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

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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 marginalization 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).
11.2 Bayesian Inference through Belief Propagation 239 θ t θ t+1 I(t) A I(t+1) Figure 11.2 Graphical Model for a HMM and its Neural Implementation. (A) Dynamic graphical model for a hidden Markov model (HMM). Each circle represents a node denoting the state variable θ t which can take on values 1, . . . , N. (B) Recurrent network for implementing on-line belief propagation for the graphical model in (A). Each circle represents a neuron encoding a state i. Arrows represent synaptic connections. The probability distribution over state values at each time step is represented by the entire population. A B Location Locations (L) Features (F) Intermediate Representation (C) Image (I) t B t+1 I(t) Feature Coding Neurons Coding Neurons Figure 11.3 A Hierarchical Graphical Model for Images and its Neural Implementation. (A) Three-level graphical model for generating simple images containing one of many possible features at a particular location. (B) Three-level network for implementing on-line belief propagation for the graphical model in (A). Arrows represent synaptic connections in the direction pointed by the arrow heads. Lines without arrow heads represent bidirectional connections. 11.2.3 Hierarchical Belief Propagation As a third example of belief propagation, consider the three-level graphical model shown in figure 11.3A. The model describes a simple process for generating images based on two random variables: L, denoting spatial locations, and F , denoting visual features (a more realistic model would involve a hierarchy of such features, sub-features, and locations). Both random variables are assumed to be discrete, with L assuming one of n values L1, . . . , Ln, and F assuming one of m different values F1, . . . , Fm. The node C denotes different combinations of features and locations, each of its values C1, . . . , Cp encoding a specific feature at a specific location. Representing all possible combinations is infeasible but it is sufficient to represent those that occur frequently and to map Image
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Page 25 and 26: 11.5 Discussion 259 [28] Pouget A,
Page 27 and 28: 11.5 Discussion 261 50% 100% D 0.05
Page 29 and 30: 11.5 Discussion 263 Normalized Firi
Page 31 and 32: Index acausal, 122 attention, 101 a
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Neural Models of Bayesian Belief Propagation Rajesh ... - Washington

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