wolffd@0: The following kinds of potentials are supported
wolffd@0: - dpot: discrete
wolffd@0: - upot: utility
wolffd@0: - mpot: Gaussian in moment form
wolffd@0: - cpot: Gaussian in canonical form
wolffd@0: - cgpot: conditional (mixture) Gaussian, a list of mpots/cpot
wolffd@0: - scgpot: stable conditional Gaussian, a list of scgcpots
wolffd@0: - scgcpot: just used by scgpot
wolffd@0: 
wolffd@0: Many of these are described in the following book
wolffd@0: 
wolffd@0: @book{Cowell99,
wolffd@0:   author = "R. G. Cowell and A. P. Dawid and S. L. Lauritzen and D. J. Spiegelhalter",
wolffd@0:   title = "Probabilistic Networks and Expert Systems",
wolffd@0:   year = 1999,
wolffd@0:   publisher = "Springer"
wolffd@0: }
wolffd@0: 
wolffd@0: CPD_to_pot converts P(Z|A,B,...) to phi(A,B,...,Z).
wolffd@0: 
wolffd@0: A table is like a dpot, except it is a structure, not an object.
wolffd@0: Code that uses tables is faster but less flexible.
wolffd@0: 
wolffd@0:                          -----------
wolffd@0: 
wolffd@0: A potential is a joint probability distribution on a set of nodes,
wolffd@0: which we call the potential's domain (which is always sorted).
wolffd@0: A potential supports the operations of multiplication and
wolffd@0: marginalization.
wolffd@0: 
wolffd@0: If the nodes are discrete, the potential can be represented as a table
wolffd@0: (multi-dimensional array). If the nodes are Gaussian, the potential
wolffd@0: can be represented as a quadratic form. If there are both discrete and
wolffd@0: Gaussian nodes, we use a table of quadratic forms. For details on the
wolffd@0: Gaussian case, see below.
wolffd@0: 
wolffd@0: For discrete potentials, the 'sizes' field specifies the number of
wolffd@0: values each node in the domain can take on. For continuous potentials,
wolffd@0: the 'sizes' field specifies the block-size of each node.
wolffd@0: 
wolffd@0: If some of the nodes are observed, extra complications arise.  We
wolffd@0: handle the discrete and continuous cases differently.  Suppose the
wolffd@0: domain is [X Y], with sizes [6 2], where X is observed to have value x.
wolffd@0: In the discrete case, the potential will have many zeros in it
wolffd@0: (T(X,:) will be 0 for all X ~= x), which can be inefficient. Instead,
wolffd@0: we set sizes to [1 2], to indicate that X has only one possible value
wolffd@0: (namely x). For continuous nodes, we set sizes = [0 2], to indicate that X no
wolffd@0: longer appears in the mean vector or covariance matrix (we must avoid
wolffd@0: 0s in Sigma, lest it be uninvertible). When a potential is created, we
wolffd@0: assume the sizes of the nodes have been adjusted to include the
wolffd@0: evidence. This is so that the evidence can be incorporated at the
wolffd@0: outset, and thereafter the inference algorithms can ignore it.
wolffd@0: 
wolffd@0:                          ------------
wolffd@0: 
wolffd@0: A Gaussian potential can be represented in terms of its
wolffd@0: moment characteristics (mu, Sigma, logp), or in terms of its canonical
wolffd@0: characteristics (g, h, K). Although the moment characteristics are
wolffd@0: more familiar, it turns out that canonical characteristics are
wolffd@0: more convenient for the junction tree algorithm, for the same kinds of
wolffd@0: reasons why backwards inference in an LDS uses the information form of
wolffd@0: the Kalman filter (see Murphy (1998a) for a discussion).
wolffd@0: 
wolffd@0: When working with *conditional* Gaussian potentials, the method proposed
wolffd@0: by Lauritzen (1992), and implemented here, requires converting from
wolffd@0: canonical to moment form before marginalizing the discrete variables,
wolffd@0: and converting back from moment to canonical form before
wolffd@0: multiplying/dividing. A new algorithm, due to Lauritzen and Jensen
wolffd@0: (1999), works exclusively in moment form, and
wolffd@0: hence is more numerically stable. It can also handle 0s in the
wolffd@0: covariance matrix, i.e., deterministic relationships between cts
wolffd@0: variables. However, it has not yet been implemented,
wolffd@0: since it requires major changes to the jtree algorithm.
wolffd@0: 
wolffd@0: In Murphy (1998b) we extend Lauritzen (1992) to handle
wolffd@0: vector-valued nodes. This means the vectors and matrices become block
wolffd@0: vectors and matrices. This manifests itself in the code as in the
wolffd@0: following example.
wolffd@0: Suppose we have a potential on nodes dom=[3,4,7] with block sizes=[2,1,3].
wolffd@0: Then nodes 3 and 7 correspond to blocks 1,3 which correspond to indices 1,2,4,5,6.
wolffd@0: >> find_equiv_posns([3 7], dom)=[1,3]
wolffd@0: >> block([1,3],blocks)=[1,2,4,5,6].
wolffd@0: 
wolffd@0: For more details, see
wolffd@0: 
wolffd@0: - "Filtering and Smoothing in Linear Dynamical Systems using the Junction Tree Algorithm",
wolffd@0:    K. Murphy, 1998a. UCB Tech Report.
wolffd@0: 
wolffd@0: - "Inference and learning in hybrid Bayesian networks",
wolffd@0:    K. Murphy. UCB Technical Report CSD-98-990, 1998b.
wolffd@0: 
wolffd@0: - "Propagation of probabilities, means and variances in mixed
wolffd@0:   graphical association models", S. L. Lauritzen, 1992, JASA 87(420):1098--1108.
wolffd@0: 
wolffd@0: - "Causal probabilistic networks with both discrete and continuous variables",
wolffd@0:   K. G. Olesen, 1993. PAMI 3(15). This discusses implementation details.
wolffd@0: 
wolffd@0: - "Stable local computation with Conditional Gaussian distributions",
wolffd@0:   S. Lauritzen and F. Jensen, 1999. Univ. Aalborg Tech Report R-99-2014.
wolffd@0:   www.math.auc.dk/research/Reports.html.