Another choice we are faced with is this—when do we stop this process? For most applications having, say, 1000 samples to use as an approximation to the true distribution is more than enough. The real question is deciding when our sampling has converged on the real distribution. There's a couple of tricks that I'll mention briefly:

  • You generally "burn in" the process by discarding some number of samples at the start, when the distribution is still heavily influenced by the starting point.
  • You can assess how things are going by looking at how many proposals are being accepted. It's common to adjust your proposal distribution to aim for a specific target probability, say 30%.1
  • Generally, you start in a bunch of different places, again to minimize the impact of where you started. This also lets you assess convergence, as we'll see shortly, because you can see if the different runs of the algorithm all produce similar answers.
  1. There is a lot of literature on this topic. As a starting point, you might look at Bédard 2008: https://doi.org/10.1016/j.spa.2007.12.005. ↩

Another choice we are faced with is this—when do we stop this process? For most applications having, say, 1000 samples to use as an approximation to the true distribution is more than enough. The real question is deciding when our sampling has converged on the real distribution. There's a couple of tricks that I'll mention briefly:

  • You generally "burn in" the process by discarding some number of samples at the start, when the distribution is still heavily influenced by the starting point.
  • You can assess how things are going by looking at how many proposals are being accepted. It's common to adjust your proposal distribution to aim for a specific target probability, say 30%.1
  • Generally, you start in a bunch of different places, again to minimize the impact of where you started. This also lets you assess convergence, as we'll see shortly, because you can see if the different runs of the algorithm all produce similar answers.
  1. There is a lot of literature on this topic. As a starting point, you might look at Bédard 2008: https://doi.org/10.1016/j.spa.2007.12.005. ↩