Consider a hidden Markov model describing a system with two types of states: a monitored state and a private state. The two types of states are dependent and evolve jointly according to a Markov process with a stationary transition probability. It is desired to reveal the monitored states to a receiver but hide the private states. For this purpose, a privacy filter is necessary which suitably perturbs the monitored states before communication with the receiver. Our objective is to design the privacy filter to optimize the tradeoff between the monitoring accuracy and privacy, measured through a time-invariant distortion measure and Shannon's equivocation, respectively. As the optimal privacy filter is difficult to compute using the dynamic programming, we adopt a suboptimal greedy approach through which the privacy filter can be computed efficiently. Here, the greedy approach has the additional advantage of not being restricted to the finite time horizon setups. Simulations show the superiority of the approach compared to a privacy filter which only adds independent noise to the observations.