 1997 preprint contains inconsistent parent/child epsilon indices in Equations (19) and(22), a finest/coarsest initialization typo and missing or inconsistent state indices in the downward recursion. This record uses the stated normalized model factorization; an asymmetric three-node finite example verifies consistent messages against complete enumeration rather than copying those displays literally.
• The denoising formula assumes the stated orthonormal transform and independent white Gaussian noise. Subtracting a noise estimate from fitted noisy variances and clipping at zero is a plug-in procedure, not a general constrained-maximum-likelihood theorem. The zero signal-variance boundary is degenerate. Estimated parameters and noise are treated as fixed, rather than integrated out.
• Reported denoising, classification and change-detection examples do not establish universal superiority. Table 1 does not make the tree best for every signal, and selected comparator numbers are imported from other work. Comparing class likelihoods alone assumes the appropriate prior-and-cost threshold is one; unequal priors or costs generally change that threshold.
• The fully inspected source is the 31-page author preprint revised August 1997. The 1998 journal article DOI 10.1109/78.668544 is corroborating bibliographic context, not an assertion that those journal PDF bytes were read. No first-invention date, benchmark reproduction or author-code execution is asserted.