at the foot of Monte Rosa

After last-year failed attempt at climbing a summit in the Monte Rosa group, we came back for hiking on the “other side” of Monte Rosa, at the Italian foot of Dufourspitze with less ambitious plans of hiking around this fantastic range. Alas the weather got set against us and thunderstorms followed one another, making altitude hikes unreasonable because of the fresh snow and frequent fog, and lower altitude walks definitely unpleasant, turning the trails into creeks.

While we had many nice chats with local people in a mixture of French and Italian, and sampled local cheese and beers, this still felt like a bit of a wasted vacations week, especially when remembering how reasonable the weather had been the week before in Scotland. However, all things considered, the previous week had seen a deadly heat wave cross the south of Europe that very week, so I should not be complaining about rain and snow! And running a whole week checking the altimeter instead of the chronometer is a nice experience.

In Dunvegan Castle gardens [jatp]

paradise island

As should be obvious from the pictures posted here in the past days, I have been away on vacations on the Scottish island of Skye, part of the Inner Hebrides. This is a place that had stood very high in my dream vacation places, mostly because of the mountain range that stands at the bottom of the island, called the Cuillins. There are 13 Munroes (tops above 3,000 feet) in that range and its entire traverse takes a very long day, including several belays. As I was there for a family vacation, we [alas!] only went up the easiest group, made of Sgùrr nan Gillean, Am Basteir, and Bruach na Frìthe, and did not climb Sgùrr nan Gillean. This was a fairly easy hike with a 900m differential, the only difficulty being in route finding. Which was made harder than needed by me first confusing a group of three hills with these Cuillins for the first third of the hike! And relating to instructions in our guidebook that covered the opposite side of the mountains. It was however a most pleasant walk, quite dry by Scottish standards (where it is often hard to separate water and soil) and with sun part of the way (it actually did not start to rain until the final half-hour). And not too many people on the path.
The other days saw easier walks at lower elevations, from a grassy and pleasant route to the top of MacLeod’s North Table [with terrific views of Western Skye and the Outer Hebrides], to a tour of the Gresornish peninsula under a pouring rain but with an amazing light (and an exciting crossing of a definitely web bog where even sheep did not do]. Overall, this was a great week in a secluded location, keeping mostly away from the few tourist traps [except for a trip to the Isle of Skye Brewery and to the compulsory Neist Point lighthouse] and I hope I can get back there one day. Although other Northern paradise islands like Mull, Harris, and the Faeroe are also beguiling..!
An unsolved puzzle about this visit to Skye is that, while there are sheep all over the island, which makes spotting any form of wildlife but midges impossible!, lamb meat is curiously absent from restaurant menus [except from the offal parts used in haggis]. The few persons we asked seemed perplexed by our question and had no convincing explanation to this absence!

Skye window [jatp]

Old Man of Storr [jatp]

ABC gas

probably ABC [and provably robust]

Two weeks ago, James Ridgway (formerly CREST) arXived a paper on misspecification and ABC, a topic on which David Frazier, Judith Rousseau and I have been working for a while now [and soon to be arXived as well].  Paper that I re-read on a flight to Amsterdam [hence the above picture], written as a continuation of our earlier paper with David, Gael, and Judith. One specificity of the paper is to use an exponential distribution on the distance between the observed and simulated sample within the ABC distribution. Which reminds me of the resolution by Bissiri, Holmes, and Walker (2016) of the intractability of the likelihood function. James’ paper contains oracle inequalities between the ABC approximation and the genuine distribution of the summary statistics, like a bound on the distance between the expectations of the summary statistics under both models. Which writes down as a sum of a model bias, of two divergences between empirical and theoretical averages, on smoothness penalties, and on a prior impact term. And a similar bound on the distance between the expected distance to the oracle estimator of θ under the ABC distribution [and a Lipschitz type assumption also found in our paper]. Which first sounded weird [to me] as I would have expected the true posterior, until it dawned on me that the ABC distribution is the one used for the estimation [a passing strike of over-Bayesianism!]. While the oracle bound could have been used directly to discuss the rate of convergence of the exponential rate λ to zero [with the sample size n], James goes into the interesting alternative direction of setting a prior on λ, an idea that dates back to Olivier Catoni and Peter Grünwald. Or rather a pseudo-posterior on λ, a common occurrence in the PAC-Bayesian literature. In one of his results, James obtains a dependence of λ on the dimension m of the summary [as well as the root dependence on the sample size n], which seems to contradict our earlier independence result, until one realises this scale parameter is associated with a distance variable, itself scaled in m.

The paper also contains a non-parametric part, where the parameter θ is the unknown distribution of the data and the summary the data itself. Which is quite surprising as I did not deem it possible to handle non-parametrics with ABC. Especially in a misspecified setting (although I have trouble perceiving what this really means).

“We can use most of the Monte Carlo toolbox available in this context.”

The theoretical parts are a bit heavy on notations and hard to read [as a vacation morning read at least!]. They are followed by a Monte Carlo implementation using SMC-ABC.  And pseudo-marginals [at least formally as I do not see how the specific features of pseudo-marginals are more that an augmented representation here]. And adaptive multiple pseudo-samples that reminded me of the Biometrika paper of Anthony Lee and Krys Latuszynski (Warwick). Therefore using indeed most of the toolbox!