https://xkcd.com/835/
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Two graphical model tutorials: this
week and next week.
Hope you’re going well in assignment 2
Final exam Fri 3 June 5:40 – 8:40 pm Canberra
○ On Wattle
○ Self-invigilated, video
recording needed ○ Instructions will be
Graphical model inference
Factor graphs
The sum-product algorithm
Other algorithms
High-level goal:
One representation for both directed and
this representation to facilitate inference.
[Frey, 1998, Kschischnang et al 2001]
undirected graphical models.
Bishop 8.4.3, 8.4.4
Factor graphs
s: a subset of variables
Joint distribution → a product of factors
Contains more info than MRF, because fa(x1, x2)
and fb(x1, x2) would be in one potential function.
Directed graph: conditionals
Undirected graph: potential functions over max cliques→ factors
Z is a factor too, over an empty set of variables.
graphs representing
same distribution
graphs representing
same distribution
Factor graphs are
Factor graph for a poly-tree
Factor graphs for directed/undirected trees, and directed poly trees retain a tree structure.
factor graphs representing the same (undirected) graph
Factorization in (c) does not correspond to any conditional independence properties.
The sum-product algorithm
Goal: evaluate local marginals over nodes or subsets of nodes. Variant: find the most probable state → max sum algorithm.
Assume (for instructional convenience): all variables
For continuous variables – perform integration, e.g.
Historic note:
belief propagation (Pearl, 1988; Lauritzen and Spiegelhalter,
inference on DAGs. It is a special case of the sum product algorithm (Frey, 1998;
Kschischnang et al., 2001)
discrete. Marginalization = sums
linear dynamic systems (e.g. Chap 13.3)
Recall from last lecture …
The sum-product algorithm
s: a subset of nodes
ne(x) : neighbouring factor nodes
Xs: the set of all variables in the subtree connected to
the variable node x via the factor node fs Fs(x,Xs): the product of all the factors in the group associated with factor fs.
A first example of
sum-product
Factorise this
F (x,X ): the product of all the factors in the group associated with factor f .
factor subgraph:
ne(fs) : neighbours of factor node fs
ne(fs)\x : neighbours of factor node fs except x
Xs: the set of all variables in the subtree connected to the variable node x via the factor node fs
= {x, x1, … , x
Two different
kinds of messages:
Computing factor-to-variable-message
● take the product of the incoming messages along all other links coming into the factor
● multiply by the factor associated with that node
● marginalize over all of the variables associated with the incoming messages
can send a message once the factor node has received incoming
messages from all
other neighbouring variable nodes
{x1, … , xM
start, how
to normalise?
Aji, S.M.; McEliece, R.J. (Mar 2000). “The generalized distributive law”. IEEE Transactions Information Theory. 46 (2): 325–343. doi:10.1109/18.825794.
Graphical model inference
Factor graphs
The sum-product algorithm
Other algorithms
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