uniform spacings

A riddle on uniform spacings!, namely when considering eight iid Uniform (0,1) variates as visiting times and three further iid Uniform (0,1) variates as server availability times, with unit service time, the question being the probability a server is available for a ninth visiting time, T⁹. Which can be decomposed into four cases:

1. at least one server becomes available between the previous visiting time and T⁹
2. at least two servers become available between the penultimate visiting time and the previous visiting time
3. two servers become available between the antepenultimate visiting time and penultimate visiting time and one server becomes available between the penultimate visiting time and the previous visiting time
4. three servers become available between the antepenultimate visiting time and the penultimate visiting time

with respective probabilities (using Devroye’s Theorem 2.1)

1/4, 8 9!3!/12!, 7 8!3!/12!, 7 8!3!/12!,

resulting in a total probability of 0.2934, compatible with a simulation assessment.

Galton and Watson voluntarily skipping some generations

A riddle on a form of a Galton-Watson process, starting from a single unit, where no one dies but rather, at each of 100 generations, Dog either opts for a Uniform number υ of additional units or increments a counter γ by this number υ, its goal being to optimise γ. The solution proposed by the Riddler does not establish his solution’s is the optimal strategy and considers anyway average gains. Solution that consists in always producing more units until the antepenultimate hour (ie incrementing only at the 99th and 100th generations),  I tried instead various logical (?) rules and compared outputs by bRute foRce, resulting in higher maxima (over numerous repeated calls) for the alternative principle

s<-function(p=.66){ 
   G=0;K=1 for(t in 1:9){ 
      i=sample(1:K,1) 
      K=K+i*(i>=K*p)
      G=G+i*(i<K*p)}
  return(c(G+sample(1:K,1),K))}

the alpinist [film review]

Watched (with supplementary oxygen) The Alpinist in the plane to Jeddah. It is a documentary (made by the same filmmakers who filmed the Dawn Wall) about the amazing Canadian alpinist Marc-André Leclerc, who died in 2018 on the Mendenhall Glacier, Alaska, in an avalanche, after achieving extraordinary complex solo climbs as eg on Mount Robbson, Cerro Torre in Patagonia. These are winter climbs, partly ice climbing, where no repetition is possible and where the objective conditions (hence dangers) may vary considerably. In that regard, these achievements could be argued to go even beyond Alex Honnold’s free solo climb of El Capitan, where Honnold practiced the route over and over before making his successful free solo attempt. (Obviously an inhuman achievement when considering the hardest bits are at least 7c!) Watching Marc-André Leclerc when mixed climbing is just as heart stopping as watching Honnold rock climbing. He must have been incredibly strong to master these monstruous icy walls and maintain his absolute vigilance in each crampon move, in each ice-pick placement. Sadly it ended up with an avalanche… I obviously enjoyed X’ing many places I had visited, like the approach walk to Robson (in 1991!), the Three Sisters of Canmore [below], and routes of Squamish [above]. (I wonder who filmed during these non-advertised climbs. For instance, he told no one except his partner when he summited Mount Robson. In some cases he was clearly self-filing at lower intensity points, but in others it could have been an helicopter (or a drone?). In this respect, but by far not only in this respect, his blog is definitely worth the read.)

~

optimal importance sampling

In Stein Π-Importance Sampling, Congye Wang et al. (mostly from Newcastle, UK) build an MCMC scheme with invariant distribution Π targeting a distribution P, showing that the optimal solution (in terms of a discrepancy) differs from P when the chain is Stein-sampled, e..g. via kernel discrepancies. In terms of densities, the solution is

the correction involving the root of a Stein kernel, introduced by Oates, Girolami, and Chopin in their 2017 Series B Read Paper. This is rather paradoxical, even though the outcome does depend on the divergence criterion. Most intriguing!!!

reciprocal importance sampling

In a recent arXival, Metodiev et al. (including my friend Adrian Raftery, who is spending the academic year in Paris) proposed a new version of reciprocal importance sampling, expanding the proposal we made with Darren Wraith (2009) of using a Uniform over an HPD region. It is called THAMES, hence the picture (of London, not Paris!), for truncated harmonic mean estimator.

“…[Robert and Wraith (2009)] method has not yet been fully developed for realistic, higher-dimensional situations. For example, we know of no simple way to compute the volume of the convex hull of a set of points in higher dimensions.”

They suggest replacing the convex hull of the HPD points with an ellipsoid ϒ derived from a Normal distribution centred at the highest of the HPD points, whose covariance matrix is estimated from the whole (?) posterior sample. Which is somewhat surprising in that this ellipsoid may as well included low probability regions when the posterior is multimodal. For instance, the estimator is biased when the posterior cancels on parts of ϒ. And with an unclear fate for the finiteness of its variance, depending on how fast the posterior gets to zero on these parts.

The central feature of the paper is selecting the radius of the ellipse that minimises the variance of the (counter) evidence. Under asymptotic normality of the posterior. This radius roughly corresponds to our HPD region in that 50% of the sample stands within. The authors also notice that separate samples should be used to estimate the ellipse and to estimate the evidence. And that a correction is necessary when the posterior support is restricted. (Examples do not include multimodal targets, apparently.)