conditional process

Zhao L, Lascoux M & Waxman D 2014 Exact simulation of conditioned Wright-Fisher models. J Theor Biol 363:419-426.

  • less attention was initially paid to soft sweeps – where selection acts on standing variation rather than on new mutations
  • the focus on hard sweeps was a consequence of the availability of tests to detect them rather than their likelihood of occurrence
  • in many organisms, for example humans, hard sweeps may be rare
  • soft sweeps may, collectively, be the more common phenomenon
  • a renewed interest has arisen in modelling and developing tests to detect soft [s]weeps
  • typically, modelling soft sweeps involves simulating the frequency trajectories of a Wright-Fisher model (Fisher, 1930 and Wright, 1931) – when conditioned on the particular starting and final allele frequencies of a given time interval
  • other reasons to be interested in these conditioned trajectories come from inferences of selection from allele frequency time-series data
  • another area, where simulated frequency trajectories are useful, is the coalescent process with selection
  • when the final state of the population was specified (conditioned), the resulting random process was shown to have the same mathematical form as an unconditioned process – but with an additional selective force
  • a conditioned Markov chain is equivalent to an unconditioned Markov chain with non-trivially modified transition probabilities
  • the only method for the generation of conditioned continuous state/continuous time trajectories is based on trajectory rejection
  • the work of Schraiber et al. (2013) uses a non-linear change of variables combined with Girsanov's theorem
  • it is by no means obvious how to extend this methodology to more complex/higher dimensional problems
  • K[f, t](t + 1|u) = Weff(t)K[f, t](t|u) ... (5)
  • we call Weff(t) the effective transition matrix
  • it can be explicitly determined and is given by
  • Weffm, n(t) = Kf, m(T|t + 1)Wm, n(t) / Kf, n(T|t) ... (6)
  • in the important case where the underlying transition matrix, W(t), is independent of time (W(t) = W) the effective transition matrix still depends on time
  • Weffm, n(t) = [WT − t − 1]f, mWm, n / [WT − t]f, n (W independent of t) ... (7)