Archive for Metropolis-Hastings algorithm

scale matters [maths as well]

Posted in pictures, R, Statistics with tags , , , , , , , , on June 2, 2021 by xi'an

A question from X validated on why an independent Metropolis sampler of a three component Normal mixture based on a single Normal proposal was failing to recover the said mixture…

When looking at the OP’s R code, I did not notice anything amiss at first glance (I was about to drive back from Annecy, hence did not look too closely) and reran the attached code with a larger variance in the proposal, which returned the above picture for the MCMC sample, close enough (?) to the target. Later, from home, I checked the code further and noticed that the Metropolis ratio was only using the ratio of the targets. Dividing by the ratio of the proposals made a significant (?) to the representation of the target.

More interestingly, the OP was fundamentally confused between independent and random-walk Rosenbluth algorithms, from using the wrong ratio to aiming at the wrong scale factor and average acceptance ratio, and furthermore challenged by the very notion of Hessian matrix, which is often suggested as a default scale.

Metropolis-Hastings via Classification [One World ABC seminar]

Posted in Statistics, University life with tags , , , , , , , , , , , , , , , on May 27, 2021 by xi'an

Today, Veronika Rockova is giving a webinar on her paper with Tetsuya Kaji Metropolis-Hastings via classification. at the One World ABC seminar, at 11.30am UK time. (Which was also presented at the Oxford Stats seminar last Feb.) Please register if not already a member of the 1W ABC mailing list.

unbalanced sampling

Posted in pictures, R, Statistics with tags , , , , , , , on May 17, 2021 by xi'an

A question from X validated on sampling from an unknown density f when given both a sample from the density f restricted to a (known) interval A , say, and a sample from f restricted to the complement of A, say. Or at least on producing an estimate of the mass of A under f, p(A)

The problem sounds impossible to solve without an ability to compute the density value at a given value, since  any convex combination αf¹+(1-α)f² would return the same two samples. Assuming continuity of the density f at the boundary point a between A and its complement, a desperate solution for p(A)/1-p(A) is to take the ratio of the density estimates at the value a, which turns out not so poor an approximation if seemingly biased. This was surprising to me as kernel density estimates are notoriously bad at boundary points.

If f(x) can be computed [up to a constant] at an arbitrary x, it is obviously feasible to simulate from f and approximate p(A). But the problem is then moot as a resolution would not even need the initial samples. If exploiting those to construct a single kernel density estimate, this estimate can be used as a proposal in an MCMC algorithm. Surprisingly (?), using instead the empirical cdf as proposal does not work.

Arianna Rosenbluth’s hit

Posted in Statistics with tags , , , , , , , , , , on February 8, 2021 by xi'an

averaged acceptance ratios

Posted in Statistics with tags , , , , , , , , , , , , , on January 15, 2021 by xi'an

In another recent arXival, Christophe Andrieu, Sinan Yıldırım, Arnaud Doucet, and Nicolas Chopin study the impact of averaging estimators of acceptance ratios in Metropolis-Hastings algorithms. (It is connected with the earlier arXival rephrasing Metropolis-Hastings in terms of involutions discussed here.)

“… it is possible to improve performance of this algorithm by using a modification where the acceptance ratio r(ξ) is integrated with respect to a subset of the proposed variables.”

This interpretation of the current proposal makes it a form of Rao-Blackwellisation, explicitly mentioned on p.18, where, using a mixture proposal, with an adapted acceptance probability, it depends on the integrated acceptance ratio only. Somewhat magically using this ratio and its inverse with probability ½. And it increases the average Metropolis-Hastings acceptance probability (albeit with a larger number of simulations). Since the ideal averaging is rarely available, the authors implement a Monte Carlo averaging version. With applications to the exchange algorithm and to reversible jump MCMC. The major application is to pseudo-marginal settings with a high complexity (in the number T of terms) and where the authors’ approach does scale efficiently with T. There is even an ABC side to the story as one illustration is made of the ABC approximation to the posterior of an α-stable sample. As an encompassing proposal for handling Metropolis-Hastings environments with latent variables and several versions of the acceptance ratios, this is quite an interesting paper that I think we will study in further detail with our students.