\[\newcommand\pos{\boldsymbol{x}} \newcommand\mom{\boldsymbol{p}} \newcommand\mass{\mathcal{M}}\]

22.1. Ensemble sampling

Note

This section is under construction. The motivation below is in place; the description of the stretch move and of the ensemble sampler in practice will be added.

Motivation: sampling from correlated distributions

If our posterior has projections that are slanted, indicating correlations, and we are doing Metropolis-Hastings (MH) sampling, how do we decide on a step size? The problem is that we want to step differently in different directions: a step long enough to explore the long axis will lead to many rejections in the orthogonal direction. So we do not want an isotropic step proposal!

If we propose steps by

\[ p(\xvec) = \frac{1}{\sqrt{(2\pi)^N |\Sigma|}} e^{-\xvec^{\intercal}\cdot \Sigma^{-1} \cdot \xvec} , \]

then we don’t need to take \(\Sigma \propto \sigma^2 \mathbb{1}_N\)!

• We have \(N(N+1)/2\) parameters in \(\Sigma\) to “tune” to reduce the correlation time.

• However, this is increasingly difficult as \(N\) increases.

One improvement is to do a linear transformation of \(\thetavec \longrightarrow \thetavec' = A\thetavec + B\) such that \(\thetavec'\) is uncorrelated with similar \(\sigma_i\)’s in each direction. Thus effectively to rotate the slanted ellipse.

Or one could use an “affine invariant” sampler such as emcee. An affine transformation is an invertible mapping from \(\mathbb{R}^N \rightarrow \mathbb{R}^N\), namely \(\yvec = A\xvec + B\), which is a combination of stretching, rotation, and translation. “Affine invariant” means that the sampler performs equally well on all affine transformations of a distribution. So emcee figures out how to make optimal steps. It does this by using the many walkers at time \(t\), which have sampled the space, to construct an appropriate affine compatible update step for \(t+1\). This is one reason to make sure there are plenty of walkers.