Probing local non-Gaussianities within a Bayesian framework
  Aims: We outline the Bayesian approach to inferring f_NL, the level of
non-Gaussianity of local type. Phrasing f_NL inference in a Bayesian framework
takes advantage of existing techniques to account for instrumental effects and
foreground contamination in CMB data and takes into account uncertainties in
the cosmological parameters in an unambiguous way.
  Methods: We derive closed form expressions for the joint posterior of f_NL
and the reconstructed underlying curvature perturbation, Phi, and deduce the
conditional probability densities for f_NL and Phi. Completing the inference
problem amounts to finding the marginal density for f_NL. For realistic data
sets the necessary integrations are intractable. We propose an exact
Hamiltonian sampling algorithm to generate correlated samples from the f_NL
posterior. For sufficiently high signal-to-noise ratios, we can exploit the
assumption of weak non-Gaussianity to find a direct Monte Carlo technique to
generate independent samples from the posterior distribution for f_NL. We
illustrate our approach using a simplified toy model of CMB data for the simple
case of a 1-D sky.
  Results: When applied to our toy problem, we find that, in the limit of high
signal-to-noise, the sampling efficiency of the approximate algorithm
outperforms that of Hamiltonian sampling by two orders of magnitude. When f_NL
is not significantly constrained by the data, the more efficient, approximate
algorithm biases the posterior density towards f_NL = 0.
