the buzz about nuzz

“…expensive in these terms, as for each root, Λ(x(s),v) (at the cost of one epoch) has to be evaluated for each root finding iteration, for each node of the numerical integral“

When using the ZigZag sampler, the main (?) difficulty is in producing velocity switch as the switches are produced as interarrival times of an inhomogeneous Poisson process. When the rate of this process cannot be integrated out in an analytical manner, the only generic approach I know is in using Poisson thinning, obtained by finding an integrable upper bound on this rate, generating from this new process and subsampling. Finding the bound is however far from straightforward and may anyway result in an inefficient sampler. This new paper by Simon Cotter, Thomas House and Filippo Pagani makes several proposals to simplify this simulation, Nuzz standing for numerical ZigZag. Even better (!), their approach is based on what they call the Sellke construction, with Tom Sellke being a probabilist and statistician at Purdue University (trivia: whom I met when spending a postdoctoral year there in 1987-1988) who also wrote a fundamental paper on the opposition between Bayes factors and p-values with Jim Berger.

“We chose as a measure of algorithm performance the largest Kolmogorov-Smirnov (KS) distance between the MCMC sample and true distribution amongst all the marginal distributions.”

The practical trick is rather straightforward in that it sums up as the exponentiation of the inverse cdf method, completed with a numerical resolution of the inversion. Based on the QAGS (Quadrature Adaptive Gauss-Kronrod Singularities) integration routine. In order to save time Kingman’s superposition trick only requires one inversion rather than d, the dimension of the variable of interest. This nuzzled version of ZIgZag can furthermore be interpreted as a PDMP per se. Except that it retains a numerical error, whose impact on convergence is analysed in the paper. In terms of Wasserstein distance between the invariant measures. The paper concludes with a numerical comparison between Nuzz and random walk Metropolis-Hastings, HMC, and manifold MALA, using the number of evaluations of the likelihood as a measure of time requirement. Tuning for Nuzz is described, but not for the competition. Rather dramatically the Nuzz algorithm performs worse than this competition when counting one epoch for each likelihood computation and better when counting one epoch for each integral inversion. Which amounts to perfect inversion, unsurprisingly. As a final remark, all models are more or less Normal, with very smooth level sets, maybe not an ideal range

 

reading classics (#10 and #10bis)

Today’s classics seminar was rather special as two students were scheduled to talk. It was even more special as both students had picked (without informing me) the very same article by Berger and Sellke (1987), Testing a point-null hypothesis: the irreconcilability of p-values and evidence, on the (deep?) discrepancies between frequentist p-values and Bayesian posterior probabilities. In connection with the Lindley-Jeffreys paradox. Here are Amira Mziou’s slides:

and Jiahuan Li’s slides:

for comparison.

It was a good exercise to listen to both talks, seeing two perspectives on the same paper, and I hope the students in the class got the idea(s) behind the paper. As you can see, there were obviously repetitions between the talks, including the presentation of the lower bounds for all classes considered by Jim Berger and Tom Sellke, and the overall motivation for the comparison. Maybe as a consequence of my criticisms on the previous talk, both Amira and Jiahuan put some stress on the definitions to formally define the background of the paper. (I love the poetic line: “To prevent having a non-Bayesian reality”, although I am not sure what Amira meant by this…)

I like the connection made therein with the Lindley-Jeffreys paradox since this is the core idea behind the paper. And because I am currently writing a note about the paradox. Obviously, it was hard for the students to take a more remote stand on the reason for the comparison, from questioning .the relevance of testing point null hypotheses and of comparing the numerical values of a p-value with a posterior probability, to expecting asymptotic agreement between a p-value and a Bayes factor when both are convergent quantities, to setting the same weight on both hypotheses, to the ad-hocquery of using a drift on one to equate the p-value with the Bayes factor, to use specific priors like Jeffreys’s (which has the nice feature that it corresponds to g=n in the g-prior,  as discussed in the new edition of Bayesian Core). The students also failed to remark on the fact that the developments were only for real parameters, as the phenomenon (that the lower bound on the posterior probabilities is larger than the p-value) does not happen so universally in larger dimensions.  I would have expected more discussion from the ground, but we still got good questions and comments on a) why 0.05 matters and b) why comparing  p-values and posterior probabilities is relevant. The next paper to be discussed will be Tukey’s piece on the future of statistics.