We present a moving target defense strategy to reduce the impact of stealthy sensor attacks on feedback systems. The defender periodically and randomly switches between thresholds from a discrete set to increase the uncertainty for the attacker and make stealthy attacks detectable. However, the defender does not know the exact goal of the attacker but only the prior of the possible attacker goals. Here, we model one period with a constant threshold as a Bayesian game and use the Bayesian Nash equilibrium concept to find the distribution for the choice of the threshold in that period, which takes the defender's uncertainty about the attacker into account. To obtain the equilibrium distribution, the defender minimizes its cost consisting of the cost for false alarms and the cost induced by the attack. We present a necessary and sufficient condition for the existence of a moving target defense and formulate a linear program to determine the moving target defense. Furthermore, we present a closed-form solution for the special case when the defender knows the attacker's goals. The results are numerically evaluated on a four-tank process.