CounterexampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Zero-evidence observations have no conditional MAP path

Statement refuted

For every finite HMM and every finite observation sequence, the posterior distribution on hidden paths is defined and therefore has a sequence MAP path.

Facts & Assumptions

Given: A finite discrete HMM with two hidden states and one observed symbol.

[F1]

Finite hidden Markov models for sequence decoding: For an observation sequence $y$ , its evidence is $p (y) = \sum_{z} p (z, y)$ ; the conditional path mass $p (z ∣ y) = p (z, y) / p (y)$ is defined only when $p (y) > 0$ , and a sequence MAP path maximizes this conditional mass.

Counterexample

Take $S = {u, v}$ , $Y = {0, 1}$ , and initial probabilities $π (u) = π (v) = 1 / 2$ . Let both transition rows be $(1 / 2, 1 / 2)$ . In each state let $B (s, 0) = 1$ and $B (s, 1) = 0$ . These are normalized initial, transition, and emission parameters. Observe the length-one sequence $y = (1)$ .

1.1givenF1algebra

For the two possible hidden paths, the joint masses are $p ((u), (1)) = (1 / 2) \cdot 0 = 0$ and $p ((v), (1)) = (1 / 2) \cdot 0 = 0$ . Thus $p ((1)) = 0 + 0 = 0$ .

2.1givenF1step 1.1algebra∎

By [F1], division by this zero evidence does not define conditional path masses, so there is no posterior MAP path for this observation. Both paths maximize the zero joint mass, but that arbitrary tie is not a conditional posterior distribution. This refutes the Statement.

Depends on

Finite hidden Markov models for sequence decoding

Used by

Nothing in the library uses this result yet.

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Sources

Manolis Kellis, 6.096 Algorithms for Computational Biology, Lecture 7: Gene Finding and HMMs, MIT OpenCourseWare (Spring 2005) (standard reference, not scraped)
Stanford CS262, Lecture 7: Hidden Markov Models (standard reference, not scraped)