  Efficient assessment of convolved hidden Markov models is discussed. The
bottom-layer is defined as an unobservable categorical first-order Markov
chain, while the middle-layer is assumed to be a Gaussian spatial variable
conditional on the bottom-layer. Hence, this layer appear as a Gaussian mixture
spatial variable unconditionally. We observe the top-layer as a convolution of
the middle-layer with Gaussian errors. Focus is on assessment of the
categorical and Gaussian mixture variables given the observations, and we
operate in a Bayesian inversion framework. The model is defined to make
inversion of subsurface seismic AVO data into lithology/fluid classes and to
assess the associated elastic material properties. Due to the spatial coupling
in the likelihood functions, evaluation of the posterior normalizing constant
is computationally demanding, and brute-force, single-site updating Markov
chain Monte Carlo algorithms converges far too slow to be useful. We construct
two classes of approximate posterior models which we assess analytically and
efficiently using the recursive Forward-Backward algorithm. These approximate
posterior densities are used as proposal densities in an independent proposal
Markov chain Monte Carlo algorithm, to assess the correct posterior model. A
set of synthetic realistic examples are presented. The proposed approximations
provides efficient proposal densities which results in acceptance probabilities
in the range 0.10-0.50 in the Markov chain Monte Carlo algorithm. A case study
of lithology/fluid seismic inversion is presented. The lithology/fluid classes
and the elastic material properties can be reliably predicted.
