Normal form backward induction for decision trees with coherent lower previsions

Huntley, Nathan; Troffaes, Matthias C. M.

doi:10.1007/s10479-011-0968-2

Cited by 13 publications

(17 citation statements)

References 22 publications

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“…The criterion we use to decide which estimates are optimal for the given gain functions is that of (Walley-Sen) maximality (Troffaes, 2007;Walley, 1991). Maximality has a number of very desirable properties that make sure it works well in optimisation contexts (De Cooman & Troffaes, 2005;Huntley & Troffaes, 2010), and it is well-justified from a behavioural point of view, as well as in a robustness approach, as we shall see presently.…”

Section: Maximal State Sequencesmentioning

confidence: 99%

An Efficient Algorithm for Estimating State Sequences in Imprecise Hidden Markov Models

Bock

Cooman

2014

jair

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We present an efficient exact algorithm for estimating state sequences from outputs or observations in imprecise hidden Markov models (iHMMs). The uncertainty linking one state to the next, and that linking a state to its output, is represented by a set of probability mass functions instead of a single such mass function. We consider as best estimates for state sequences the maximal sequences for the posterior joint state model conditioned on the observed output sequence, associated with a gain function that is the indicator of the state sequence. This corresponds to and generalises finding the state sequence with the highest posterior probability in (precise-probabilistic) HMMs, thereby making our algorithm a generalisation of the one by Viterbi. We argue that the computational complexity of our algorithm is at worst quadratic in the length of the iHMM, cubic in the number of states, and essentially linear in the number of maximal state sequences. An important feature of our imprecise approach is that there may be more than one maximal sequence, typically in those instances where its precise-probabilistic counterpart is sensitive to the choice of prior. For binary iHMMs, we investigate experimentally how the number of maximal state sequences depends on the model parameters. We also present an application in optical character recognition, demonstrating that our algorithm can be usefully applied to robustify the inferences made by its precise-probabilistic counterpart.

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Section: Maximal State Sequencesmentioning

confidence: 99%

An Efficient Algorithm for Estimating State Sequences in Imprecise Hidden Markov Models

Bock

Cooman

2014

jair

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“…see Barker and Wilson ( 2012 ) and Cao ( 2014 ). Another approach that we focus on in the present paper is to assume that a decision maker cannot uniquely assign the probabilities to the possible events (Huntley and Troffaes 2012 ; Jaffray 2007 ; Walley 1991 ). This was dubbed ambiguity by Borgonovo and Marinacci ( 2015 ); however, other terms are sometimes used (e.g.…”

Section: Introductionmentioning

confidence: 99%

“…However, they note that it might be difficult for a user to decide, firstly, what probability density function should be assigned to a particular node and, secondly, how those probabilities should be correlated. Huntley and Troffaes ( 2012 ) present several ideas for choice functions, i.e. criteria the decision maker may use to select a subset of strategies.…”

Section: Introductionmentioning

confidence: 99%

“…The approach we propose is most suitable when a decision problem is solved once but the optimal strategy is then applied in numerous individual cases. First-order uncertainty can be addressed by calculating the expected value; however, the distributional uncertainty cannot be averaged-out, see the discussion above, as well as in Ben-Tal and Nemirovski ( 2000 ), Høyland and Wallace ( 2001 ), Huntley and Troffaes ( 2012 ) and Kouvelis and Yu ( 2013 ). For example, let us consider a bank designing a multi-step debt recovery process.…”

Section: Introductionmentioning

confidence: 99%

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A framework for sensitivity analysis of decision trees

Kamiński

Jakubczyk

Szufel

2017

Cent Eur J Oper Res

391

189

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In the paper, we consider sequential decision problems with uncertainty, represented as decision trees. Sensitivity analysis is always a crucial element of decision making and in decision trees it often focuses on probabilities. In the stochastic model considered, the user often has only limited information about the true values of probabilities. We develop a framework for performing sensitivity analysis of optimal strategies accounting for this distributional uncertainty. We design this robust optimization approach in an intuitive and not overly technical way, to make it simple to apply in daily managerial practice. The proposed framework allows for (1) analysis of the stability of the expected-value-maximizing strategy and (2) identification of strategies which are robust with respect to pessimistic/optimistic/mode-favoring perturbations of probabilities. We verify the properties of our approach in two cases: (a) probabilities in a tree are the primitives of the model and can be modified independently; (b) probabilities in a tree reflect some underlying, structural probabilities, and are interrelated. We provide a free software tool implementing the methods described.

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“…The latter models, called credal networks, offer a direct sensitivity-analysis interpretation: a credal network is a collection of Bayesian networks, all over the same variables and with the same graph, whose parameters are consistent with constraints (e.g., interval specifications) modeling a limited ability in the assessment of sharp estimates. Something similar has been also done with decision trees [16,17,19], while the situation is different for IDs. The early attempts of Fertig and Breese [13] first, and Zaffalon [11] after, to extend these models to non-sharp quantification are among the few works in this direction.…”

Section: Introductionmentioning

confidence: 99%

Evaluating interval-valued influence diagrams

Cabañas

Antonucci

Cano

et al. 2017

International Journal of Approximate Reasoning

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Influence diagrams are probabilistic graphical models used to represent and solve sequential decision problems under uncertainty. Sharp numerical values are required to quantify probabilities and utilities. This might be an issue with real models, whose parameters are typically obtained from expert judgments or partially reliable data. We consider an interval-valued quantification of the parameters to gain realism in the modeling and evaluate the sensitivity of the inferences with respect to perturbations in the sharp values of the parameters. An extension of the classical influence diagrams formalism to support such interval-valued potentials is presented. The variable elimination and arc reversal inference algorithms are generalized to cope with these models. At the price of an outer approximation, the extension keeps the same complexity as with sharp values. Numerical experiments show improved performances with respect to previous methods. As a natural application, we propose these models for practical sensitivity analysis in traditional influence diagrams. The maximum perturbation level on single or multiple parameters preserving the optimal strategy can be computed. This allows the identification of the parameters deserving a more careful elicitation.

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Normal form backward induction for decision trees with coherent lower previsions

Cited by 13 publications

References 22 publications

An Efficient Algorithm for Estimating State Sequences in Imprecise Hidden Markov Models

An Efficient Algorithm for Estimating State Sequences in Imprecise Hidden Markov Models

A framework for sensitivity analysis of decision trees

Evaluating interval-valued influence diagrams

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