ensemble Metropolis-Hastings

A question on X validated about ensemble MCMC samplers had me try twice to justify the Metropolis-Hasting ratio the authors used. To recap, ensemble sampling moves a cloud of points (just like our bouncy particle sampler) one point X at a time by using another point Z as a pivot or origin and moving randomly X along the line [XZ]. In the paper,  the distribution of the rescaling is symmetric in the sense that f(z)=f(1/z). I indeed started by perceiving the basic step of the sampler as a Metropolis-within-Gibbs step along a random direction. But it did not work as the direction depends on the current X. I then wondered at a possible importance sampling interpretation compensating for the change of scale, but it was leading to the wrong power anyway. Before hitting the fact that this was actually a change of radius in the space with origin Z, leaving the angular coordinates invariant. Which explained for the power (n-1) in the Metropolis ratio, in agreement with a switch to polar coordinates.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Google photo

You are commenting using your Google account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s

This site uses Akismet to reduce spam. Learn how your comment data is processed.

%d bloggers like this: