  In this paper, a general framework is formalised to characterise the value of
information (VoI) in hidden Markov models. Specifically, the VoI is defined as
the mutual information between the current, unobserved status at the source and
a sequence of observed measurements at the receiver, which can be interpreted
as the reduction in the uncertainty of the current status given that we have
noisy past observations of a hidden Markov process. We explore the VoI in the
context of the noisy Ornstein-Uhlenbeck process and derive its closed-form
expressions. Moreover, we study the effect of different sampling policies on
VoI, deriving simplified expressions in different noise regimes and analysing
the statistical properties of the VoI in the worst case. In simulations, the
validity of theoretical results is verified, and the performance of VoI in the
Markov and hidden Markov models is also analysed. Numerical results further
illustrate that the proposed VoI framework can support the timely transmission
in status update systems, and it can also capture the correlation property of
the underlying random process and the noise in the transmission environment.
