## Another slice

No this is not yet another post-Christmas/NY ‘Og entry about food! Ian Murray, Ryan Adams and David MacKay posted a small piece on arXiv on Tuesday where they advocate a new type of slice sampler in cases when the posterior distribution on the parameter $f$ is associated with a Gaussian prior,

$\pi(f|x) \propto \mathcal{N}(f|0,\Sigma) L(f|x)$

and where the update in the Markov chain is based on an elliptic update,

$f^\prime = f \cos \theta + \nu \sin\theta,\quad\nu\sim\mathcal{N}(0,\Sigma)$,

except that $\theta$ is also updated at each MCMC step by a slice sampler. The resulting algorithm is a slice sampler in that it does not reject new values of $f^\prime$.

I find the proposal interesting, especially because it incorporates a “cyber-parameter” like $\theta$ within the Markov chain, but I wonder how widely the efficiency of the algorithm persists. Indeed, simulating from the prior cannot be very efficient when the likelihood strongly differs from the Gaussian prior. A lack of rejection is not a positive property per se and Gibbs sampling (incl. slice sampling) is notoriously slow for this very lack…