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70 Artificial Intelligence for the Internet of Everything

every good regulator/controller of a system must be a model of that system
(Conant & Ross Ashby, 1970; Smith, 1959). The basic assumption under-
lying human decision making in dynamic contexts is that well-trained and
motivated humans behave in a normative, rational manner subject to their
sensory and neuro-motor limitations, as well as perceived task objectives
(e.g., Kleinman, Baron, & Levison, 1971; Pattipati, Kleinman, &
Ephrath, 1983).

4.2.3 Formal Definitions

Given an agent and its generative model of environment m, we formally
assert that a purpose-driven adaptive system “behaves rationally” if it max-
imizes the model evidence, a probability distribution p(ojm) over observations
o conditioned on the model of the environment m, or equivalently mini-
mizes a measure of surprise:

Surprise o, mÞ ¼ lnpoj mÞ:
ð
ð
A direct optimization of model evidence or surprise is intractable due to
marginalization over all possible hidden states of the world (Friston, 2012).
Recently, researchers conjectured that the only tractable way to optimize
surprise is to minimize the variational free energy F(o,b), an information-
theoretic function of outcomes o, and an internal state of the agent defined
as a probability density b over (hidden) causes of these outcomes (Friston,
Thornton, & Clark, 2012):
ފ Hq sj bފ ,
Fo, bÞ ¼ E q lnps, oj mð½ ½ ð
ð
|fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflffl{zfflfflfflfflffl}
average energy entropy
where:
• p(s,ojm)isa generative density representing the joint probability of world
states s and observations o based on an agent model m;
• q(sjb)isa recognition density that defines an agent’s beliefs about the hidden
states s given internal state of agent b;
• E q [ ] is the expected value over recognition density, i.e., E q [ lnp(s,
P
s q(sjb)lnp(s,ojm); and
ojm)]¼
P
• H[ ]istheentropyoftherecognitiondensity,i.e.,H[q(sjb)]¼ s q(sjb)
lnq(sjb).
By rewriting the free energy function, we can obtain several interpretations
of how adaptive agents “behave.” First, free energy is equal to the sum of