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Hamelryck probability distribution encoded in a BN is the product of factors one for each variable , that each consist of the conditional probability distribution of the variable given its parents. For example, consider a discrete node which can adopt two values that has two parents which can adopt three and four values, respectively. The resulting CPT will be a 4 x 3 x 2 table, with 12 unique parameters because the probabilities need to sum to one. The case of a Gaussian node with one discrete parent is typically identical to the mixture model described in Sect.

The example also clarifies that different BNs can give rise to the same probability distribution: in Fig. In our example, the graph is fully connected: all nodes are connected to each other. The strength of BNs lies in the fact that one can leave out connections, which induces conditional independencies in the joint probability distribution. Hence, a BN is a carrier of conditional independence relations among a set of variables.

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In other words, it defines the set of possible factorizations for a joint probability distribution. One of the appealing properties of BNs is that it is easy to examine these independence relationships by inspecting the graph [62]. This includes conditional independencies that are not limited to individual nodes. Consider the BN in Fig.

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The graph of this BN is clearly not fully connected, and it is easy to see that this can speed up inference related calculations tremendously. According to the rules specified above, d and e are conditionally independent given a, as two arrows in the single path between d and e meet tail-to- tail in a. However, when we make use of the conditional independencies as encoded in the BN, it is easy to see that this computation can be performed much more efficiently. The strength of BNs lies in the fact that it is possible to construct algorithms that perform these clever ways of inference in an automated way.

This will be discussed in Sect.

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However, many sequential variables, such as protein 38 T. The arrows denote non-zero transition probabilities, which are shown next to the arrows, right The transition matrix associated with the shown state diagram.

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Zeros in the matrix correspond to missing arrows in the state diagram sequences, do not have a temporal character. Each sequence position, which corresponds to one slice in the DBN, is represented by one hidden or latent node h that has one observed node x as child.

Formally, this corresponds to creating additional nodes and edges. The parameters associated with these nodes and edges are identical to those in the previous slices.

Hence, a DBN can be fully specified by two slices: an initial slice at the first position and the slice at the second position. Sequences of any length can then be modelled by adding additional slices that are identical to the slice at position one. In the HMM example shown in Fig. Why are two slices needed, instead of one? This node has no parents, and thus its conditional probability distribution will have a different number of parameters than the - shared - probability distribution of the consecutive hidden nodes, which do have a parent.

Hence, we need to specify a starting slice and one consecutive slice to specify the model fully. Such a diagram is shown in Fig. Figure 1. Given the superficial similarity of a DBN graph and the HMM state diagram, it is important to understand the distinction. Typically, the state diagram of an HMM, which specifies the possible state transitions but not their probabilities, is decided on before parameter estimation. Inference of the state diagram itself, as opposed to inference of the transition probabilities, is called structure learning in the HMM community; see for example [] for structure learning in secondary structure prediction.

Such models quickly become intractable [,]. There is no need to develop custom inference algorithms for every variant under this view. One of the attractive aspects of BNs is that they 40 T. In addition, several important sampling operations are trivial in BNs. Generating samples from the joint probability distribution encoded in the BN can be done using ancestral sampling [62]. In order to do this, one first orders all the nodes such that there are no arrows from any node to any lower numbered node. In such an ordering, any node will always have a higher index than its parents.

Sampling is initiated starting at the node with the lowest index which has no parents , and proceeds in a sequential manner to the nodes with higher indices. At any point in this procedure, the values of the parents of the node to be sampled are available. When the node with the highest index is reached, we have obtained a sample from the joint probability distribution. Typically, the nodes with low indices are latent variables, while the nodes with higher indices are observed. Ancestral sampling can easily be illustrated by sampling in a Gaussian mixture model.

In such a model, the hidden node h is assigned index zero, and the observed node o is assigned index one. For most BNs with nodes that represent discrete variables, this will be the most common node type and associated sampling method. Next, we sample a value for the observed node o, conditional upon the sampled values of its parents.

The pair y, h is a sample from the joint probability distribution represented by the Gaussian mixture model. Ancestral sampling can also be used when some nodes are observed, as long as all observed nodes either have no parents, or only observed parents. The observed nodes are simply clamped to their observed value instead of resampled.

However, if some observed nodes have one or more unobserved parents, sampling of the parent nodes needs to take into account the values of the observed children, which cannot be done with ancestral sampling. In such cases, one can resort to Monte Carlo sampling techniques [62,] such as Gibbs sampling [], as discussed in Sect. In general. In MRF, the joint probability distribution is a normalized product of potential functions, which assign a positive value to a set of nodes.

For the MRF in Fig. Factor graphs consists of two nodes types: factor nodes and variable nodes. The factor nodes represent positive functions that act on a subset of the variable nodes. The edges in the graph denote which factors act on which variables. The factor nodes are shown in grey. Then, each conditional distribution in the BN gives rise to one factor node, which connects each child node with its parents Fig.

A MRF can be represented as a factor graph by using one variable node in the factor graph for each node in the MRF, and using one factor node in the factor graph for each clique potential in the MRF.

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Each factor node is then connected to the variables in the clique. In other cases, one wants to find the hidden node values that maximize ;; h x, 0. In some cases, the goal of inference is to obtain quantities such as entropy or mutual information, which belong to the realm of information 1 An Overview of Bayesian Inference and Graphical Models 43 theory. The latter are always approximative, while the former can be exact or approximative. Message-passing algorithms occur in Chap. As mentioned before, factor graphs provide a unified view on all graphical models, including Bayesian networks and Markov random fields.

Many algorithms that were developed independently in statistical physics, machine learning, signal processing and communication theory can be understood in terms of message passing in factor graphs. These algorithms include the forward-backward and Viterbi algorithms in HMMs, the Kalman filter, and belief propagation in Bayesian networks [62, , ], In message passing algorithms applied to factor graphs, variable nodes V send messages to factor nodes F, and vice versa. Each message that is sent is a vector over all the possible states of the variable node. Messages from a factor node F to a variable node V contain information on the probabilities of the states of V according to F.

Messages from a variable node V to a factor node F contain information on the probabilities of the states of V based on all its neighboring nodes except F.

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The sum-product algorithm is a message passing algorithm that infers the local marginals of the hidden nodes; the max-sum algorithm finds the values of the hidden nodes that result in the highest probability. The application of the sum- product algorithm to Bayesian networks results in the belief propagation algorithm, originally proposed by Pearl [].