Adaptive Metropolis

There have been several arXived entries on adaptive MCM on the past days. One is an adaptive extension to the recent Read Paper by Christophe Andrieu, Arnaud Doucet and Roman Holenstein, Particle Markov chain Monte Carlo where Silva, Giordani, Kohn and Pitt manage to use an adapted mixture of normals as their proposal within non-linear state-space models. They also obtain unbiased estimators of the likelihood, which may have an appeal in ABC settings! To see this extension appearing a few weeks after the original paper is amazing as well. A second paper by Matti Vihola considers the impact of removing the stabilising term in the Haario-Saaksman-Tamminen original paper

S_n = \widehat \Sigma_n + \varepsilon I

on the convergence of the corresponding adaptative Metropolis algorithm. The change is in using instead a stochastic approximation update

S_{n+1} = (1-\eta_n) S_n + \eta_n (x_{n+1}-\hat\mu_n)^\text{T}(x_{n+1}-\hat\mu_n)

where \eta_n decreases to zero at a proper speed and \hat\mu_n is the empirical mean updated the same way. The paper is highly technical but shows the almost sure explosion of the resulting sequence under a flat target, an ergodic for a double Laplace target and a unimodal proposal, and a more general version under assumptions on the target and for a proposal suggested by Gareth Roberts and Jeff Rosenthal (2009)

q(z) = (1-\beta) \varphi_{S_n}(z) + \beta q_0(z)

which is akin to a renewal process in that the static q_0 part is not adaptative and thus regulates the behaviour of the whole chain. At last, Yves Atachadé and Gersende Fort posted the second half of their paper on limit theorems for some adaptive MCMC algorithms with subgeometric kernels, yet another fairly technical work that relates to Andrieu and Moulines (2006) and Saaksman and Vihola (2008). The adaptivity is controlled by retroprojections and contains as a special case stochastic approximation schemes, the main assumptions being a drift condition on the core kernel

P_\theta V(x) = V(x) -c V(x)^{1-\alpha}(x) +b

and a diminishing adaptation condition common to all adaptive MCMC papers.

3 Responses to “Adaptive Metropolis”

  1. […] to stochastic approximation methods, making the link to recent works by Christophe Andrieu, Heikki Haario, Faming Liang, Eric Moulines, Enro Saksman, and co-authors. Martin and Ghosh also reinterpret […]

  2. Andrew Gelman pointed out at a paper of his, with Cristian Pasaricà, to appear in Statistica Sinica, about a new scaling adaptive strategy for random walk Metropolis samplers.

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