## convergence speeds

Posted in pictures, Running, Statistics, Travel, University life with tags , , , , , , on December 5, 2013 by xi'an

While waiting for Jean-Michel to leave a thesis defence committee he was part of, I read this recently arXived survey by Novak and Rudolf, Computation of expectations by Markov chain Monte Carlo methods. The first part hinted at a sort of Bernoulli factory problem: when computing the expectation of f against the uniform distribution on G,

For x ∈ G we can compute f (x) and G is given by a membership oracle, i.e. we are able to check whether any x is in G or not.

However, the remainder of the paper does not get (in) that direction but recalls instead convergence results for MCMC schemes under various norms. Like spectral gap and Cheeger’s inequalities. So useful for a quick reminder, e.g. to my Monte Carlo Statistical Methods class Master students, but altogether well-known. The paper contains some precise bounds on the mean square error of the Monte Carlo approximation to the integral. For instance, for the hit-and-run algorithm, the uniform bound (for functions f bounded by 1) is

$9.5\cdot 10^{7}\dfrac{dr}{\sqrt{n}}+6.4\cdot 10^{15}\dfrac{d^2r^2}{n}$

where d is the dimension of the space and r a scale of the volume of G. For the Metropolis-Hastings algorithm, with (independent) uniform proposal on G, the bound becomes

$\dfrac{2C\alpha_dr^d}{n}+\dfrac{4C^2\alpha_d^2r^{2d}}{n^2}\,,$

where C is an upper bound on the target density (no longer the uniform). [I rephrased Theorem 2 by replacing vol(G) with the containing hyper-ball to connect both results, αd being the proportionality constant.] The paper also covers the case of the random walk Metropolis-Hastings algorithm, with the deceptively simple bound

$1089\dfrac{(d+1)\max\{\alpha,\sqrt{d+1}\}}{\sqrt{n}}+8.38\cdot 10^5\dfrac{(d+1)\max\{\alpha^2,d+1\}}{n}$

but this is in the special case when G is the ball of radius d. The paper concludes with a list of open problems.

## running along Lake Ontario

Posted in pictures, Running, Travel with tags , , , , on November 1, 2013 by xi'an

## seminars at CMU and University of Toronto

Posted in Statistics, Travel, University life with tags , , , , , , , , , , , , on October 29, 2013 by xi'an

Here are the slides for my seminar talks at Carnegie Mellon University (Pittsburgh) and the University of Toronto, tomorrow and the day after, respectively:

## spam

Posted in Statistics, University life with tags , , , , , on February 11, 2013 by xi'an

Another focussed spam in the mail:

Dear Dr. Christian P. Robert,

How are you?

I read your interesting article of “Error and inference: an outsider stand on a frequentist philosophy“, and I know you are an active professional in this field.

Now, I am writing you to call for new papers, on behalf of Review of Economics & Finance, which is an English quarterly journal in Canada.

This journal is currently indexed by EconLit of American Economic Association (AEA), EBSCO, RePEc, National Bibliography of Canada, Library and Archives Canada, DOAJ, Ulrich, and so on.

The publication fee is CAD$450, if your paper is qualified for publication after refereeing. The submission fee of$50 is NOT applied to you by March 6th, 2013.

Thank you for your consideration. Have a rewarding month!

At least, they are quite honest about the cost of publishing there. But they should check on which paper they pick rather than using a robot that takes a paper not talking about economics or finance… (And why on Earth this line about the rewarding month?!)

## Banff sunset

Posted in Mountains, pictures, Travel, University life with tags , , , , on March 27, 2012 by xi'an