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Optimal scaling of random walk Metropolis algorithms with discontinuous target densities. (English) Zbl 1259.60082

The authors discuss the optimal scaling of random walk Metropolis (RWM) Markov chain Monte Carlo algorithms in higher dimensions for target distributions with discontinuous densities confined to the \(d\)-dimensional hypercube \([0,1]^d\). In particular, the authors are interested in i.i.d. product densities of the form \[ \pi_d(x^d)=\prod_{i=1}^d f(x_i^d)\quad \text{with}\quad f(x)\propto \exp(g(x))1_{[0,1]}(x),\;x\in\mathbb{R}, \] where \(g\) is twice differentiable on \([0,1]\) with bounded first derivative. To approximate the distribution the following RWM algorithm is considered.
Draw \(X_0^d\) from \(\pi^d\) and for \(i,t=1,\dots\) let \(Z_{ti}\) be i.i.d. according to \(U[-1,1]\), and set \(Z^d_t=(Z_{t1},\dotsc,Z_{td})\). Then propose \(X_{t+1}^d=X_t^d+\sigma^d Z_t^d\) with probability \(1\wedge \pi^d(X_{t+1}^d)/\pi^d(X_t^d)\) and \(X_{t+1}^d=X_t^d\) otherwise, where \(\sigma_d=l/d\) and \(l>0\) is a free parameter.
The main result of the paper states that the appropriately scaled first component of the chain \(X^d\) converges weakly for \(d\to\infty\) suitably chained to the solution of an reflected Ornstein-Uhlenbeck process. The speed of the mixing is of order \(d^2\). Moreover, it is shown that the average optimal acceptance rate is given by \(\exp(-2)\). In a later section of the paper, the authors discuss extensions of the result to target distributions with marginal densities supported on the whole positive axis as well as targets beyond the i.i.d. product structure. On a technical level the result is similar to results for targets with continuous densities, however, the methods of proof differ substantially as is discussed thoroughly by the authors in the introduction.

MSC:

60J22 Computational methods in Markov chains
60F05 Central limit and other weak theorems
60J05 Discrete-time Markov processes on general state spaces
65C05 Monte Carlo methods
65C40 Numerical analysis or methods applied to Markov chains
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References:

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