Hidden semi-Markov model
Model Families AIMech:HiddenSemiMarkovModel DRAFT
Aliases: HSMM, Explicit-duration hidden Markov model, Variable-duration hidden Markov model
A latent-state sequence-model family that assigns explicit distributions to state sojourn durations, with state changes governed by an embedded transition process and observations conditioned on the hidden states.
Identity Notes & Concept Scope
The inspected construction is a discrete-time, finite-state hidden semi-Markov model with state-specific duration distributions and conditionally independent emissions. It separates entry into a state from persistence within that state, rather than obtaining every dwell time from repeated self-transitions of an ordinary time-homogeneous HMM. The record is a model family, not the Yu–Kobayashi forward–backward algorithm. General continuous-time, duration-dependent destination and hybrid variants require their own evidence; they are not automatically covered by the detailed mechanism here. No normalized identity match was found across all 442 pinned AIO labels and all scoped synonyms and 388 preceding native IDs, names and aliases for the three selected concepts. Full-checkout fixed-string searches included ignored and hidden files in both worktree and main, yielding 8 and 8 matched lines; all 8 unique contents were read. Hits are prior pending leads and primary-reference context, not records of these identities. All 78 semantic label/synonym hits and 22 fully annotated exact AIO CLASS nodes with outgoing edges were read. Twenty-six contextual YAMLs were byte-verified after prior full reading and three freshly read, totaling 22 generated and seven native contextual records. Short acronym identities were compared across the complete normalized record inventory. Absence is bounded to the pinned source, maintained collection and inspected semantic neighborhood.
Claims & Grounded Evidence
1 Claim
Mechanism
#method
In the inspected formulation, let q_t be one of M hidden states and let r_t be its remaining duration, including the current time step. On entry into state m, draw d from a normalized state-specific distribution p_m(d), with d in 1,...,D, and set r_t=d. If r_t>1, the next state remains m and the residual duration decreases by one. If r_t=1, choose a different state n using the normalized entry-transition row a_mn, with a_mm=0, and draw its duration from p_n. Emit each observation independently conditional on the hidden path using b_m(o_t). The pair (q_t,r_t) is a first-order Markov state; q_t alone need not be. The source starts a new sojourn at t=1, giving alpha_1(m,d)=pi_m*p_m(d)*b_m(o_1). Forward recursion sums deterministic countdown paths and new entries; grouping incoming entry mass once per state avoids repeating that transition sum for each duration. Its terminal backward value beta_T(m,d)=1 sums over all remaining durations, so the finite observation window can end inside a sojourn. Likelihoods and smoothed state marginals follow by summing duration states with this endpoint convention.
⚠️ Curated Limitations & Negative Results
- An explicit duration distribution is not required to be non-geometric, but it permits duration laws whose exit hazard changes with elapsed dwell time. A finite maximum D is a modeling restriction; truncating an unbounded geometric law changes it.
- The new-sojourn initial law is different from sampling at an arbitrary time in a stationary process, which needs an appropriate residual-life distribution. Requiring a sojourn to end at T is also a different observation/endpoint event from the source’s right-censored window.
- The source excludes self-transitions at segment boundaries. Allowing adjacent same-labeled segments changes the interpretation and identifiability of the duration law, and needs explicit conventions.
- The printed parameter re-estimation sums use transition/entry quantities at t=1 although their displayed definitions refer to t-1. Initial-sojourn duration responsibilities need a separate boundary term; transitions between observed time steps start at t=2. This record does not certify the literal printed re-estimation formulas as a complete implementation.
- The statewise posterior maximizer in Equation(14) is marginal decoding, not a globally most probable hidden path. Numerical scaling, impossible observations and zero expected-count parameter rows require handling in an implementation.
- The paper’s one-day web-traffic example, choices of M and D and reported initialization behavior are application-specific. No universal convergence, predictive-accuracy or runtime guarantee for all semi-Markov variants is asserted.
Artificial Intelligence Ontology (AIO) Grounding
Mapped AIO Classes
| AIO Target Class | Mapping Relation | Mapping Rationale |
|---|---|---|
AIO:BayesianNetwork
Bayesian Network
|
broad_match | The inspected finite-duration model has a normalized directed acyclic representation over duration-augmented hidden states and observation children. The Bayesian-network family is broader than this explicit-sojourn model. |
AIO:MarkovChain
Markov Chain
|
related_match | The embedded state-entry process and the state-plus-residual-duration process are Markov representations. The hidden state label at each observation time generally is not first-order Markov, so the mapping does not assert that marginal process as a Markov chain. |
Evidence Sources & Primary Literature
1 Source| Source ID | Publication / Document | Type / Access | Reference & Checksum |
|---|---|---|---|
method-source
|
Shun-Zheng Yu, Hisashi Kobayashi
|
primary paper
full_text
|
ID: doi:10.1109/LSP.2002.806705
sha256:e625e483c462…
|
Curation Provenance
Curator: Codex
Method: ai_assisted
Model: GPT-6
Created: 2026-10-11T20:49:12Z
Updated: 2026-10-11T20:49:12Z
Source File:
data/curated/HiddenSemiMarkovModel.yaml