deal (version 1.1-1)

localmaster: Local master

Description

Calculates the joint distribution of a node and its parents from the joint prior.

Usage

localmaster(family,nw,prior=jointprior(nw))
printmaster(nw,prior=jointprior(nw))

Arguments

nw

The network.

family

Indices of node and parents of the node.

prior

A joint prior, see jointprior.

Details

Called by cond.node. For the discrete part of the network, the master is the marginal distribution of the discrete nodes in the family. For the mixed part of the network, for each configuration $i$ of the discrete variables in family, the joint parameter priors are given by jointprior as $$p(m_i|{\mathrm{\Sigma }}_i)=N({\mu }_i,{\mathrm{\Sigma }}_i/{\nu }_i)$$ $$p({\mathrm{\Sigma }}_i)=IW({\rho }_i,{\mathrm{\Phi }}_i)$$ where IW denotes the inverse Wishart distribution. Then, the local master for configuration $i$ is deduced for the family $A$ as $${\mathrm{\Sigma }}_{A\cap \mathrm{\Gamma }|i_{A\cap \mathrm{\Delta }}}\sim IW({\rho }_{i_{A\cap \mathrm{\Delta }}},{\tilde{\mathrm{\Phi }}}_{A\cap \mathrm{\Gamma }|i_{A\cap \mathrm{\Delta }}})$$ $$m_{A\cap \mathrm{\Gamma }|i_{A\cap \mathrm{\Delta }}}|{\mathrm{\Sigma }}_{A\cap \mathrm{\Gamma }|i_{A\cap \mathrm{\Delta }}}\sim N({\overline{\mu }}_{A\cap \mathrm{\Gamma }|i_{A\cap \mathrm{\Delta }}},{\mathrm{\Sigma }}_{A\cap \mathrm{\Gamma }|i_{A\cap \mathrm{\Delta }}}/{\nu }_{A\cap \mathrm{\Delta }})$$ where $\Gamma$ is the set of continuous nodes and $\Delta$ is the set of discrete nodes. Furthermore, $${\rho }_{i_{A\cap \mathrm{\Delta }}}=\sum _{j:j_{A\cap \mathrm{\Delta }}=i_{A\cap \mathrm{\Delta }}}{\rho }_j$$ and likewise for $\nu_{i_{A\cap\Delta}}$ and $\Phi_{i_{A\cap\Delta}}$. Finally, $${\overline{\mu }}_{A\cap \mathrm{\Delta }=(\sum _{j:j_{A\cap \mathrm{\Delta }}=i_{A\cap \mathrm{\Delta }}}{\mu }_j{\nu }_j)/{\nu }_{i_{A\cap \mathrm{\Delta }}}}$$ $${\tilde{\mathrm{\Phi }}}_{A\cap \mathrm{\Gamma }|i_{A\cap \mathrm{\Delta }}}={\mathrm{\Phi }}_{i_{A\cap \mathrm{\Delta }}}+\sum _{j:j_{A\cap \mathrm{\Delta }}=i_{A\cap \mathrm{\Delta }}}{\nu }_j({\mu }_j-{\overline{\mu }}_{i_{A\cap \mathrm{\Delta }}})({\mu }_j-{\overline{\mu }}_{i_{A\cap \mathrm{\Delta }}}{)}^{\mathrm{\top }}$$

References

Further information about Deal can be found at: http://www.math.auc.dk/novo/deal.

See Also

cond.node