## Bernoulli

• Bernoulli
• Volume 17, Number 3 (2011), 987-1014.

### On nonlinear Markov chain Monte Carlo

#### Abstract

Let $\mathscr{P}(E)$ be the space of probability measures on a measurable space $(E,\mathcal{E})$. In this paper we introduce a class of nonlinear Markov chain Monte Carlo (MCMC) methods for simulating from a probability measure $\pi\in\mathscr{P}(E)$. Nonlinear Markov kernels (see [Feynman–Kac Formulae: Genealogical and Interacting Particle Systems with Applications (2004) Springer]) $K:\mathscr{P}(E)\times E\rightarrow\mathscr{P}(E)$ can be constructed to, in some sense, improve over MCMC methods. However, 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; our proof technique is via the Poisson equation and Foster–Lyapunov conditions. We investigate the performance of our approximations with some simulations.

#### Article information

Source
Bernoulli, Volume 17, Number 3 (2011), 987-1014.

Dates
First available in Project Euclid: 7 July 2011

https://projecteuclid.org/euclid.bj/1310042853

Digital Object Identifier
doi:10.3150/10-BEJ307

Mathematical Reviews number (MathSciNet)
MR2817614

Zentralblatt MATH identifier
1241.60037

#### Citation

Andrieu, Christophe; Jasra, Ajay; Doucet, Arnaud; Del Moral, Pierre. On nonlinear Markov chain Monte Carlo. Bernoulli 17 (2011), no. 3, 987--1014. doi:10.3150/10-BEJ307. https://projecteuclid.org/euclid.bj/1310042853

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