We propose a rigorous, non-perturbative, Bayesian framework which enables one jointly to test Gaussianity and estimate the power spectrum of CMB anisotropies. It makes use of the Hilbert space of an harmonic oscillator to set up an exact likelihood function, dependent on the power spectrum and on a set of parameters α_i, which are zero for Gaussian processes. The latter can be expressed as series of cumulants; indeed they perturbatively reduce to cumulants. However they have the advantage that

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... heir variation is essentially unconstrained. Any truncation(i.e.: finite set of α_i) therefore still produces a proper distribution - something which cannot be said of the only other such tool on offer, the Edgeworth expansion. We apply our method to Very Small Array (VSA) simulations based on signal Gaussianity, showing that our algorithm is indeed not biased.