As a coincidence, I received my copy of JRSS Series B with the Read Paper by Mathieu Gerber and Nicolas Chopin on sequential quasi Monte Carlo just as I was preparing an arXival of a few discussions on the paper! Among the [numerous and diverse] discussions, a few were of particular interest to me [I highlighted members of the University of Warwick and of Université Paris-Dauphine to suggest potential biases!]:
- Mike Pitt (Warwick), Murray Pollock et al. (Warwick) and Finke et al. (Warwick) all suggested combining quasi Monte Carlo with pseudomarginal Metropolis-Hastings, pMCMC (Pitt) and Rao-Bklackwellisation (Finke et al.);
- Arnaud Doucet pointed out that John Skilling had used the Hilbert (ordering) curve in a 2004 paper;
- Chris Oates, Dan Simpson and Mark Girolami (Warwick) suggested combining quasi Monte Carlo with their functional control variate idea;
- Richard Everitt wondered about the dimension barrier of d=6 and about possible slice extensions;
- Zhijian He and Art Owen pointed out simple solutions to handle a random number of uniforms (for simulating each step in sequential Monte Carlo), namely to start with quasi Monte Carlo and end up with regular Monte Carlo, in an hybrid manner;
- Hans Künsch points out the connection with systematic resampling à la Carpenter, Clifford and Fearnhead (1999) and wonders about separating the impact of quasi Monte Carlo between resampling and propagating [which vaguely links to one of my comments];
- Pierre L’Ecuyer points out a possible improvement over the Hilbert curve by a preliminary sorting;
- Frederik Lindsten and Sumeet Singh propose using ABC to extend the backward smoother to intractable cases [but still with a fixed number of uniforms to use at each step], as well as Mateu and Ryder (Paris-Dauphine) for a more general class of intractable models;
- Omiros Papaspiliopoulos wonders at the possibility of a quasi Markov chain with “low discrepancy paths”;
- Daniel Rudolf suggest linking the error rate of sequential quasi Monte Carlo with the bounds of Vapnik and Ĉervonenkis (1977).
The arXiv document also includes the discussions by Julyan Arbel and Igor Prünster (Turino) on the Bayesian nonparametric side of sqMC and by Robin Ryder (Dauphine) on the potential of sqMC for ABC.