LemmaStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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The finite hidden Markov joint law is normalized

Statement

For every finite discrete HMM and each integer $T \geq 1$ , the joint masses $p (z, y)$ in the Definition sum to $1$ over all hidden paths $z \in S^{T}$ and observation sequences $y \in Y^{T}$ .

Facts & Assumptions

Given: A finite discrete HMM $(S, Y, π, A, B)$ and $T \geq 1$ .

[F1]

Finite hidden Markov models for sequence decoding: The initial weights satisfy $\sum_{s \in S} π (s) = 1$ , each transition row satisfies $\sum_{s^{'} \in S} A (s, s^{'}) = 1$ , each emission row satisfies $\sum_{y \in Y} B (s, y) = 1$ , and $p (z, y) = π (z_{1}) B (z_{1}, y_{1}) \prod_{t = 2}^{T} A (z_{t - 1}, z_{t}) B (z_{t}, y_{t})$ .

Proof

1.1givenF1algebra

Fix a hidden path $z$ . Finite distributivity and the emission row sums in [F1] give $\sum_{y \in Y^{T}} p (z, y) = π (z_{1}) \prod_{t = 2}^{T} A (z_{t - 1}, z_{t})$ , because summing over each coordinate $y_{t}$ contributes the factor $\sum_{y_{t} \in Y} B (z_{t}, y_{t}) = 1$ .

2.1givenF1step 1.1algebra∎

Sum the expression in step 1.1 over all $z \in S^{T}$ . Summing first over $z_{T}$ , then $z_{T - 1}$ down to $z_{2}$ , removes one transition factor at a time by the corresponding row sum in [F1]. The remaining sum is $\sum_{z_{1} \in S} π (z_{1}) = 1$ . Thus the total joint mass is $1$ .

Depends on

Finite hidden Markov models for sequence decoding

Used by

Finite-HMM Viterbi decoding is exact sequence MAP Theorem

Cited to discharge well-definedness by Finite hidden Markov models for sequence decoding.

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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)