  We consider the problem of distributed binary hypothesis testing in a
parallel network topology where sensors independently observe some phenomenon
and send a finite rate summary of their observations to a fusion center for the
final decision. We explicitly consider a scenario under which (integer) rate
messages are sent over an error free multiple access channel, modeled by a sum
rate constraint at the fusion center. This problem was previously studied by
Chamberland and Veeravalli, who provided sufficient conditions for the
optimality of one bit sensor messages. Their result is however crucially
dependent on the feasibility of having as many one bit sensors as the (integer)
sum rate constraint of the multiple access channel, an assumption that can
often not be satisfied in practice. This prompts us to consider the case of an
a-priori limited number of sensors and we provide sufficient condition under
which having no two sensors with rate difference more than one bit, so called
rate balancing, is an optimal strategy with respect to the Bhattacharyya
distance between the hypotheses at the input to the fusion center. We further
discuss explicit observation models under which these sufficient conditions are
satisfied.
