why is this algorithm simulating a Normal variate?

A backward question from X validated as to why the above is a valid Normal generator based on exponential generations. Which can be found in most textbooks (if not ours). And in The Bible, albeit as an exercise. The validation proceeds from the (standard) Exponential density dominating the (standard) Normal density and, according to Devroye, may have originated from von Neumann himself. But with a brilliant reverse engineering resolution by W. Huber on X validated. While a neat exercise, it requires on average 2.64 Uniform generations per Normal generation, against a 1/1 ratio for Box-Muller (1958) polar approach, or 1/0.86 for the Marsaglia-Bray (1964) composition-rejection method. The apex of the simulation jungle is however Marsaglia and Tsang (2000) ziggurat algorithm. At least on CPUs since, Note however that “The ziggurat algorithm gives a more efficient method for scalar processors (e.g. old CPUs), while the Box–Muller transform is superior for processors with vector units (e.g. GPUs or modern CPUs)” according to Wikipedia.

To draw a comparison between this Normal generator (that I will consider as von Neumann’s) and the Box-Müller polar generator,

#Box-Müller
bm=function(N){
  a=sqrt(-2*log(runif(N/2)))
  b=2*pi*runif(N/2)
  return(c(a*sin(b),a*cos(b)))
}

#vonNeumann
vn=function(N){
  u=-log(runif(2.64*N))
  v=-2*log(runif(2.64*N))>(u-1)^2
  w=(runif(2.64*N)<.5)-2
  return((w*u)[v])
}

here are the relative computing times

> system.time(bm(1e8))
utilisateur     système      écoulé 
     7.015       0.649       7.674 
> system.time(vn(1e8))
utilisateur     système      écoulé 
     42.483       5.713      48.222 

normal variates in Metropolis step

A definitely puzzled participant on X validated, confusing the Normal variate or variable used in the random walk Metropolis-Hastings step with its Normal density… It took some cumulated efforts to point out the distinction. Especially as the originator of the question had a rather strong a priori about his or her background:

“I take issue with your assumption that advice on the Metropolis Algorithm is useless to me because of my ignorance of variates. I am currently taking an experimental course on Bayesian data inference and I’m enjoying it very much, i believe i have a relatively good understanding of the algorithm, but i was unclear about this specific.”

despite pondering the meaning of the call to rnorm(1)… I will keep this question in store to use in class when I teach Metropolis-Hastings in a couple of weeks.

simulation by hand

A rather weird question on X validated this week was about devising a manual way to simulate (a few) normal variates. By manual I presume the author of the question means without resorting to a computer or any other business machine. Now, I do not know of any real phenomenon that is exactly and provably Normal. As analysed in a great philosophy of science paper by Aidan Lyon, the standard explanations for a real phenomenon to be Normal are almost invariably false, even those invoking the Central Limit Theorem. Hence I cannot think of a mechanical device that would directly return Normal generations from a Normal distribution with known parameters. However, since it is possible to simulate by hand Uniform U(0,1) variates [up to a given precision] using a chronometre or a wheel, calls to versions of the Box-Müller algorithm that do not rely on logarithmic or trigonometric functions are feasible, for instance by generating two Exponential variates, x and y, until 2y>(1-x)², x being the output. And generating Exponential variates is easy provided a radioactive material with known half-life is available, along with a Geiger counter. Or, if not, by calling von Neumann’s exponential generator. As detailed in Devroye’s simulation book.

After proposing this solution, I received a comment from the author of the question towards a simpler solution based, e.g., on the Central Limit Theorem. Presumably for simple iid random variables such as coin tosses or dice experiments. While I used the CLT for simulating Normal variables in my very early days [just after programming on punched cards!], I do not think this is a very good or efficient method, as the tails grow very slowly to normality. By comparison, using the same amount of coin tosses to create a sufficient number of binary digits of a Uniform variate produces a computer-precision exact Uniform variate, which can be exploited in Box-Müller-like algorithms to return exact Normal variates… Even by hand if necessary. [For some reason, this question attracted a lot of traffic and an encyclopaedic answer on X validated, despite being borderline to the point of being proposed for closure.]