Order Effects on Measurements in Multi-Agent Hypothesis Testing
In multi-agent systems, agents observe data, and use them to make inferences and take actions. As a result sensing and control naturally interfere, more so from a real-time perspective. A natural consequence is that in multi-agent systems there are propositions based on the set of observed events that might not be simultaneously veriﬁable, which leads to the need for probability structures that allow such incompatible events. We revisit the structure of events in a multi-agent system and we introduce the necessary new models that incorporate such incompatible events in the formalism. These models are essential for building non-commutative probability models, which are diﬀerent than the classical models based on the Kolmogorov construction. From this perspective, we revisit the concepts of event-state-operation structure and the needed relation- ship of incompatibility from the literature and use them as a tool to study the needed new algebraic structure of the set of events. We present an example from multi-agent hypothesis testing where the set of events does not form a Boolean algebra, but forms an ortholattice. A possible construction of
a ‘noncommutative probability space’, accounting for incompatible events is discussed. We formulate and solve the binary hypothesis testing problem in the noncommutative probability space. We illustrate the occurrence of ‘order eﬀects’ in the multi-agent hypothesis testing problem by computing the minimum probability of error that can be achieved with diﬀerent orders of measurements.