  We present a principled Bayesian framework for signal reconstruction, in
which the signal is modelled by basis functions whose number (and form, if
required) is determined by the data themselves. This approach is based on a
Bayesian interpretation of conventional sparse reconstruction and
regularisation techniques, in which sparsity is imposed through priors via
Bayesian model selection. We demonstrate our method for noisy 1- and
2-dimensional signals, including astronomical images. Furthermore, by using a
product-space approach, the number and type of basis functions can be treated
as integer parameters and their posterior distributions sampled directly. We
show that order-of-magnitude increases in computational efficiency are possible
from this technique compared to calculating the Bayesian evidences separately,
and that further computational gains are possible using it in combination with
dynamic nested sampling. Our approach can also be readily applied to neural
networks, where it allows the network architecture to be determined by the data
in a principled Bayesian manner by treating the number of nodes and hidden
layers as parameters.
