eral respects. While Direct Sampling focuses on the shape of the data likelihood alone, we are concerned with the characteristics of the entire posterior density. Our method bypasses the need for Bernstein polynomial approximations, which are integral to the Direct Sampling algorithm. Finally, while Direct Sampling takes proposal draws from the prior (which may conflict with the data), our method samples proposals from a separate density that is ideally a good approximation to the target posterior density itself. 

## 2.4.3 Markov chain Monte Carlo 

We have already mentioned the key advantage of our method over traditional MCMC: generating independent samples that can be collected in parallel. We do not need to be concerned with issues like autocorrelation, convergence of estimation chains, and so forth. Without delving into a discussion of all possible variations and improvements to MCMC that have been proposed in the last few decades, there have been some attempts to parallelize MCMC that deserve some mention. For a deeper analysis, see Suchard et al. (2010). 

It is possible to run multiple independent MCMC chains that start from different starting points. Once all of those chains have converged to the posterior distribution, we can estimate the posterior by combining samples from all of the chains. The numerical efficiency of that approxi- mation should be higher than if we used only one chain, because there should be no correlation between samples collected in different chains. However, each chain still needs to converge to the posterior independently, and only after that convergence could we start collecting samples. If it takes a long time for one chain to converge, it will take at least that long for all chains to converge. Thus, the potential for parallelization is much greater for our method than for MCMC. 

Another approach to parallelization is to exploit parallel processing power for individual steps in an algorithm. One example is a parallel implementation of a multivariate slice sampler (MSS), as in Tibbits et al. (2010). The benefits of parallelizing the MSS come from parallel evaluation of the target density at each of the vertices of the slice, and from more efficient use of resources to execute linear algebra operations (e.g, Cholesky decompositions). But the MSS itself remains a Markovian algorithm, and thus will still generate dependent draws. Using parallel technology to generate a single draw from a distribution is not the same as generating all of the required draws 

