Spectral Analytic Comparisons for Data Augmentation

• Activity Number: 247
• Type: Contributed
• Date/Time: Monday, August 1, 2011 : 2:00 PM to 3:50 PM
• Sponsor: IMS
• Author(s): Vivekananda Roy*+
• Companies: Iowa State University
• Keywords: data augmentation algorithm ; spectrum ; convergence rate ; Markov chain ; compact operator ; eigenvalue

Abstract

The data augmentation (DA) algorithm, though very useful in practice, often suffers from slow convergence. Hobert and Marchev (2008) recently introduced an alternative to DA algorithm, which we call sandwich DA (SDA) algorithm since it involves an extra move that is sandwiched between the two conditional draws of the DA algorithm. The SDA chain often converges much faster than the DA chain. In this paper we consider theoretical comparisons of DA and SDA algorithms. In particular, we prove that SDA is always as good as DA in terms of having smaller operator norm. If the Markov operator corresponding to the DA chain is compact and the extra move that is required in SDA is idempotent, which is often the case in practice, then the SDA is also compact and the spectrum of the SDA dominates that of the DA chain in the sense that all (ordered) eigenvalues of SDA are smaller than or equal to those corresponding eigenvalues of DA. We also present a necessary and sufficient condition that the extra move in SDA should satisfy for the operator norm of SDA to be strictly less than that of DA. We then consider some examples.