  In Bayesian inverse problems, one aims at characterizing the posterior
distribution of a set of unknowns, given indirect measurements. For
non-linear/non-Gaussian problems, analytic solutions are seldom available:
Sequential Monte Carlo samplers offer a powerful tool for approximating complex
posteriors, by constructing an auxiliary sequence of densities that smoothly
reaches the posterior. Often the posterior depends on a scalar hyper-parameter.
In this work, we show that properly designed Sequential Monte Carlo (SMC)
samplers naturally provide an approximation of the marginal likelihood
associated with this hyper-parameter for free, i.e. at a negligible additional
computational cost. The proposed method proceeds by constructing the auxiliary
sequence of distributions in such a way that each of them can be interpreted as
a posterior distribution corresponding to a different value of the
hyper-parameter. This can be exploited to perform selection of the
hyper-parameter in Empirical Bayes approaches, as well as averaging across
values of the hyper-parameter according to some hyper-prior distribution in
Fully Bayesian approaches. For FB approaches, the proposed method has the
further benefit of allowing prior sensitivity analysis at a negligible
computational cost. In addition, the proposed method exploits particles at all
the (relevant) iterations, thus alleviating one of the known limitations of SMC
samplers, i.e. the fact that all samples at intermediate iterations are
typically discarded. We show numerical results for two distinct cases where the
hyper-parameter affects only the likelihood: a toy example, where an SMC
sampler is used to approximate the full posterior distribution; and a brain
imaging example, where a Rao-Blackwellized SMC sampler is used to approximate
the posterior distribution of a subset of parameters in a conditionally linear
Gaussian model.
