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Jasper De Bock & Gert de Cooman
27 July 2011
State sequence prediction in imprecise hidden Markov models Jasper - - PowerPoint PPT Presentation
State sequence prediction in imprecise hidden Markov models Jasper - - PowerPoint PPT Presentation
State sequence prediction in imprecise hidden Markov models Jasper De Bock & Gert de Cooman 27 July 2011 Jasper De Bock & Gert de Cooman Jasper De Bock & Gert de Cooman Research group SYSTeMS Jasper De Bock Gert de Cooman Research
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Jasper De Bock & Gert de Cooman
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Jasper De Bock Gert de Cooman Research group
SYSTeMS
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Jasper De Bock Gert de Cooman Research group
SYSTeMS
Filip Hermans Erik Quaeghebeur Keivan Shariatmadar Arthur Van Camp
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State sequence prediction in imprecise hidden Markov models
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State sequence prediction in imprecise hidden Markov models
The imprecise hidden Markov model
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8
Jasper De Bock
X1 X2 X3 O1 O3 O2
S1 (O1|X1) S3 (O3|X3) S2 (O2|X2)
Imprecise hidden Markov model
A sequence of hidden state variables A sequence of observable variables
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Jasper De Bock
X1 X2 X3 O1 O3 O2
S1 (O1|X1) S3 (O3|X3) S2 (O2|X2) Q2 (X2|X1) Q1 (X1) Q2 (X3|X2)
Imprecise hidden Markov model
A sequence of hidden state variables A sequence of observable variables
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Jasper De Bock
X1 X2 X3 O1 O3 O2
S1 (O1|X1) S3 (O3|X3) S2 (O2|X2) Q2 (X2|X1) Q1 (X1) Q2 (X3|X2)
Imprecise hidden Markov model
All local models are coherent lower previsions
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Jasper De Bock
X1 X2 X3 O1 O3 O2
S1 (O1|X1) S3 (O3|X3) S2 (O2|X2) Q2 (X2|X1) Q1 (X1) Q2 (X3|X2)
Imprecise hidden Markov model
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State sequence prediction in imprecise hidden Markov models
Epistemic Irrelevance
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Jasper De Bock
X1 X2 X3 O1 O3 O2
Epistemic irrelevance
Conditional on its mother variable, the non-parent non- descendants of any variable in the tree are epistemically irrelevant to this variable and its descendants
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State sequence prediction in imprecise hidden Markov models
Recursive construction
- f a joint model for
the imprecise hidden Markov model
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Recursive construction of a joint model
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Recursive construction of a joint model
- Marginal extension
- Independent natural extension
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Recursive construction of a joint model
- Marginal extension
- Independent natural extension
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State sequence prediction in imprecise hidden Markov models
Conditioning the model
- n the observations
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State sequence prediction in imprecise hidden Markov models
Conditioning the model
- n the observations
Generalised Bayes rule: An extension of the Bayes rule to imprecise probabilities
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State sequence prediction in imprecise hidden Markov models
Maximal state sequences
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State sequence prediction in imprecise hidden Markov models
Maximal state sequences
Strict partial ordening: Maximal state sequences: We predict the state sequence by calculating a set of optimal sequences Notion of optimality: maximality
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State sequence prediction in imprecise hidden Markov models
Maximal state sequences
Strict partial ordening: Maximal state sequences: We predict the state sequence by calculating a set of optimal sequences Notion of optimality: maximality
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EstiHMM:
an efficient algorithm to determine the maximal state sequences in an imprecise hidden Markov model
State sequence prediction in imprecise hidden Markov models
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EstiHMM:
an efficient algorithm to determine the maximal state sequences in an imprecise hidden Markov model
State sequence prediction in imprecise hidden Markov models
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EstiHMM: an efficient algorithm to determine the maximal sequences
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- Principle of optimality
EstiHMM: an efficient algorithm to determine the maximal sequences
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- Principle of optimality
EstiHMM: an efficient algorithm to determine the maximal sequences
X2 X3 O2 O3
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- Principle of optimality
EstiHMM: an efficient algorithm to determine the maximal sequences
X1 X2 X3 O1 O2 O3
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- Principle of optimality
- Deriving an alternative
Roptimality criterion
EstiHMM: an efficient algorithm to determine the maximal sequences
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- Principle of optimality
- Deriving an alternative
Roptimality criterion
EstiHMM: an efficient algorithm to determine the maximal sequences
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- Principle of optimality
- Deriving an alternative
Roptimality criterion
- A recursive approach
EstiHMM: an efficient algorithm to determine the maximal sequences
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- Principle of optimality
- Deriving an alternative
Roptimality criterion
- A recursive approach
EstiHMM: an efficient algorithm to determine the maximal sequences Complexity
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- Principle of optimality
- Deriving an alternative
Roptimality criterion
- A recursive approach
EstiHMM: an efficient algorithm to determine the maximal sequences
- Theoretical analysis
Complexity
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- Principle of optimality
- Deriving an alternative
Roptimality criterion
- A recursive approach
EstiHMM: an efficient algorithm to determine the maximal sequences
- Theoretical analysis
- Linear in the number of
Rmaximal sequences
Complexity
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- Principle of optimality
- Deriving an alternative
Roptimality criterion
- A recursive approach
EstiHMM: an efficient algorithm to determine the maximal sequences
- Theoretical analysis
- Linear in the number of
Rmaximal sequences
- Empirical confirmation
Complexity
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A first experiment
State sequence prediction in imprecise hidden Markov models
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A first experiment
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A first experiment
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