Sampling Techniques: Estimating Evolutionary Rates and Generating Molecular Structures

This chapter first discusses the general problem of characterizing ensemble properties of biological systems through integration by sampling, along with the difficulties of sampling in high-dimensional space. It briefly examines the approach of rejection sampling, and discusses two general frameworks in Monte Carlo sampling, namely, the Markov chain Monte Carlo (MCMC) or Metropolis Monte Carlo method and the sequential Monte Carlo method. The chapter also discusses basic concepts such as sampling from a desired target distribution, properties of Markov chains, time reversibility, detailed balance, and the stationary state. This is followed by the example of estimating evolutionary substitution rates of amino acids. The chapter then describes the general principle of importance sampling, the approach of sequentially building up the target distribution, and the technique of resampling for variance reduction. Finally, it presents the applications in generating self-avoiding walks for studying chain polymers and calculating RNA loop entropy.