  Incorporating information about the target distribution in proposal
mechanisms generally produces efficient Markov chain Monte Carlo algorithms (or
at least, algorithms that are more efficient than uninformed counterparts). For
instance, it has proved successful to incorporate gradient information in
fixed-dimensional algorithms, as seen with algorithms such as Hamiltonian Monte
Carlo. In trans-dimensional algorithms, Green (2003) recommended to sample the
parameter proposals during model switches from normal distributions with
informative means and covariance matrices. These proposal distributions can be
viewed as asymptotic approximations to the parameter distributions, where the
limit is with regard to the sample size. Models are typically proposed using
uninformed uniform distributions. In this paper, we build on the approach of
Zanella (2020) for discrete spaces to incorporate information about
neighbouring models. We rely on approximations to posterior model probabilities
that are asymptotically exact. We prove that, in some scenarios, the samplers
combining this approach with that of Green (2003) behave like ideal ones that
use the exact model probabilities and sample from the correct parameter
distributions, in the large-sample regime. We show that the implementation of
the proposed samplers is straightforward in some cases. The methodology is
applied to a real-data example. The code is available online.
