Neural Models of Bayesian Belief Propagation Rajesh ... - Washington

238 11 Neural Models of Bayesian Belief Propagation Rajesh P. N. Rao
in figure 11.2A. The input that is observed at time t (= 1, 2, . . .) is represented by
the random variable I(t), which can either be discrete-valued or a real-valued
vector such as an image or a speech signal. The input is assumed to be generated
by a hidden cause or “state” θ(t), which can assume one of N discrete
values 1, . . . , N. The state θ(t) evolves over time in a Markovian manner, depending
only on the previous state according to the transition probabilities
given by P (θ(t) = i|θ(t − 1) = j) = P (θt i |θt−1 j ) for i, j = 1 . . . N. The observation
I(t) is generated according to the probability P (I(t)|θ(t)).
The belief propagation algorithm can be used to compute the posterior probability
of the state given current and past inputs (we consider here only the
“forward” propagation case, corresponding to on-line state estimation). As in
the previous example, the node θt performs a marginalization over neighboring
variables, in this case θt−1 and I(t). The first marginalization results in a
probability vector whose ith component is
is
j P (θt i |θt−1 j )m t−1,t
j where m t−1,t
j
the jth component of the message from node θt−1 to θt . The second marginal-
ization is from node I(t) and is given by
I(t) P (I(t)|θt i
lar input I ′ is observed, this sum becomes
)P (I(t)). If a particu-
I(t) P (I(t)|θt i )δ(I(t), I′ ) = P (I ′ |θ t i ),
where δ is the delta function which evaluates to 1 if its two arguments are equal
and 0 otherwise. The two “messages” resulting from the marginalization along
the arcs from θ t−1 and I(t) can be multiplied at node θ t to yield the following
message to θ t+1 :
m t,t+1
i
= P (I ′ |θ t i)
If m 0,1
i
Bayes rule that m t,t+1
i
j
P (θ t i|θ t−1
j )m t−1,t
j
(11.4)
= P (θi) (the prior distribution over states), then it is easy to show using
= P (θt i , I(t), . . . , I(1)).
Rather than computing the joint probability, one is typically interested in
calculating the posterior probability of the state, given current and past inputs,
i.e., P (θt i |I(t), . . . , I(1)). This can be done by incorporating a normalization step
at each time step. Define (for t = 1, 2, . . .):
m t i = P (I ′ |θ t i)
P (θ t i|θ t−1
j )m t−1,t
j
(11.5)
j
m t,t+1
i = m t i/n t , (11.6)
where nt =
is easy to see that:
j mt j
. If m0,1
i
= P (θi) (the prior distribution over states), then it
m t,t+1
i = P (θ t i|I(t), . . . , I(1)) (11.7)
This method has the additional advantage that the normalization at each time
step promotes stability, an important consideration for recurrent neuronal networks,
and allows the likelihood function P (I ′ |θt i ) to be defined in proportional
terms without the need for explicitly calculating its normalization factor (see
section 11.4 for an example).