  In this paper, we consider the binary hypothesis testing problem with two
observers. There are two possible states of nature (or hypotheses).
Observations are collected by two observers. The observations are statistically
related to the true state of nature. Given the observations, the objective of
both observers is to find out what is the true state of nature. We present four
different approaches to address the problem. In the first (centralized)
approach, the observations collected by both observers are sent to a central
coordinator where hypothesis testing is performed. In the second approach, each
observer performs hypothesis testing based on locally collected observations.
Then they exchange binary information to arrive at a consensus. In the third
approach, each observer constructs an aggregated probability space based on the
observations collected by it and the decision it receives from the alternate
observer and performs hypothesis testing in the new probability space. In this
approach also they exchange binary information to arrive at consensus. In the
fourth approach, if observations collected by the observers are independent
conditioned on the hypothesis we show the construction of the aggregated sample
space can be skipped. In this case, the observers exchange real-valued
information to achieve consensus. Given the same fixed number of samples, n, n
sufficiently large, for the centralized (first) and decentralized (second)
approaches, it has been shown that if the observations collected by the
observers are independent conditioned on the hypothesis, then the minimum
probability that the two observers agree and are wrong in the decentralized
approach is upper bounded by the minimum probability of error achieved in the
centralized approach.
