Notes on: Qin, Q., & Hobert, J. P. (2017): Asymptotically stable drift and minorization for markov chains with application to albert and chib's algorithm

Table of Contents

Notation

  • qin17_asymp_stabl_drift_minor_markov_f07b932b12c91bca424fd428d383495413b5b6a9.png samples size
  • qin17_asymp_stabl_drift_minor_markov_fefe9e556d399665a26a37824ec578cbffb0cabe.png covariates
  • qin17_asymp_stabl_drift_minor_markov_176130422efb42c4c3e6d5f8b226ab96567a2ab0.png denotes the data (i.e. responses and covariates)
  • qin17_asymp_stabl_drift_minor_markov_b519e07011744f28e192701c809e6d78dce1a0a1.png is a sequence of data
  • qin17_asymp_stabl_drift_minor_markov_e8324354751e1665822bd2155382e058ecaa7d2c.png
  • qin17_asymp_stabl_drift_minor_markov_8b75bcdb57274317a585353619ac869fca5e1642.png denotes the corresponding probability measure
  • qin17_asymp_stabl_drift_minor_markov_40d6d70e2c8f80b5085ee8bb9710ec13a63457db.png denotes an intractable probability density function (pdf)
  • qin17_asymp_stabl_drift_minor_markov_bc419f68d9d35fccdec82a2769da4a1784d4d9fd.png with qin17_asymp_stabl_drift_minor_markov_9e47faa96d468cca71dc58fc8a06f4b5200df197.png denotes the Markov transition function (Mtf) of an irreducible, aperiodic, Harris recurrent Markov chain with invariant probability measure qin17_asymp_stabl_drift_minor_markov_376212dcdcab088a0e0939c933fb80900c016ebd.png

Definitions

convergence complexity analysis
ascertain how the convergence rate of a MCMC scales with sample size qin17_asymp_stabl_drift_minor_markov_f07b932b12c91bca424fd428d383495413b5b6a9.png and/or number of covariates qin17_asymp_stabl_drift_minor_markov_fefe9e556d399665a26a37824ec578cbffb0cabe.png
drift
minorization
d&m method
refers to the method of using drift and minorization

The chain is called geometrically ergodic if there exists qin17_asymp_stabl_drift_minor_markov_37fd77d77fb139871836325ea9784ebf51348ed0.png and qin17_asymp_stabl_drift_minor_markov_44593a524ade3cbc3e000acca427cde41c955893.png such that

qin17_asymp_stabl_drift_minor_markov_3428ff0d54d350386eb0e83c5fbf217b55c47031.png

where qin17_asymp_stabl_drift_minor_markov_a52bc4b28058ca2b77daadabf88737c95e8d64e9.png denotes the total variation norm and qin17_asymp_stabl_drift_minor_markov_b8f52f56dc29d46f9c245478f08fc9e078578416.png is the m-step Mtf.

The geometric convergence rate of a chain is given by

qin17_asymp_stabl_drift_minor_markov_dfe2a15dbe00fedff8c836163f7d87bb78a5080c.png

The chain is geometrically ergodic if and only if qin17_asymp_stabl_drift_minor_markov_0388a8968513f878fee3f30cbf5269c5e3dd8b15.png.

Overview

  • Attempts to answer question "What can we say about the convergence properties of the Markov chains as qin17_asymp_stabl_drift_minor_markov_d61f568f7cbf68be029e968c1124ebaf741c0df9.png?"