  We investigate sets of random variables that can be arranged sequentially
such that a given variable only depends conditionally on its immediate
predecessor. For such sets, we show that the full joint probability
distribution may be expressed exclusively in terms of uni- and bivariate
marginals. Under the assumption that the CMB power spectrum likelihood only
exhibits correlations within a banded multipole range, \Delta l, we apply this
expression to two outstanding problems in CMB likelihood analysis. First, we
derive a statistically well-defined hybrid likelihood estimator, merging two
independent (e.g., low- and high-l) likelihoods into a single expression that
properly accounts for correlations between the two. Applying this expression to
the WMAP likelihood, we verify that the effect of correlations on cosmological
parameters in the transition region is negligible in terms of cosmological
parameters for WMAP; the largest relative shift seen for any parameter is
0.06\sigma. However, because this may not hold for other experimental setups
(e.g., for different instrumental noise properties or analysis masks), but must
rather be verified on a case-by-case basis, we recommend our new hybridization
scheme for future experiments for statistical self-consistency reasons. Second,
we use the same expression to improve the convergence rate of the Blackwell-Rao
likelihood estimator, reducing the required number of Monte Carlo samples by
several orders of magnitude, and thereby extend it to high-l applications.
