## Archive for Norway

## precisely!

Posted in pictures, Travel with tags Donald Trump, immigration, Norway, Stephen King, Trondheim, twitter, USA on January 12, 2018 by xi'an## a new paradigm for improper priors

Posted in Books, pictures, Statistics, Travel with tags Alfréd Rényi, Andrei Kolmogorov, axioms of probability, convergence of Gibbs samplers, improper priors, σ-algebra, marginalisation paradoxes, Norway, Trondheim on November 6, 2017 by xi'an**G**unnar Taraldsen and co-authors have arXived a short note on using improper priors from a new perspective. Generalising an earlier 2016 paper in JSPI on the same topic. Which both relate to a concept introduced by Rényi (who himself attributes the idea to Kolmogorov). Namely that random variables measures are to be associated with arbitrary measures [not necessarily σ-finite measures, the later defining σ-finite random variables], rather than those with total mass one. Which allows for an alternate notion of conditional probability in the case of σ-finite random variables, with the perk that this conditional probability distribution is itself of mass 1 (a.e.). Which we know happens when moving from prior to proper posterior.

I remain puzzled by the 2016 paper though as I do not follow the meaning of a *random variable* associated with an *infinite mass probability measure*. If the point is limited to construct posterior probability distributions associated with improper priors, there is little value in doing so. The argument in the 2016 paper is however that one can then define a conditional distribution in marginalisation paradoxes à la Stone, Dawid and Zidek (1973) where the marginal does not exist. Solving with this formalism the said marginalisation paradoxes as conditional distributions are only defined for σ-finite random variables. Which gives a fairly different conclusion that either Stone, Dawid and Zidek (1973) [with whom I agree, namely that there is no paradox because there is no “joint” distribution] or Jaynes (1973) [with whom I less agree!, in that the use of an invariant measure to make the discrepancy go away is not a particularly strong argument in favour of this measure]. The 2016 paper also draws an interesting connection with the study by Jim Hobert and George Casella (in Jim’s thesis) of [null recurrent or transient] Gibbs samplers with no joint [proper] distribution. Which in some situations can produce proper subchains, a phenomenon later exhibited by Alan Gelfand and Sujit Sahu (and Xiao-Li Meng as well if I correctly remember!). But I see no advantage in following this formalism, as it does not impact whether the chain is transient or null recurrent, or anything connected with its implementation. Plus a link to the approximation of improper priors by sequences of proper ones by Bioche and Druihlet I discussed a while ago.

## fiducial inference

Posted in Books, Mountains, pictures, Running, Statistics, Travel with tags cross validated, fiducial distribution, fiducial inference, fjord, σ-algebra, measure theory, Norway, probability theory, Ronald Fisher, Trondheim, Trondheimsfjord on October 30, 2017 by xi'an**I**n connection with my recent tale of the many ε’s, I received from Gunnar Taraldsen [from Tronheim, Norge] a paper [jointly written with Bo Lindqvist and just appeared on-line in JSPI] on conditional fiducial models.

“The role of the prior and the statistical model in Bayesian analysis is replaced by the use of the fiducial model x=R(θ,ε) in fiducial inference. The fiducial is obtained in this case without a prior distribution for the parameter.”

Reading this paper after addressing the X validated question made me understood better the fundamental wrongness of fiducial analysis! If I may herein object to Fisher himself… Indeed, when writing x=R(θ,ε), as the representation of the [observed] random variable x as a deterministic transform of a parameter θ and of an [unobserved] random factor ε, the two random variables x and ε are based on the same random preimage ω, i.e., x=x(ω) and ε=ε(ω). Observing x hence sets a massive constraint on the preimage ω and on the conditional distribution of ε=ε(ω). When the fiducial inference incorporates another level of randomness via an independent random variable ε’ and inverts x=R(θ,ε’) into θ=θ(x,ε’), assuming there is only one solution to the inversion, it modifies the nature of the underlying σ-algebra into something that is incompatible with the original model. Because of this sudden duplication of the random variates. While the inversion of this equation x=R(θ,ε’) gives an idea of the possible values of θ when ε varies according to its [prior] distribution, it does not account for the connection between x and ε. And does not turn the original parameter into a random variable with an implicit prior distribution.

As to conditional fiducial distributions, they are defined by inversion of x=R(θ,ε), under a certain constraint on θ, like C(θ)=0, which immediately raises a Pavlovian reaction in me, namely that since the curve C(θ)=0 has measure zero under the original fiducial distribution, how can this conditional solution be uniquely or at all defined. Or to avoid the Borel paradox mentioned in the paper. If I get the meaning of the authors in this section, the resulting fiducial distribution will actually depend on the choice of σ-algebra governing the projection.

“A further advantage of the fiducial approach in the case of a simple fiducial model is that independent samples areproduced directly from independent sampling from [the fiducial distribution]. Bayesian simulations most often come as dependent samples from a Markov chain.”

This side argument in “favour” of the fiducial approach is most curious as it brings into the picture computational aspects that do not have any reason to be there. (The core of the paper is concerned with the unicity of the fiducial distribution in some univariate settings. Not with computational issues.)

## Adam Ondra completes a first 9c

Posted in Kids, Mountains, pictures with tags 9c, Adam Ondra, extremes, Flatanger, Hanshelleren Cave, Norway, sport climbing on September 30, 2017 by xi'an**I**n my office hangs this poster of Adam Ondra climbing at sunset an impressive overhang with a little Czech town at the foot of the cliff, already in the shade. Impressive because of the view and of the climb which at 7c is a whole grade (and then some) beyond my reach. But now Ondra has managed to climb the first 9c in the world, which is universes beyond the impressive and beyond the fathomable, with passages only manageable feet first. Which actually makes a lot of sense, the way he explains it. The route is currently called *Project Hard *and is located* *in Hanshelleren Cave, Flatanger, Norway.

## fiducial on a string

Posted in Books, pictures, Statistics, Travel, University life with tags Bayesian Fiducial & Frequentist Conference, fiducial inference, Harvard University, Norway, NTU, Philosophy of Science, Teddy Seidenfeld on June 26, 2017 by xi'an**A** very short note in arXiv today by Gunnar Taraldsen and Bo Henry Lindqvist (NTU, Norway). With the above title. I find the note close to unreadable, I must say, as the notations are not all or well- defined. The problem starts from Teddy Seidenfeld [whom I met in Harvard around Dutch book arguments] arguing about the lack of unicity of fiducial distributions in a relatively simple setting. Actually the note is also inspired from Bayes, Fiducial and Frequentist, and comments from Teddy, a talk I apparently missed by taking a flight back home too early!

What I find surprising in this note is that the “fiducial on a string” is a conditional distribution on the parameter space restricted to a curve, derived from the original fiducial distribution by a conditioning argument. Except that since the conditioning is on a set of measure zero, this conditional is not only not-unique, but it is completely undefined and arbitrary, since changing it does not modify the properties of the joint distribution.

## ABC postdoc in Olso

Posted in Kids, Mountains, pictures, Travel, University life with tags ABC, French elections, Norway, Oslo, postdoc, postdoctoral position, Scandinavia, UiO, University of Oslo on April 26, 2017 by xi'anJukka Corander sent me the announcement that he is opening a 3 year postdoctoral position at the University of Oslo, to work with him and his team on ABC projects. This sounds quite an exciting offer, plus gives the nominee the opportunity to live in the most enjoyable city of Oslo for several years in fairly comfy conditions! **The deadline is May 31.** (If I was at a stage of my career where applying made sense, I would definitely candidate. Not even waiting for the outcome of the French elections on May 7!)

## Abel Prize goes to Yves Meyer

Posted in Books, pictures, University life with tags Abel Prisen, Abel Prize, École Normale de Cachan, Gauss Prize, mathematics, Norway, prizes, wavelets, Yves Meyer on March 21, 2017 by xi'an**J**ust heard the great news that the Abel Prize for 2017 goes to Yves Meyer! Yves Meyer is an emeritus professor at École Normale de Cachan and has produced fundamental contributions to number theory, operator theory and harmonic analysis. He is one of the originators of the theory of wavelets and multiresolution analysis. Among other recognitions and prizes, he was an invited speaker at the International Congress of Mathematicians in 1970 (Nice), in 1983 (Warsaw), and in 1990 (Kyoto), and was awarded the Gauß Prize in 2010. Congratulations and total respect to Yves Meyer!!!