K. Murphy. Inference and learning in hybrid bayesian networks. T.R. 990, U.C.Berkeley, Dept. Comp. Sci, 1998.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....of the function is a ridge along the line X Gamma 4 and the width of the ridge is determined by the variance of P (Y j X) which is 1 in this case. To deal with the problem of representing linear CPDs, we use a different representation called canonical form or canonical characteristics [Lau92, Mur98] A canonical form represents a function of the form e where Q is some quadratic function. More precisely, we define C (X;K;h; g) or C (K; h; g) where we omit the random variables) C (X;K;h; g) KX h X g (3.5) We can represent every Gaussian as a canonical form. Rewriting ....

....possible to enter evidence into a canonical form, i.e. set the values of some of its variables. The result is a canonical form over the unobserved variables. Assume the canonical form C (K; h; g) is given by Equation 3. 8, then setting Y = y results in the canonical form C (X;K ) given by [Mur98] KXX = hX Gamma KXY y = g h Y y Gamma KY Y y (3.9) Since we can perform all these operations with canonical forms, we can adapt the clique tree algorithm from Section 2.3.1 to linear Gaussians, representing the factors as canonical forms. There are two issues that must be ....

K. Murphy. Inference and learning in hybrid bayesian networks. Technical Report CSD-98-990, Department of Computer Science, U.C. Berkeley, 1998.

....it is possible to regard X also as a multidimensional random variable, but for the purpose of the article it is sufficient to use a one dimensional distribution. If not only continuous variables are used or if non linearities are required, these needs are met by hybrid BNs as described in [3] 7] [5] and [4] The set of nodes of a hybrid BN contains both discrete and continuous nodes. Discrete nodes having only discrete predecessors are handled as usual. i.e. each node X i stores the conditional probabilities P (X i jP a(i) in a table which is used for calculation of joint and marginal ....

Kevin P. Murphy, `Inference and learning in hybrid bayesian networks', Technical report, University of California, Computer Science Division (EECS), (January 1998).

....the covariance of the input node to a maximum to tell the system, that there is no a priori information about the correct value of the input signal. The other covariances remain unchanged to keep the accurate modeling behavior. For a further improvement we used the EM algorithm (see [2] 13] [15] or [14] to learn the mean of the input nodes. We tested the reference action of the control loop on systems with proportional response with second order delay. This systems are described by a differential equation of second order. 5 Experiments Our system was tested with a Bayesian network ....

Kevin P. Murphy. Inference and Learning in Hybrid Bayesian Networks. Technical report, University of California, Computer Science Division (EECS), January 1998.

....a threshold value of 0:5 in logarithmic (base 2) scale. It is clear that by discretizing the measured expression levels we are loosing information. An alternative to discretization is using (semi)parametric density models for representing conditional probabilities in the networks we learn (e.g. [23, 26, 30]) However, a bad choice of the parametric family can strongly bias the learning algorithm. We believe that discretization provides a reasonably unbiased approach for dealing with this type of data. We are currently exploring the appropriateness of several density models for this type of data. 4 ....

Kevin Murphy. Inference and learning in hybrid Bayesian networks. Technical Report CSD-98-990, U.C. Berkeley, 1998.

....instead of a table of scalars. Finally, if some variables are discrete random variables and some are discrete utility variables, we can represent the potential as a pair of tables; this is useful for in uence diagrams. All of these types of potentials are described in [CDLS99] see also [Mur98b] CG potentials can represent nite mixtures of Gaussians. Unfortunately, this representation is not closed. That is, if we have a potential with domain (D; C) where D is a discrete variable with k possible values, and C is a continuous variable, then P D (D; C) is still a mixture of k ....

K. P. Murphy. Inference and learning in hybrid Bayesian networks. Technical Report 990, U.C. Berkeley, Dept. Comp. Sci, 1998.

....the conditional distributions can be represented as conditional probability tables, called CPTs. See Table 2 for an example. However, we can also allow the nodes to be continuous and employ conditional Gaussians. Both CPTs and Gaussian parameters can be learned from training data using EM. See [13] for more details. There are two computational tasks that must be performed in order to use these networks as classifiers. After the network topology has been specified, the first task is to obtain the local CPT for each variable conditioned on its parent(s) Once the CPTs have been specified ....

K. P. Murphy. Inference and learning in hybrid Bayesian networks. Technical Report 990, U.C. Berkeley, Dept. Comp. Sci, 1998.

.... probability on the remaining hidden nodes must be expressable in one of three forms: as a table, as a Gaussian, or as a conditional Gaussian (a table of Gaussians) since these are the only forms of clique potential that support the operations of marginalization and multiplication in closed form [LW89, Mur98a]. This means that, if X is hidden, it must have a Gaussian distribution, and if X and Q are hidden, there cannot be an arc from X to Q. If X or Q or both are observed, however, we can imagine having an arc from X to Q where Q has e.g. a softmax distribution. The situation is summarized in Table ....

K. P. Murphy. Inference and learning in hybrid Bayesian networks. Technical Report 990, U.C. Berkeley, Dept. Comp. Sci, 1998.

....the conditional distributions can be represented as conditional probability tables, called CPTs. See Table 2 for an example. However, we can also allow the nodes to be continuous and employ conditional Gaussians. Both CPTs and Gaussian parameters can be learned from training data using EM. See [13] for more details. There are two computational tasks that must be performed in order to use these networks as classifiers. After the network topology has been specified, the first task is to obtain the local CPT for each variable conditioned on its parent(s) Once the CPTs have been specified ....

K. P. Murphy. Inference and learning in hybrid Bayesian networks. Technical Report 990, U.C. Berkeley, Dept. Comp. Sci, 1998.

.... probability on the remaining hidden nodes must be expressable in one of three forms: as a table, as a Gaussian, or as a conditional Gaussian (a table of Gaussians) since these are the only forms of clique potential that support the operations of marginalization and multiplication in closed form [Mur98a]. This means that, if X is hidden, it must have a Gaussian distribution, and if X and Q are hidden, there cannot be an arc from X to Q. If X or Q or both are observed, however, we can imagine having an arc from X to Q where Q has e.g. a softmax distribution. The situation is summarized in Table ....

K. P. Murphy. Inference and learning in hybrid Bayesian networks. Technical Report 990, U.C. Berkeley, Dept. Comp. Sci, 1998.

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K. Murphy. Inference and learning in hybrid bayesian networks. T.R. 990, U.C.Berkeley, Dept. Comp. Sci, 1998.

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K. Murphy. Inference and learning in hybrid bayesian networks. Technical Report 990, U.C.Berkeley, Dept. Comp. Sci, 1998.

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