IDEAS home Printed from https://ideas.repec.org/a/spr/jotpro/v33y2020i2d10.1007_s10959-019-00903-2.html
   My bibliography  Save this article

On Hitting Time, Mixing Time and Geometric Interpretations of Metropolis–Hastings Reversiblizations

Author

Listed:
  • Michael C. H. Choi

    (The Chinese University of Hong Kong)

  • Lu-Jing Huang

    (Fujian Normal University)

Abstract

Given a target distribution $$\mu $$μ and a proposal chain with generator Q on a finite state space, in this paper, we study two types of Metropolis–Hastings (MH) generator $$M_1(Q,\mu )$$M1(Q,μ) and $$M_2(Q,\mu )$$M2(Q,μ) in a continuous-time setting. While $$M_1$$M1 is the classical MH generator, we define a new generator $$M_2$$M2 that captures the opposite movement of $$M_1$$M1 and provide a comprehensive suite of comparison results ranging from hitting time and mixing time to asymptotic variance, large deviations and capacity, which demonstrate that $$M_2$$M2 enjoys superior mixing properties than $$M_1$$M1. To see that $$M_1$$M1 and $$M_2$$M2 are natural transformations, we offer an interesting geometric interpretation of $$M_1$$M1, $$M_2$$M2 and their convex combinations as $$\ell ^1$$ℓ1 minimizers between Q and the set of $$\mu $$μ-reversible generators, extending the results by Billera and Diaconis (Stat Sci 16(4):335–339, 2001). We provide two examples as illustrations. In the first one, we give explicit spectral analysis of $$M_1$$M1 and $$M_2$$M2 for Metropolized independent sampling, while in the second example, we prove a Laplace transform order of the fastest strong stationary time between birth–death $$M_1$$M1 and $$M_2$$M2.

Suggested Citation

  • Michael C. H. Choi & Lu-Jing Huang, 2020. "On Hitting Time, Mixing Time and Geometric Interpretations of Metropolis–Hastings Reversiblizations," Journal of Theoretical Probability, Springer, vol. 33(2), pages 1144-1163, June.
    • Handle: RePEc:spr:jotpro:v:33:y:2020:i:2:d:10.1007_s10959-019-00903-2
    DOI: 10.1007/s10959-019-00903-2
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10959-019-00903-2
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10959-019-00903-2?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. James Allen Fill, 2009. "On Hitting Times and Fastest Strong Stationary Times for Skip-Free and More General Chains," Journal of Theoretical Probability, Springer, vol. 22(3), pages 587-600, September.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Choi, Michael C.H. & Huang, Zhipeng, 2023. "Generalized Markov chain tree theorem and Kemeny’s constant for a class of non-Markovian matrices," Statistics & Probability Letters, Elsevier, vol. 193(C).

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Yu Gong & Yong-Hua Mao & Chi Zhang, 2012. "Hitting Time Distributions for Denumerable Birth and Death Processes," Journal of Theoretical Probability, Springer, vol. 25(4), pages 950-980, December.
    2. James Allen Fill & Vince Lyzinski, 2014. "Hitting Times and Interlacing Eigenvalues: A Stochastic Approach Using Intertwinings," Journal of Theoretical Probability, Springer, vol. 27(3), pages 954-981, September.
    3. Laurent Miclo & Pierre Patie, 2021. "On interweaving relations," Post-Print hal-03159496, HAL.
    4. Yong-Hua Mao & Feng Wang & Xian-Yuan Wu, 2015. "Large Deviation Behavior for the Longest Head Run in an IID Bernoulli Sequence," Journal of Theoretical Probability, Springer, vol. 28(1), pages 259-268, March.
    5. Miclo, Laurent & Arnaudon, Marc & Coulibaly-Pasquier, Koléhè, 2024. "On Markov intertwining relations and primal conditioning," TSE Working Papers 24-1509, Toulouse School of Economics (TSE).
    6. L. Avena & A. Gaudillière, 2018. "Two Applications of Random Spanning Forests," Journal of Theoretical Probability, Springer, vol. 31(4), pages 1975-2004, December.
    7. Persi Diaconis & Laurent Miclo, 2009. "On Times to Quasi-stationarity for Birth and Death Processes," Journal of Theoretical Probability, Springer, vol. 22(3), pages 558-586, September.
    8. Chen Jia, 2019. "Sharp Moderate Maximal Inequalities for Upward Skip-Free Markov Chains," Journal of Theoretical Probability, Springer, vol. 32(3), pages 1382-1398, September.
    9. James Allen Fill, 2009. "The Passage Time Distribution for a Birth-and-Death Chain: Strong Stationary Duality Gives a First Stochastic Proof," Journal of Theoretical Probability, Springer, vol. 22(3), pages 543-557, September.
    10. Marc Arnaudon & Koléhè Coulibaly-Pasquier & Laurent Miclo, 2024. "On Markov Intertwining Relations and Primal Conditioning," Journal of Theoretical Probability, Springer, vol. 37(3), pages 2425-2456, September.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:jotpro:v:33:y:2020:i:2:d:10.1007_s10959-019-00903-2. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.