## what does more efficient Monte Carlo mean?

“I was just thinking that there might be a magic trick to simulate directly from this distribution without having to go for less efficient methods.”

In a simple question on X validated a few days ago [about simulating from x²φ(x)] popped up the remark that the person asking the question wanted a direct simulation method for higher efficiency. Compared with an accept-reject solution. Which shows a misunderstanding of what “efficiency” means on Monte Carlo situations. If it means anything, I would think it is reflected in the average time taken to return one simulation and possibly in the worst case. But there is no reason to call an inverse cdf method more efficient than an accept reject or a transform approach since it all depends on the time it takes to make the inversion compared with the other solutions… Since inverting the closed-form cdf in this example is much more expensive than generating a Gamma(½,½), and taking plus or minus its root, this is certainly the case here. Maybe a ziggurat method could be devised, especially since x²φ(x)<φ(x) when |x|≤1, but I am not sure it is worth the effort!

March 18, 2017 at 3:03 am

[…] What does more efficient Monte Carlo mean? […]

March 17, 2017 at 12:54 am

So this person’s questions really isn’t that interesting, but as you allude to, there it is the big computational (Bayesian) question of our time. Can we tell before hand when method x will be more efficient (in terms of time, or more mathematically flops) than method y. It makes me wonder if we can’t rope in some algebraists to actually study this (like they did when differential algebra identified which integrals can be expressed by elementary functions).