Abstract
This chapter is devoted to describing the class of the adaptive rejection sampling (ARS) schemes. These (theoretically) universal methods are very efficient samplers that update the proposal density whenever a generated sample is rejected in the RS test. In this way, they can produce i.i.d. samples from the target with an increasing acceptance rate that can converge to 1. As a by-product, these techniques also generate a sequence of proposal pdfs converging to the true shape of the target density. Another advantage of the ARS samplers is that, when they can be applied, the user only has to select a set of initial conditions. After the initialization, they are completely automatic, self-tuning algorithms (i.e., no parameters need to be adjusted by the user) regardless of the specific target density. However, the need to construct a suitable sequence of proposal densities restricts the practical applicability of this methodology. As a consequence, ARS schemes are often tailored to specific classes of target distributions. Indeed, the construction of the proposal is particularly hard in multidimensional spaces. Hence, ARS algorithms are usually designed only for drawing from univariate densities.
In this chapter we discuss the basic adaptive structure shared by all ARS algorithms. Then we look into the performance of the method, characterized by the acceptance probability (which increases as the proposal is adapted), and describe various extensions of the standard ARS approach which are aimed either at improving the efficiency of the method or at covering a broader class of target pdfs. Finally, we consider a hybrid method that combines the ARS and Metropolis-Hastings schemes.
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Notes
- 1.
We assign a name to the function V (x) to ease the treatment, so that we can refer directly to it. The name potential function recalls a physical interpretation. In statistical mechanics, for instance, the potential energy function V is central to the evaluation of many thermodynamic properties, where V is used as log-density [31]. To be specific, the estimation of the thermodynamic properties demands the computation of integrals involving the function \(\exp {(-V)}\). This interpretation of the log-pdf as a potential is also evoked explicitly in the Hamiltonian MCMC techniques [32].
- 2.
Note that with simple inspections it is always possible to know the interval including the mode (for instance, considering the signs of the slopes of the secant lines passing through the support points).
- 3.
- 4.
The case when the set of simple estimates is empty is similar to the case μ →±∞, hence the construction of the linear functions is also similar, but it needs a special care. For further details, see [23].
- 5.
Note that \([\bar {\pi }_t(x)- p(x)]\geq 0\) for all \(x\in \mathcal {D}\).
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Martino, L., Luengo, D., Míguez, J. (2018). Adaptive Rejection Sampling Methods. In: Independent Random Sampling Methods. Statistics and Computing. Springer, Cham. https://doi.org/10.1007/978-3-319-72634-2_4
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