In this letter, we borrow from the inference techniques developed for unbounded state-cardinality (nonparametric) variants of the HMM and use them to develop a tuning-parameter free, black-box inference procedure for explicit-state-duration hidden Markov models (EDHMM). EDHMMs are HMMs that have latent states consisting of both discrete state-indicator and discrete state-duration random variables. In contrast to the implicit geometric state duration distribution possessed by the standard HMM, EDHMMs allow the direct parameterization and estimation of per-state duration distributions. As most duration distributions are defined over the positive integers, truncation or other approximations are usually required to perform EDHMM inference.

In order to exploit word-oriented operations for linear-feedback shift registers (LFSRs), Tsaban and Vishne [2002] introduced the notion of linear transformation shift registers (TSRs). An implementation of their primitive TSR generating algorithm shows that the LFSR are paired for all transformations. We prove that the characteristic polynomials of a pair of LFSRs are either both irreducible or both reducible for all transformations. This allows some time improvement when finding primitive TSRs. The authors give a full enumeration of all primitive TSRs with transformations of order 8 and LFSRs of order 3, 4, 5, and 6.