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On nonlinear Markov chain Monte Carlo. (English) Zbl 1241.60037

Markov chain Monte Carlo (MCMC) algorithms are developed for simulating from complicated distributions, for example, when the target distribution has multiple modes and/or possesses strong dependencies between subcomponents of the state space. In the article, nonlinear kernels of the form \[ K_{\mu}(x,dy)=(1-\varepsilon)K(x,dy)+\varepsilon\Phi(\mu)(dy) \] are introduced, where \(K(x,dy)\) is the original kernel with complicated target distribution and \(\epsilon\Phi(\mu)(dy)\) is added in order to improve algorithmic performance.
Such nonlinear kernels cannot be simulated exactly, so approximations of the nonlinear kernels are constructed using auxiliary or potentially self-interacting chains. Several nonlinear kernels are presented, and it is demonstrated that, under some conditions, the associated approximations exhibit a strong law of large numbers; the proof technique uses the Poisson equation and Foster-Lyapunov conditions. The performance of the approximations is investigated with some simulations.

MSC:

60J22 Computational methods in Markov chains
65C05 Monte Carlo methods
65C40 Numerical analysis or methods applied to Markov chains
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