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That's a good question. If your distribution isn't multimodal but still suffers from having its mass concentrated in one place, and if the variable you're integ
by murbard2 10y ago
That's a good question. If your distribution isn't multimodal but still suffers from having its mass concentrated in one place, and if the variable you're integrating isn't fat-tailed, then MCMC will help you by sticking around the right area once it finds it.
- yummyfajitas 10y agoI don't even think that's true. Consider a distribution in R^2 that is concentrated on [0,1] x [0,100]. If you have an O(1) jump size it'll take time O(100^2) for the MCMC to fully sample the support of this distribution. If you had a jump size of O(100), then you'll be rejecting 99% of your jumps due to the narrowness in the x-direction. This is hardly an uncommon scenario.
- murbard2 10y agoIt's easy to design examples that trip MCMC, but they also trip straight Monte Carlo integration even more. You can, of course, construct examples where MCMC will do worse than simple Monte Carlo integration, but these are uncommon. They mostly illustrate the difficulty of picking an appropriate jump size.
- yummyfajitas 10y agoIf you directly draw samples from the underlying distribution, you don't have to worry at all about whether the markov chain or other process (e.g. HMC) is fully sampling the distribution. MC has no rejections and always samples the entire distribution. You simply don't need to worry about trajectories not going everywhere they should.
- deleted 10y ago[deleted]