Approximate Bayesian Computation in state space models

While it took quite a while (!), with several visits by three of us to our respective antipodes, incl. my exciting trip to Melbourne and Monash University two years ago, our paper on ABC for state space models was arXived yesterday! Thanks to my coauthors, Gael Martin, Brendan McCabe, and  Worapree Maneesoonthorn,  I am very glad of this outcome and of the new perspective on ABC it produces.  For one thing, it concentrates on the selection of summary statistics from a more econometrics than usual point of view, defining asymptotic sufficiency in this context and demonstrated that both asymptotic sufficiency and Bayes consistency can be achieved when using maximum likelihood estimators of the parameters of an auxiliary model as summary statistics. In addition, the proximity to (asymptotic) sufficiency yielded by the MLE is replicated by the score vector. Using the score instead of the MLE as a summary statistics allows for huge gains in terms of speed. The method is then applied to a continuous time state space model, using as auxiliary model an augmented unscented Kalman filter. We also found in the various state space models tested therein that the ABC approach based on the marginal [likelihood] score was performing quite well, including wrt Fearnhead’s and Prangle’s (2012) approach… I like the idea of using such a generic object as the unscented Kalman filter for state space models, even when it is not a particularly accurate representation of the true model. Another appealing feature of the paper is in the connections made with indirect inference.

first pho in Saigon

a paradox in decision-theoretic interval estimation (solved)

In 1993, we wrote a paper [with George Casella and Gene/Juinn Hwang] on the paradoxical consequences of using the loss function

(published in Statistica Sinica, 3, 141-155) since it led to the following property: for the standard normal mean estimation problem, the regular confidence interval is dominated by the modified confidence interval equal to the empty set when s² is too large… This was first pointed out by Jim Berger and the most natural culprit is the artificial loss function where the first part is unbounded while the second part is bounded by k. Recently, Paul Kabaila—whom I met in both Adelaide, where he quite appropriately commented about the abnormal talk at the conference!,  and Melbourne, where we met with his students after my seminar at the University of Melbourne—published a paper (first on arXiv then in Statistics and Probability Letters) where he demonstrates that the mere modification of the above loss into

solves the paradox:! For Jeffreys’ non-informative prior, the Bayes (optimal) estimate is the regular confidence interval. besides doing the trick, this nice resolution explains the earlier paradox as being linked to a lack of invariance in the (earlier) loss function. This is somehow satisfactory since Jeffreys’ prior also is the invariant prior in this case.

60k versus 65k

An add in Melbourne took a while to click in (for me!): it represented a woman projected on the hood of a car with the legend 65k and the same woman prostrated in front of the same car with the legend 60k… I was seeing this ad every day when driven to Monash and could not see the point as I was interpreting 60k as 60kg! So it sounded like a weird campaign for a new diet… After a while, I eventually got the point that it was a campaign towards speed reduction and against drivers thinking that 60km/h does not differ much from 65km/h. (I could not find a reproduction of the campaign posters on the official site.) Besides this misinterpretation, I find the message rather unclear and unconvincing: while driving more slowly obviously gives a driver more time to react, the 60k/65k opposition could be replaced with a 55k/60k opposition and would not make less or more sense. Furthermore, the variability in driver’s reactions and car behaviours is likely to influence the consequences of an impact as significantly as a reduction of 5km/h…

St Kilda’s wines

Following my courses in Monash, we celebrated (!) by having a nice dinner at St Kilda, one of Melbourne beaches. The restaurant had an incredible collection of French wines, including a whole range of wines from the upper Rhône valley, like Côte Rotie, Saint Joseph (my favourite!), and Cornas. Those were however priced at three and even four digits (in dollars or Euros!), and we sampled instead the local version of those, which were truly pleasant (although too young and missing a few hours of oxygenation!)

fairy penguins

During our Aussie vacation week, we visited Phillip Island, south of Melbourne, as Kangaroo Island just proved too complicated to reach within our limited time-span The island has many interesting features (although it is heavily inhabited, even in the winter), including volcanic feature and plenty of wildlife (like the goose above). However, the major attraction of this place is the daily (and “insanely popular” to quote from Lonely Planet Australia) ritual of fairy penguins, a special type of tiny penguins (33cm!) that go fishing in Melbourne Bay for a day or two and return to their burrow at dusk. The “fairy” denomination comes from the fact that they appear to pop out from the sea in a magical way, being “suddenly” there on the shore… It was an interesting experience to sit and watch hundreds of birds rush through the beach and climb the hills toward their burrow by group of five to ten penguins.

However, I was quite puzzled (and, to be completely honest, rather incensed!) by the commercialisation of the “penguin parade”: it is impossible to see the landing without paying a fee to the company owning the only beach where this occurs… The argument is that this is a non-profit organisation and that the money goes to research and rehabilitation of the penguin habitat (and this is presumably the case!), but I feel queasy about the whole process of putting lights on the landing beach and walkways next to the penguins’ path to their burrow (some are even located right below the visitors’ centre, with lights and openings to see inside the burrow!). Hundreds of (“oohing-and-aahing” to quote again Lonely Planet!) tourists sit every evening to watch the penguins, with a certain amount of noise and movements. I find this mix of protection (as indeed this part of the island is full of replanted grass and man-made burrows) and raw business (the side business of selling hot food and junk products to tourists is less of an issue as it does not impact the penguins) quite counter-natural. (I do not think a national park would tolerate such an intrusive behaviour… However, I did not find much protest about this on the web.)

Melbourne trams

(Melbourne has one of the largest tram networks in the World, some specimens below.)