Convergence rate of Markov chain methods for genomic motif discovery
Abstract
We analyze the convergence rate of a simplified version of a popular Gibbs sampling method used for statistical discovery of gene regulatory binding motifs in DNA sequences. This sampler satisfies a very strong form of ergodicity (uniform). However, we show that, due to multimodality of the posterior distribution, the rate of convergence often decreases exponentially as a function of the length of the DNA sequence. Specifically, we show that this occurs whenever there is more than one true repeating pattern in the data. In practice there are typically multiple such patterns in biological data, the goal being to detect the most wellconserved and frequentlyoccurring of these. Our findings match empirical results, in which the motifdiscovery Gibbs sampler has exhibited such poor convergence that it is used only for finding modes of the posterior distribution (candidate motifs) rather than for obtaining samples from that distribution. Ours are some of the first meaningful bounds on the convergence rate of a Markov chain method for sampling from a multimodal posterior distribution, as a function of statistical quantities like the number of observations.
 Publication:

arXiv eprints
 Pub Date:
 March 2013
 arXiv:
 arXiv:1303.2814
 Bibcode:
 2013arXiv1303.2814W
 Keywords:

 Mathematics  Statistics Theory
 EPrint:
 Published in at http://dx.doi.org/10.1214/12AOS1075 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)