Adaptive-rejection-sampling

Gilks, W. R. and Wild, P. (1992) Adaptive rejection sampling for Gibbs sampling. 

ADAPTIVE-REJECTION-SAMPLING FROM A LOG-CONCAVE DISTRIBUTION 35 (Fourth column)... 

Adaptive-rejection-sampling for multiple parameters would be based on the idea of bounding the logarithm of a log-concave target by tangent hyperplanes. While this would work in theory, the boundaries of the envelope function would become extremely complicated. In practice, adaptive-rejection-sampling only works for a single dimension parameter. 

Adaptive-rejection-sampling is a method for finding a random draw from a unsealed posterior (taiget) distribution that is log-concave. Log-concave means that the second derivative of its logarithm is always non-positive. 

Adaptive-rejection-sampling would be extremely complicated for multiple parameters, so it is used only for single parameters. 

The main use of acceptance-rejection-sampling and adaptive-rejection-sampling will be to sample single parameters as part of a larger Markov-chain Monte Carlo process. 

If the node is a single parameter and its conditional distribution is log-concave, we can draw an observation from the conditional distribution using the adaptive rejection sampling algorithm described in Chapter 2. Generally it takes only a few steps before we get an accepted draw from the conditional distribution, since we are tightening the candidate density with every unaccepted draw. 

We can show that the full conditional distributions in (2.29) through (2.31) are log-concave in each of these parameters. Therefore, we can use the adaptive rejection algorithm of Gilks and Wild (1992) or the localized Metropolis algorithm discussed in Chen et al. (2000) to sample (y, 0, Oq). 

Among stochastic simulation methods, one can distinguish the flexibihty of adaptive rejection Metropolis sampling - ARMS (see Gilks et al. (1996)) and Griddy-Gibbs sampler - GGS (see Ritter Taimer (1992)). These approaches allows for handling FCDs expensive to be sampled from, the so-caUed nonstandard FCDs. 

Meyer, R., Cai, B. Perron, F. 2008. Adaptive rejection Metropolis sampling using Lagrange interpolation polynomials of degree 2. Computational Statistics and Data Analysis, 52(7) 3408-3423. 

In other cases, we only know the proportional form of the full conditional density for that node. We would use one of the direct methods from Chapter 2 such as acceptance-rejection-sampling or adaptive-iejection-sampling to sample from that node. 

Figure 9.13. FT-Raman spectra of a mildly fluorescent, impure sample of ortho dinitrobenzene before (A) and after (B) correction for instrumental response. Modulation at A is caused by the laser rejection filter. (Adapted from Reference 4, p. 104.)...

Actually, since 4% is the miniTniiTn concentration required by law, not the average concentration, we must adapt our calculations to this fact. A simple way of doing this is to take 4% as the lower limit of an interval about the mean. Assuming that a s 0.15%, which is the standard deviation of all the vedues in Table 2.1, and adding three times this quantity to 4%, we obtain 4.45% as a new mean to be tested. If /z = 4.45%, then 99.87% of the observations are expected to occur above 4% (because this limit is now three standard deviations below the mean). Now, we only need to repeat the test taking 4.45% as the new mean. If we cannot reject this hypothesis, 99.87% of the samples are expected to have a concentration of acetic acid of 4% or more, and therefore be in compliance with the law. 

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