Adaptive rejection Metropolis sampling using Lagrange interpolation polynomials of degree 2

Department of Statistics, University of Auckland, Private Bag 92019, Auckland, New Zealand
Computational Statistics & Data Analysis (Impact Factor: 1.4). 03/2008; 52(7):3408-3423. DOI: 10.1016/j.csda.2008.01.005
Source: DBLP


A crucial problem in Bayesian posterior computation is efficient sampling from a univariate distribution, e.g. a full conditional distribution in applications of the Gibbs sampler. This full conditional distribution is usually non-conjugate, algebraically complex and computationally expensive to evaluate. We propose an alternative algorithm, called ARMS2, to the widely used adaptive rejection sampling technique ARS [Gilks, W.R., Wild, P., 1992. Adaptive rejection sampling for Gibbs sampling. Applied Statistics 41 (2), 337–348; Gilks, W.R., 1992. Derivative-free adaptive rejection sampling for Gibbs sampling. In: Bernardo, J.M., Berger, J.O., Dawid, A.P., Smith, A.F.M. (Eds.), Bayesian Statistics, Vol. 4. Clarendon, Oxford, pp. 641–649] for generating a sample from univariate log-concave densities. Whereas ARS is based on sampling from piecewise exponentials, the new algorithm uses truncated normal distributions and makes use of a clever auxiliary variable technique [Damien, P., Walker, S.G., 2001. Sampling truncated normal, beta, and gamma densities. Journal of Computational and Graphical Statistics 10 (2) 206–215]. Furthermore, we extend this algorithm to deal with non-log-concave densities to provide an enhanced alternative to adaptive rejection Metropolis sampling, ARMS [Gilks, W.R., Best, N.G., Tan, K.K.C., 1995. Adaptive rejection Metropolis sampling within Gibbs sampling. Applied Statistics 44, 455–472]. The performance of ARMS and ARMS2 is compared in simulations of standard univariate distributions as well as in Gibbs sampling of a Bayesian hierarchical state-space model used for fisheries stock assessment.

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    • "Iterations (while n < N ): 2. Build a proposal, πt(x), given the set St = {s 1 , . . . , sm t }, using a convenient procedure (e.g. the ones described in [28], [30] or the simpler ones proposed in Section III-C). 3. Draw x ∼ e πt(x) ∝ πt(x) and u ∼ U ([0] [1]). "

    Full-text · Dataset · Jul 2015
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    • "(a) illustrates an example of this construction. More sophisticated approaches to build W t (x) (e.g., using quadratic segments when possible [12] "
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    ABSTRACT: Adaptive Rejection Metropolis Sampling (ARMS) is a well-known MCMC scheme for generating samples from one-dimensional target distributions. ARMS is widely used within Gibbs sampling, where automatic and fast samplers are of-ten needed to draw from univariate full-conditional densities. In this work, we propose an alternative adaptive algorithm (IA 2 RMS) that overcomes the main drawback of ARMS (an uncomplete adaptation of the proposal in some cases), speed-ing up the convergence of the chain to the target. Numerical results show that IA 2 RMS outperforms the standard ARMS, providing a correlation among samples close to zero. Index Terms— Monte Carlo methods, Gibbs sampler, adaptive rejection Metropolis sampling (ARMS).
    Full-text · Conference Paper · Jan 2014
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    • "Using the method described in Section 3.4, we did check to see if any inflection points were missed. In this case, we set C sub = [0] [7] /B 1 and observed that |l (x)| < 71 ∀ x ∈ C rmnd . In each of the final samples generated, it was the case that min x j ∈C rmnd {|l (x j )|} > 71 × max x j ,x j−1 ∈C rmnd (|x j − x j−1 |) , and so no inflection points were overlooked. "
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    ABSTRACT: The need to simulate from a univariate density arises in several settings, particularly in Bayesian analysis. An especially efficient algorithm which can be used to sample from a univariate density, f X , is the adaptive accept-reject algorithm. To implement the adaptive accept-reject algorithm, the user has to envelope T • f X , where T is some transformation such that the density g(x) ∝ T −1 (α + βx) is easy to sample from. Successfully enveloping T • f X , however, requires that the user identify the number and location of T • fX 's inflection points. This is not always a trivial task. In this paper we propose an adaptive accept-reject algorithm which relieves the user of precisely identifying the location of T • f X 's inflection points. This new algorithm is shown to be efficient and can be used to sample from any density such that its support is bounded and its log is three-times differentiable.
    Preview · Article · Feb 2010 · Journal of Statistical Computation and Simulation
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