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The verb 'to do' plays a vital part in our understanding of the world, and it goes hand-in-hand with words such as active, action and agent. But the physical sciences describe only mechanical happenings, not acts. Their theoretical language is, in essence, a strict mathematical formalism applied to the description of variables usually quantitative ones that can - at least in principle - be measured by mechanical instruments. In such a language, what is the definition of an agent? Of an act? In contrast to previous approaches, which attempt to discriminate between agent and non-agent systems, we pursue a more Dennettian approach that attempts only to characterise the explanatory logic of intentional agentive interpretations of a physical system; we wish to do so purely in terms of the formal relations that hold between variables in a dynamical system or stochastic process. Our approach is straightforward: we use Pearl's causal formalism to identify physical variables at the causal boundary between 'agent' and 'environment', and identify these with variables in Bayesian decision theory; this provides a rigorous bridge between mathematical models of physics and mathematical models of rational decision-making.

The Bayesian stance: Equations for ‘as-if’ sensorimotor agency