  We introduce an exact Bayesian approach to search for non-Gaussianity of
local type in Cosmic Microwave Background (CMB) radiation data. Using simulated
CMB temperature maps, the newly developed technique is compared against the
conventional frequentist bispectrum estimator. Starting from the joint
probability distribution, we obtain analytic expressions for the conditional
probabilities of the primordial perturbations given the data, and for the level
of non-Gaussianity, f_nl, given the data and the perturbations. We propose
Hamiltonian Monte Carlo sampling as a means to derive realizations of the
primordial fluctuations from which we in turn sample f_nl. Although being
computationally expensive, this approach allows us to exactly construct the
full target posterior probability distribution. When compared to the
frequentist estimator, applying the Bayesian method to Gaussian CMB maps
provides consistent results. For the analysis of non-Gaussian maps, however,
the error bars on f_nl do not show excess variance within the Bayesian
framework. This finding is of particular relevance in the light of upcoming
high precision CMB measurements obtained by the Planck satellite mission.
