Blind Categorical Deconvolution in Two-Level Hidden Markov Models

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A convolved two-level hidden Markov model is defined as an observed top level representing convolutions of an unobserved middle level of responses to an unobserved bottom level containing a Markov chain of categorical classes. The associated model parameters include a Markov chain transition matrix, response levels and variances, a convolutional kernel, and an observation error variance. The convolutional kernel and the error variance are defined to be unknown. Focus is on the joint assessment of the unknown model parameters and the sequence of categorical classes given the observed top level. This is termed blind categorical deconvolution and is cast in a Bayesian inversion setting. An approximate posterior model based on an approximate likelihood model in factorizable form is defined. The approximate model, including the likelihoods for the unknown model parameters, can be exactly assessed by a recursive algorithm. A sequence of approximations is defined such that tradeoffs between accuracy and computational demands can be made. The model parameters are assessed by approximate maximum-likelihood estimation, whereas the inversion is represented by the approximate posterior model. A limited empirical study demonstrates that reliable model parameter assessments and inversions can be made from the approximate model. An example of blind seismic deconvolution is also presented and discussed.