Archive for the pictures Category

Signori degli Anelli

Posted in Kids, pictures, Travel, Wines with tags , , , , on May 30, 2015 by xi'an

rin6rin3rin5rip2rip1rin9rin8rin4rin1

Some rings and their stand from the walls and doors of the city of Siena.

Chianti

Posted in pictures, Running, Travel, Wines with tags , , , , , on May 29, 2015 by xi'an

chianti6

Toscana [#3]

Posted in Mountains, pictures, Running, Travel, Wines with tags , , , , , on May 28, 2015 by xi'an

huse

escaping the frame…

Posted in Kids, pictures, Travel with tags , , , , , on May 27, 2015 by xi'an

ofdawal

pigeons

Posted in pictures, Travel with tags , , , on May 26, 2015 by xi'an

pigeons

non-reversible MCMC [comments]

Posted in Books, Mountains, pictures, Statistics, University life with tags , , , , , on May 26, 2015 by xi'an

[Here are comments made by Matt Graham that I thought would be more readable in a post format. The beautiful picture of the Alps above is his as well. I do not truly understand what Matt’s point is, as I did not cover continuous time processes in my discussion…]

In terms of interpretation of the diffusion with non-reversible drift component, I think this can be generalised from the Gaussian invariant density case by

dx = [ – (∂E/∂x) dt + √2 dw ] + S’ (∂E/∂x) dt

where ∂E/∂x is the usual gradient of the negative log (unnormalised) density / energy and S=-S’ is a skew symmetric matrix. In this form it seems the dynamic can be interpreted as the composition of an energy and volume conserving (non-canonical) Hamiltonian dynamic

dx/dt = S’ ∂E/∂x

and a (non-preconditioned) Langevin diffusion

dx = – (∂E/∂x) dt + √2 dw

As an alternative to discretising the combined dynamic, it might be interesting to compare to sequential alternation between ‘Hamiltonian’ steps either using a simple Euler discretisation

x’ = x + h S’ ∂E/∂x

or a symplectic method like implicit midpoint to maintain reversibility / volume preservation and then a standard MALA step

x’ = x – h (∂E/∂x) + √2 h w, w ~ N(0, I)

plus MH accept. If only one final MH accept step is done this overall dynamic will be reversible, however if a an intermediary MH accept was done after each Hamiltonian step (flipping the sign / transposing S on a rejection to maintain reversibility), the composed dynamic would in general be non-longer reversible and it would be interesting to compare performance in this case to that using a non-reversible MH acceptance on the combined dynamic (this alternative sidestepping the issues with finding an appropriate scale ε to maintain the non-negativity condition on the sum of the vorticity density and joint density on a proposed and current state).

With regards to your point on the positivity of g(x,y)+π(y)q(y,x), I’m not sure if I have fully understood your notation correctly or not, but I think you may have misread the definition of g(x,y) for the discretised Ornstein-Uhlenbeck case (apologies if instead the misinterpretation is mine!). The vorticity density is defined as the skew symmetric component of the density f of F(dx, dy) = µ(dx) Q(x, dy) with respect to the Lebesgue measure, where µ(dx) is the true invariant distribution of the Euler-Maruyama discretised diffusion based proposal density Q(x, dy) rather than g(x, y) being defined in terms of the skew-symmetric component of π(dx) Q(x, dy) which in general would lead to a vorticity density which does not meet the zero integral requirement as the target density π is not invariant in general with respect to Q.

an even more senseless taxi-ride

Posted in Books, Kids, pictures, Travel, University life with tags , , , , , , , , , on May 24, 2015 by xi'an

I was (exceptionally) working in (and for) my garden when my daughter shouted down from her window that John Nash had just died. I thus completed my tree trimming and went to check about this sad item of news. What I read made the news even sadder as he and his wife had died in a taxi crash in New Jersey, apparently for not wearing seat-belts, a strategy you would think far from minimax… Since Nash was in Norway a few days earlier to receive the 2015 Abel Prize, it may even be that the couple was on its way home back from the airport.  A senseless death for a Beautiful Mind.

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