Access

You are not currently logged in.

Access your personal account or get JSTOR access through your library or other institution:

login

Log in to your personal account or through your institution.

If You Use a Screen Reader

This content is available through Read Online (Free) program, which relies on page scans. Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.

Exponential Convergence of Langevin Distributions and Their Discrete Approximations

Gareth O. Roberts and Richard L. Tweedie
Bernoulli
Vol. 2, No. 4 (Dec., 1996), pp. 341-363
DOI: 10.2307/3318418
Stable URL: http://www.jstor.org/stable/3318418
Page Count: 23
  • Read Online (Free)
  • Subscribe ($19.50)
  • Cite this Item
Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Exponential Convergence of Langevin Distributions and Their Discrete Approximations
Preview not available

Abstract

In this paper we consider a continuous-time method of approximating a given distribution π using the Langevin diffusion d Lt= d Wt+1/2∇ log π ( Lt) dt. We find conditions under this diffusion converges exponentially quickly to π or does not: in one dimension, these are essentially that for distributions with exponential tails of the form π(x) ∝ exp(-γ |x|β), 0 < β < ∞, exponential convergence occurs if and only if β ≥ 1. We then consider conditions under which the discrete approximations to the diffusion converge. We first show that even when the diffusion itself converges, naive discretizations need not do so. We then consider a 'Metropolis-adjusted' version of the algorithm, and find conditions under which this also converges at an exponential rate: perhaps surprisingly, even the Metropolized version need not converge exponentially fast even if the diffusion does. We briefly discuss a truncated form of the algorithm which, in practice, should avoid the difficulties of the other forms.

Page Thumbnails

  • Thumbnail: Page 
[341]
    [341]
  • Thumbnail: Page 
342
    342
  • Thumbnail: Page 
343
    343
  • Thumbnail: Page 
344
    344
  • Thumbnail: Page 
345
    345
  • Thumbnail: Page 
346
    346
  • Thumbnail: Page 
347
    347
  • Thumbnail: Page 
348
    348
  • Thumbnail: Page 
349
    349
  • Thumbnail: Page 
350
    350
  • Thumbnail: Page 
351
    351
  • Thumbnail: Page 
352
    352
  • Thumbnail: Page 
353
    353
  • Thumbnail: Page 
354
    354
  • Thumbnail: Page 
355
    355
  • Thumbnail: Page 
356
    356
  • Thumbnail: Page 
357
    357
  • Thumbnail: Page 
358
    358
  • Thumbnail: Page 
359
    359
  • Thumbnail: Page 
360
    360
  • Thumbnail: Page 
361
    361
  • Thumbnail: Page 
362
    362
  • Thumbnail: Page 
363
    363