  In this paper, we study a multiple-input multiple-output (MIMO) integrated
sensing and communication (ISAC) system where one multi-antenna base station
(BS) sends information to a user with multiple antennas in the downlink and
simultaneously senses the location parameter of a target based on its reflected
echo signals received back at the BS receive antennas. We focus on the case
where the location parameter to be sensed is unknown and random, for which the
prior distribution information is available for exploitation. First, we propose
to adopt the posterior Cram\'er-Rao bound (PCRB) as the sensing performance
metric with prior information, which quantifies a lower bound of the
mean-squared error (MSE). Since the PCRB is in a complicated form, we derive a
tight upper bound of it to draw more insights. Based on this, we analytically
show that by exploiting the prior distribution information, the PCRB is always
no larger than the CRB averaged over random location realizations without prior
information exploitation. Next, we formulate the transmit covariance matrix
optimization problem to minimize the sensing PCRB under a communication rate
constraint. We obtain the optimal solution and derive useful properties on its
rank. Then, by considering the derived PCRB upper bound as the objective
function, we propose a low-complexity suboptimal solution in semi-closed form.
Numerical results demonstrate the effectiveness of our proposed designs in MIMO
ISAC exploiting prior information.
