### Abstract

In this chapter, we investigate the ideal of polynomial invariants satisfied by the probabilities of observations in a hidden Markov model. Two main techniques for computing this ideal are employed. First, we describe elimination using Gröbner bases. This technique is only feasible for small models and yields invariants that may not be easy to interpret. Second, we present a technique using linear algebra refined by two gradings of the ideal of relations. Finally, we classify some of the invariants found in this way. The hidden Markov model The hidden Markov model was described in Section 1.4.3 (see also Figure 1.5) as the algebraic statistical model defined by composing the fully observed Markov model F with the marginalization ρ, giving a map, where Θ_{1} is the subset of Θ defined by requiring row sums equal to one. Here we will write the hidden Markov model as a composition of three maps, ρ ∘ F ∘ g, beginning in a coordinate space Θ″ ⊂ ℂ^{d} which parameterizes the d = l(l − 1) + l(l′ − 1)-dimensional linear subspace Θ_{1} lying in the l ^{2} + ll′-dimensional space Θ, so that Θ_{1} = g(Θ″). These maps are shown in the following diagrams: In the bottom row of the diagram, we have phrased the hidden Markov model in terms of rings by considering the ring homomorphism g*, F* and ρ*.

Original language | English (US) |
---|---|

Title of host publication | Algebraic Statistics for Computational Biology |

Publisher | Cambridge University Press |

Pages | 237-249 |

Number of pages | 13 |

ISBN (Electronic) | 9780511610684 |

ISBN (Print) | 0521857007, 9780521857000 |

DOIs | |

State | Published - Jan 1 2005 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

*Algebraic Statistics for Computational Biology*(pp. 237-249). Cambridge University Press. https://doi.org/10.1017/CBO9780511610684.015

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*Algebraic Statistics for Computational Biology.*Cambridge University Press, pp. 237-249. https://doi.org/10.1017/CBO9780511610684.015

**Equations defining hidden markov models.** / Bray, Nicolas; Morton, Jason Ryder.

Research output: Chapter in Book/Report/Conference proceeding › Chapter

TY - CHAP

T1 - Equations defining hidden markov models

AU - Bray, Nicolas

AU - Morton, Jason Ryder

PY - 2005/1/1

Y1 - 2005/1/1

N2 - In this chapter, we investigate the ideal of polynomial invariants satisfied by the probabilities of observations in a hidden Markov model. Two main techniques for computing this ideal are employed. First, we describe elimination using Gröbner bases. This technique is only feasible for small models and yields invariants that may not be easy to interpret. Second, we present a technique using linear algebra refined by two gradings of the ideal of relations. Finally, we classify some of the invariants found in this way. The hidden Markov model The hidden Markov model was described in Section 1.4.3 (see also Figure 1.5) as the algebraic statistical model defined by composing the fully observed Markov model F with the marginalization ρ, giving a map, where Θ1 is the subset of Θ defined by requiring row sums equal to one. Here we will write the hidden Markov model as a composition of three maps, ρ ∘ F ∘ g, beginning in a coordinate space Θ″ ⊂ ℂd which parameterizes the d = l(l − 1) + l(l′ − 1)-dimensional linear subspace Θ1 lying in the l 2 + ll′-dimensional space Θ, so that Θ1 = g(Θ″). These maps are shown in the following diagrams: In the bottom row of the diagram, we have phrased the hidden Markov model in terms of rings by considering the ring homomorphism g*, F* and ρ*.

AB - In this chapter, we investigate the ideal of polynomial invariants satisfied by the probabilities of observations in a hidden Markov model. Two main techniques for computing this ideal are employed. First, we describe elimination using Gröbner bases. This technique is only feasible for small models and yields invariants that may not be easy to interpret. Second, we present a technique using linear algebra refined by two gradings of the ideal of relations. Finally, we classify some of the invariants found in this way. The hidden Markov model The hidden Markov model was described in Section 1.4.3 (see also Figure 1.5) as the algebraic statistical model defined by composing the fully observed Markov model F with the marginalization ρ, giving a map, where Θ1 is the subset of Θ defined by requiring row sums equal to one. Here we will write the hidden Markov model as a composition of three maps, ρ ∘ F ∘ g, beginning in a coordinate space Θ″ ⊂ ℂd which parameterizes the d = l(l − 1) + l(l′ − 1)-dimensional linear subspace Θ1 lying in the l 2 + ll′-dimensional space Θ, so that Θ1 = g(Θ″). These maps are shown in the following diagrams: In the bottom row of the diagram, we have phrased the hidden Markov model in terms of rings by considering the ring homomorphism g*, F* and ρ*.

UR - http://www.scopus.com/inward/record.url?scp=84907289956&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84907289956&partnerID=8YFLogxK

U2 - 10.1017/CBO9780511610684.015

DO - 10.1017/CBO9780511610684.015

M3 - Chapter

AN - SCOPUS:84907289956

SN - 0521857007

SN - 9780521857000

SP - 237

EP - 249

BT - Algebraic Statistics for Computational Biology

PB - Cambridge University Press

ER -