Markov chain quasi-Monte Carlo

“It is known that Tausworthe generators can be viewed as polynomial Korobov lattice point sets with a denominator polynomial p(x) and a numerator polynomial q(x) over IF₂“

A recently arXived paper by Shin Harase, “A table of short-period Tausworthe generators for Markov chain quasi-Monte Carlo”, discusses the use of [quasi-Monte Carlo] Tausworthe generators rather than iid uniform sampling. As shown by Owen and Tribble, it is indeed legit to replace a sequence of iid (pseudo-random) uniforms with its quasi-Monte Carlo (qMC) version if the sequence keeps a sufficient degree of uniformity. The current paper optimises the parameters of the Tausworthe generators in terms of the t-value of the generator, an indicator of the uniform occupancy of the qMC sequence.

For a range of values of m, if 2^m-1 is the period of the pseudo-random generator, the author obtains the optimal weights in the Tausworthe generator, which is a linear feedback shift register generator over {0,1}, ie shifting all the bits of the current uniform realisation by linear combination modulo 2. The comparison with other qMC and MC is provided on a Gibbs sampler for a bidimensional Gaussian target, which presents the advantage of requiring exactly one uniform per simulation and the disadvantage of … requiring exactly one uniform per simulation! Since this is harder to envision for simulation methods requiring a random number of uniforms.

Regarding the complexity of the approach, I do not see any gap between using these Tausworthe generators and something like the Mersenne generator. I just wonder at the choice of m, that is, whether or not it makes sense to pick any value lower than 2³² for the period.

Congruential generators all are RANDUs!

In case you did not read all the slides of Regis Lebrun’s talk on pseudo-random generators I posted yesterday, one result from Marsaglia’s (in a 1968 PNAS paper) exhibited my ignorance during Regis’ Big’ MC seminar on Thursday. Marsaglia indeed showed that all multiplicative congruential generators

lie on a series of hyperplanes whose number gets ridiculously small as the dimension d increases! If you turn the ‘s into uniforms and look at the d dimensional vectors

they are on a small number of hyperplanes, at most , which gives 41 hyperplanes when … So in this sense all generators share the same poor property as the infamous RANDU which is such that that is always over one of 16 hyperplanes, an exercise we use in both Introducing Monte Carlo Methods with R and Monte Carlo Statistical Methods (but not in our general audience out solution manual). I almost objected to the general result being irrelevant as the ‘s share ‘s, but of course the subsequence also share enjoys this property!