We are interested in populations in which the fitness of different genetic types fluctuates in time and space, driven by temporal and spatial fluctuations in the environment. For simplicity, our population is assumed to be composed of just two genetic types. Short bursts of selection acting in opposing directions drive to maintain both types at intermediate frequencies, while the fluctuations due to 'genetic drift' work to eliminate variation in the population.We consider first a population with no spatial structure, modelled by an adaptation of the Lambda (or generalised) Fleming-Viot process, and derive a stochastic differential equation as a scaling limit. This amounts to a limit result for a Lambda-Fleming-Viot process in a rapidly fluctuating random environment. We then extend to a population that is distributed across a spatial continuum, which we model through a modification of the spatial Lambda-Fleming-Viot process with selection. In this setting we show that the scaling limit is a stochastic partial differential equation. As is usual with spatially distributed populations, in dimensions greater than one, the 'genetic drift' disappears in the scaling limit, but here we retain some stochasticity due to the fluctuations in the environment, resulting in a stochastic p.d.e. driven by a noise that is white in time but coloured in space.We discuss the (rather limited) situations under which there is a duality with a system of branching and annihilating particles. We also write down a system of equations that captures the frequency of descendants of particular subsets of the population and use this same idea of 'tracers', which we learned from HALLATSCHEK and NELSON (2008, [23]) and DURRETT and FAN (2016, [13]), in numerical experiments with a closely related model based on the classical Moran model.

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We consider Markov chain Monte Carlo (MCMC) algorithms for Bayesian high‐dimensional regression with continuous shrinkage priors. A common challenge with these algorithms is the choice of the number of iterations to perform. This is critical when each iteration is expensive, as is the case when dealing with modern data sets, such as genome‐wide association studies with thousands of rows and up to hundreds of thousands of columns. We develop coupling techniques tailored to the setting of high‐dimensional regression with shrinkage priors, which enable practical, non‐asymptotic diagnostics of convergence without relying on traceplots or long‐run asymptotics. By establishing geometric drift and minorization conditions for the algorithm under consideration, we prove that the proposed couplings have finite expected meeting time. Focusing on a class of shrinkage priors which includes the ‘Horseshoe’, we empirically demonstrate the scalability of the proposed couplings. A highlight of our findings is that less than 1000 iterations can be enough for a Gibbs sampler to reach stationarity in a regression on 100,000 covariates. The numerical results also illustrate the impact of the prior on the computational efficiency of the coupling, and suggest the use of priors where the local precisions are Half‐t distributed with degree of freedom larger than one.

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