We describe an efficient and exact method that enables global Bayesian analysis of cosmic microwave background (CMB) data. The method reveals the joint posterior density (or likelihood for flat priors) of the power spectrum $C_ell$ and the CMB signal. Foregrounds and instrumental parameters can be simultaneously inferred from the data. The method allows the specification of a wide range of foreground priors. We explicitly show how to propagate the non-Gaussian dependency structure of the $C_ell$ posterior through to the posterior density of the parameters. If desired, the analysis can be coupled to theoretical (cosmological) priors and can yield the posterior density of cosmological parameter estimates directly from the time-ordered data. The method does not hinge on special assumptions about the survey geometry or noise properties, etc. It is based on a Monte Carlo approach and hence parallelizes trivially. No trace or determinant evaluations are necessary. The feasibility of this approach rests on the ability to solve the systems of linear equations which arise. These are of the same size and computational complexity as the map-making equations. We describe a pre-conditioned conjugate gradient technique that solves this problem and demonstrate in a numerical example that the computational time required for each Monte Carlo sample scales as $n_p^{3/2}$ with the number of pixels $n_p$. We test our method using the COBE-DMR data and explore the non-Gaussian joint posterior density of the COBE-DMR $C_ell$ in several projections.
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