Multilevel rejection sampling for approximate Bayesian computation

Abstract

Likelihood-free methods, such as approximate Bayesian computation, are powerful tools for practical inference problems with intractable likelihood functions. Markov chain Monte Carlo and sequential Monte Carlo variants of approximate Bayesian computation can be effective techniques for sampling posterior distributions in an approximate Bayesian computation setting. However, without careful consideration of convergence criteria and selection of proposal kernels, such methods can lead to very biased inference or computationally inefficient sampling. In contrast, rejection sampling for approximate Bayesian computation, despite being computationally intensive, results in independent, identically distributed samples from the approximated posterior. An alternative method is proposed for the acceleration of likelihood-free Bayesian inference that applies multilevel Monte Carlo variance reduction techniques directly to rejection sampling. The resulting method retains the accuracy advantages of rejection sampling while significantly improving the computational efficiency.

Using experimental data and information criteria to guide model selection for reaction–diffusion problems in mathematical biology

“…As a result, we apply an ABC variant of a Markov chain Monte Carlo (MCMC) sampler (Marjoram et al, 2003) (see Appendix C). While we find that the ABC MCMC sampler works well for this problem, other advanced ABC-based Monte Carlo schemes are also possible, such as sequential Monte Carlo (Sisson et al, 2007) and multilevel Monte Carlo (Warne et al, 2018). Using the ABC MCMC sampler, the four parameter posterior marginal PDFs derived from Equation (20) are estimated using the same data and observation error as in Section 4.2.…”

Simulation and inference algorithms for stochastic biochemical reaction networks: from basic concepts to state-of-the-art

“…Such a benchmark would be significantly more computationally intensive than the comparison of Monte Carlo methods for the forwards problem (Figure 3). The computational statistics literature demonstrates that ABCMCMC [94], ABCSMC [11,121] and ABCMLMC [61,73,144] can be tuned to provide very competitive results on a given inference problem. However, a large number of trial computations are often required to achieve this tuning, or more complex adaptive schemes need to be exploited [30,111].…”

“…The formulation of ABCSMC using a sequence of discrepancy thresholds hints that MLMC ideas could also be applicable. Recently, a variety of MLMC methods for ABC have been proposed [61,73,144]. All of these approaches are similar in their application of the multilevel telescoping summation to compute expectations with respect to ABC posterior distributions,…”

“…,j )). This results in a coupled marginal pair (k ,j , k −1 ,j ) (i) since it is equivalent to using the inverse transform method to generate k Thus, the ABC multilevel Monte Carlo (ABCMLMC) [144] method proceeds though computing the Monte Carlo estimate to Equation (33),…”

“…However, additional bias is introduced by the approximately correlated ABC rejection sampler since the telescoping summation is not strictly satisfied. In practice, this affects the choice of the acceptance threshold sequence [144]. Table 1: Comparison of ABC methods for the mono-molecular inference problem.…”