The sequential importance sampling (SIS) algorithm has gained considerable popularity for its empirical success. One of its noted applications is to the binary contingency tables problem, an important problem in statistics, where the goal is to estimate the number of 0/1 matrices with prescribed row and column sums. We give a family of examples in which the… (More)
Markov chain Monte Carlo (MCMC) algorithms play a critical role in the Bayesian approach to phylogenetic inference. We present a theoretical analysis of the rate of convergence of many of the widely used Markov chains. For N characters generated from a uniform mixture of two trees, we prove that the Markov chains take an exponentially long (in N) number of… (More)
Transitions at CpG dinucleotides, referred to as "CpG substitutions", are a major mutational input into vertebrate genomes and a leading cause of human genetic disease. The prevalence of CpG substitutions is due to their mutational origin, which is dependent on DNA methylation. In comparison, other single nucleotide substitutions (for example those… (More)
Using data from primates, we show that molecular clocks in sites that have been part of a CpG dinucleotide in recent past (CpG sites) and non-CpG sites are of markedly different nature, reflecting differences in their molecular origins. Notably, single nucleotide substitutions at non-CpG sites show clear generation-time dependency, indicating that most of… (More)
Different genes often have different phylogenetic histories. Even within regions having the same phylogenetic history, the mutation rates often vary. We investigate the prospects of phylogenetic reconstruction when all the characters are generated from the same tree topology, but the branch lengths vary (with possibly different tree shapes). Furthering work… (More)
In recent years there has been considerable progress on the analysis of Markov chains for generating a random coloring of an input graph. These improvements have come in conjunction with refinements of the coupling technique, which is a classical tool in probability theory. We survey results on generating random colorings, and related technical improvements.
In this lecture, we will introduce Markov chains and show a potential algorithmic use of Markov chains for sampling from complex distributions. For a finite state space Ω, we say a sequence of random variables (X t) on Ω is a Markov chain if the sequence is Markovian in the following sense, for all t, We consider transitions which are independent of the… (More)
Improved inapproximability results for counting independent sets in the hard-core model. J24. DanielŠtefankovič, Eric Vigoda. Fast convergence of MCMC algorithms for phylogenetic reconstruction with homogeneous data on closely related species.
Research Interests • Computational complexity. Exact and approximate counting algorithms and hardness.