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In this paper, we consider a robust state estimation problem for uncertain discrete-time, homogeneous, first-order, finite-state finite-alphabet hidden Markov models (HMMs). A class of time-varying uncertain HMMs is considered in which the uncertainty is sequentially described by a regular conditional relative entropy constraint on perturbed regular conditional probability measures given the observation sequence. For this class of uncertain HMMs, the robust state estimation problem is formulated as a constrained optimization problem. Using a Lagrange multiplier technique and a duality relationship for regular conditional relative entropy, the above problem is converted into an unconstrained optimization problem and a problem related to partial information risk-sensitive filtering. Furthermore, a measure transformation technique and an information state method are employed to solve this equivalent problem related to risk-sensitive filtering.