Page 92 - Artificial Intelligence for the Internet of Everything

78 Artificial Intelligence for the Internet of Everything

computes max-marginal distributions by iteratively passing belief messages
between variable and factor nodes in the factor graph (Fig. 4.4A) using
the following update equations:
• Variable-to-factor message updates involve multiplication of all except one
of the incoming beliefs:
Y
c2NiðÞnj
m i!j d i ¼ m c!i d i ðÞ
ðÞ
• Factor-to-variable message updates are conducted by maximizing compo-
nent functions:
Y
m j!i d i ðÞ ¼ maxφ d j v2NjðÞni m v!j d v
j
ðÞ
d j nd i
In the above, N(i) denotes the neighbor nodes of node i in a factor graph. For
a team of agents, some of the computations above are conducted locally by a
single agent, while the message passing across the links between variables and
factors that are assigned to different agents need to be physically passed
among the agents (Fig. 4.4B). These messages define formal local-global
decision making, while the collaborative process is formally specified as
the belief messages sent between connected agents.
Variable-to-factor messages contain the beliefs about the variable, inter-
preted as experience messages. Factor-to-variable messages are the prob-
ability distributions over decision values at a corresponding variable node,
corresponding to influence messages. The free energy principle and
approximate inference with a max-product algorithm only requires the
agents to understand the dependencies of their local decisions on the deci-
sions of other agents; they must also be capable of creating, sharing and inter-
preting the experience and influence messages.
After a max-product algorithm converges, the agents obtain the max-
marginal probability distributions, and use these distributions to sample

1 2 3 4
1 2 3 4
3
2
3 1
2
1
(A) (B)
Fig. 4.4 Message passing in belief propagation. (A) Passing beliefs in factor graph. (B)
Collaboration by message passing.