Archive for macha tea

Kick-Kac teleportation

Posted in Books, pictures, Statistics with tags , , , , , , , , on January 23, 2022 by xi'an

Randal Douc, Alain Durmus, Aurélien Enfroy, and Jimmy Olson have arXived their Kick-Kac teleportation paper, which was presented by Randal at CIRM last semester. It is based on Kac’s theorem, which states that, for a Markov chain with invariant distribution π, under (π) stationarity, the average tour between two visits to an accessible set C is also stationary. Which can be used for approximating π(h) if π(C) is known (or well-estimated). Jim Hobert and I exploited this theorem in our 2004 perfect sampling paper. The current paper contains a novel proof of the theorem under weaker conditions. (Note that the only condition on C is that it is accessible, rather than a small set. Which becomes necessary for geometric ergodicity, see condition (A4).)

What they define as the Kick-Kac teleportation (KKT) process is the collection of trajectories between two visits to C. Their memoryless version requires perfect simulations from π restricted to the set C. With a natural extension based on a Markov kernel keeping π restricted to the set C stationary. And a further generalisation allowing for lighter tails that also contains the 2005 paper by Brockwell and Kadane as a special case.

The ability of generating from a different kernel Q at each visit to C allows for different dynamics (as in other composite kernels). In their illustrations, the authors use lowest density regions for C, which is rather surprising to me. Except that it allows for a better connection between modes of the target π: the higher performances of the KKT algorithms against the considered alternatives are apparently dependent on the ability of the kernel Q to explore other modes with sufficient frequency.

transport Monte Carlo

Posted in Books, pictures, Statistics, Travel with tags , , , , , , , , , , , , , , , on August 31, 2020 by xi'an

Read this recent arXival by Leo Duan (from UF in Gainesville) on transport approaches to approximate Bayesian computation, in connection with normalising flows. The author points out a “lack of flexibility in a large class of normalizing flows”  to bring forward his own proposal.

“…we assume the reference (a multivariate uniform distribution) can be written as a mixture of many one-to-one transforms from the posterior”

The transportation problem is turned into defining a joint distribution on (β,θ) such that θ is marginally distributed from the posterior and β is one of an infinite collection of transforms of θ. Which sounds quite different from normalizing flows, to be sure. Reverting the order, if one manages to simulate β from its marginal the resulting θ is one of the transforms. Chosen to be a location-scale modification of β, s⊗β+m. The weights when going from θ to β are logistic transforms with Dirichlet distributed scales. All with parameters to be optimised by minimising the Kullback-Leibler distance between the reference measure on β and its inverse mixture approximation, and resorting to gradient descent. (This may sound a wee bit overwhelming as an approximation strategy and I actually had to make a large cup of strong macha to get over it, but this may be due to the heat wave occurring at the same time!) Drawing θ from this approximation is custom-made straightforward and an MCMC correction can even be added, resulting in an independent Metropolis-Hastings version since the acceptance ratio remains computable. Although this may defeat the whole purpose of the exercise by stalling the chain if the approximation is poor (hence suggesting this last step being used instead as a control.)

The paper also contains a theoretical section that studies the approximation error, going to zero as the number of terms in the mixture, K, goes to infinity. Including a Monte Carlo error in log(n)/n (and incidentally quoting a result from my former HoD at Paris 6, Paul Deheuvels). Numerical experiments show domination or equivalence with some other solutions, e.g. being much faster than HMC, the remaining $1000 question being of course the on-line evaluation of the quality of the approximation.

enjoy a cuppa for International Tea Day

Posted in Mountains, pictures, Travel, Wines with tags , , , , , , , , , , on May 21, 2020 by xi'an

 

art brut

Posted in Kids, Travel with tags , , , , , on November 17, 2019 by xi'an

Kudzu [クズ] delicacies [jatp]

Posted in Kids, pictures, Travel with tags , , , , , , , , , on August 12, 2019 by xi'an



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