Robust hidden Markov LQG problems

Lars Peter Hansen, Ricardo Mayer, Thomas Sargent

    Research output: Contribution to journalArticle

    Abstract

    For linear quadratic Gaussian problems, this paper uses two risk-sensitivity operators defined by Hansen and Sargent (2007b) to construct decision rules that are robust to misspecifications of (1) transition dynamics for state variables and (2) a probability density over hidden states induced by Bayes' law. Duality of risk sensitivity to the multiplier version of min-max expected utility theory of Hansen and Sargent (2001) allows us to compute risk-sensitivity operators by solving two-player zero-sum games. Because the approximating model is a Gaussian probability density over sequences of signals and states, we can exploit a modified certainty equivalence principle to solve four games that differ in continuation value functions and discounting of time t increments to entropy. The different games express different dimensions of concerns about robustness. All four games give rise to time consistent worst-case distributions for observed signals. But in Games I-III, the minimizing players' worst-case densities over hidden states are time inconsistent, while Game IV is an LQG version of a game of Hansen and Sargent (2005) that builds in time consistency. We show how detection error probabilities can be used to calibrate the risk-sensitivity parameters that govern fear of model misspecification in hidden Markov models.

    Original languageEnglish (US)
    Pages (from-to)1951-1966
    Number of pages16
    JournalJournal of Economic Dynamics and Control
    Volume34
    Issue number10
    DOIs
    StatePublished - Oct 2010

    Fingerprint

    Game
    Mathematical operators
    Probability Density
    Error detection
    Hidden Markov models
    Time Consistency
    Equivalence Principle
    Utility Theory
    Parameter Sensitivity
    Detection Probability
    Discounting
    Model Misspecification
    Zero sum game
    Entropy
    Misspecification
    Expected Utility
    Error Probability
    Bayes
    Decision Rules
    Operator

    Keywords

    • Certainty equivalence
    • Entropy
    • Hidden Markov models
    • Kalman filter
    • Misspecification
    • Robustness

    ASJC Scopus subject areas

    • Economics and Econometrics
    • Applied Mathematics
    • Control and Optimization

    Cite this

    Robust hidden Markov LQG problems. / Hansen, Lars Peter; Mayer, Ricardo; Sargent, Thomas.

    In: Journal of Economic Dynamics and Control, Vol. 34, No. 10, 10.2010, p. 1951-1966.

    Research output: Contribution to journalArticle

    Hansen, LP, Mayer, R & Sargent, T 2010, 'Robust hidden Markov LQG problems', Journal of Economic Dynamics and Control, vol. 34, no. 10, pp. 1951-1966. https://doi.org/10.1016/j.jedc.2010.05.004
    Hansen, Lars Peter ; Mayer, Ricardo ; Sargent, Thomas. / Robust hidden Markov LQG problems. In: Journal of Economic Dynamics and Control. 2010 ; Vol. 34, No. 10. pp. 1951-1966.
    @article{4af04aa8316e4e8f801d12ab578c655c,
    title = "Robust hidden Markov LQG problems",
    abstract = "For linear quadratic Gaussian problems, this paper uses two risk-sensitivity operators defined by Hansen and Sargent (2007b) to construct decision rules that are robust to misspecifications of (1) transition dynamics for state variables and (2) a probability density over hidden states induced by Bayes' law. Duality of risk sensitivity to the multiplier version of min-max expected utility theory of Hansen and Sargent (2001) allows us to compute risk-sensitivity operators by solving two-player zero-sum games. Because the approximating model is a Gaussian probability density over sequences of signals and states, we can exploit a modified certainty equivalence principle to solve four games that differ in continuation value functions and discounting of time t increments to entropy. The different games express different dimensions of concerns about robustness. All four games give rise to time consistent worst-case distributions for observed signals. But in Games I-III, the minimizing players' worst-case densities over hidden states are time inconsistent, while Game IV is an LQG version of a game of Hansen and Sargent (2005) that builds in time consistency. We show how detection error probabilities can be used to calibrate the risk-sensitivity parameters that govern fear of model misspecification in hidden Markov models.",
    keywords = "Certainty equivalence, Entropy, Hidden Markov models, Kalman filter, Misspecification, Robustness",
    author = "Hansen, {Lars Peter} and Ricardo Mayer and Thomas Sargent",
    year = "2010",
    month = "10",
    doi = "10.1016/j.jedc.2010.05.004",
    language = "English (US)",
    volume = "34",
    pages = "1951--1966",
    journal = "Journal of Economic Dynamics and Control",
    issn = "0165-1889",
    publisher = "Elsevier",
    number = "10",

    }

    TY - JOUR

    T1 - Robust hidden Markov LQG problems

    AU - Hansen, Lars Peter

    AU - Mayer, Ricardo

    AU - Sargent, Thomas

    PY - 2010/10

    Y1 - 2010/10

    N2 - For linear quadratic Gaussian problems, this paper uses two risk-sensitivity operators defined by Hansen and Sargent (2007b) to construct decision rules that are robust to misspecifications of (1) transition dynamics for state variables and (2) a probability density over hidden states induced by Bayes' law. Duality of risk sensitivity to the multiplier version of min-max expected utility theory of Hansen and Sargent (2001) allows us to compute risk-sensitivity operators by solving two-player zero-sum games. Because the approximating model is a Gaussian probability density over sequences of signals and states, we can exploit a modified certainty equivalence principle to solve four games that differ in continuation value functions and discounting of time t increments to entropy. The different games express different dimensions of concerns about robustness. All four games give rise to time consistent worst-case distributions for observed signals. But in Games I-III, the minimizing players' worst-case densities over hidden states are time inconsistent, while Game IV is an LQG version of a game of Hansen and Sargent (2005) that builds in time consistency. We show how detection error probabilities can be used to calibrate the risk-sensitivity parameters that govern fear of model misspecification in hidden Markov models.

    AB - For linear quadratic Gaussian problems, this paper uses two risk-sensitivity operators defined by Hansen and Sargent (2007b) to construct decision rules that are robust to misspecifications of (1) transition dynamics for state variables and (2) a probability density over hidden states induced by Bayes' law. Duality of risk sensitivity to the multiplier version of min-max expected utility theory of Hansen and Sargent (2001) allows us to compute risk-sensitivity operators by solving two-player zero-sum games. Because the approximating model is a Gaussian probability density over sequences of signals and states, we can exploit a modified certainty equivalence principle to solve four games that differ in continuation value functions and discounting of time t increments to entropy. The different games express different dimensions of concerns about robustness. All four games give rise to time consistent worst-case distributions for observed signals. But in Games I-III, the minimizing players' worst-case densities over hidden states are time inconsistent, while Game IV is an LQG version of a game of Hansen and Sargent (2005) that builds in time consistency. We show how detection error probabilities can be used to calibrate the risk-sensitivity parameters that govern fear of model misspecification in hidden Markov models.

    KW - Certainty equivalence

    KW - Entropy

    KW - Hidden Markov models

    KW - Kalman filter

    KW - Misspecification

    KW - Robustness

    UR - http://www.scopus.com/inward/record.url?scp=77956884459&partnerID=8YFLogxK

    UR - http://www.scopus.com/inward/citedby.url?scp=77956884459&partnerID=8YFLogxK

    U2 - 10.1016/j.jedc.2010.05.004

    DO - 10.1016/j.jedc.2010.05.004

    M3 - Article

    VL - 34

    SP - 1951

    EP - 1966

    JO - Journal of Economic Dynamics and Control

    JF - Journal of Economic Dynamics and Control

    SN - 0165-1889

    IS - 10

    ER -