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https://xkcd.com/1056/
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Two graphical model tutorials: week 10 and week 11
Final exam Fri 3 June
5:40 – 8:40 pm Canberra
¡ð On Wattle
¡ð Self-invigilated, video
recording needed
¡ð Instructions will be

Markov random
Conditional independence
Relation to directed graphs
Inference in
graphical models (part 1) –
and factorisation
chains and trees
Bishop 8.3, 8.4.1-8.4.2

Random fields

Definition: Undirected
graph separation

Separation in
undirected vs directed

Conditional
independence in MRFs

Factorisation in

Cliques in graphs
A clique is a subset of nodes in a graph such that there exists an edge between pairs of nodes in the subset. (The nodes in a clique are fully connected.)
maximal clique of a graph
clique. (No other nodes of the graph
destroying the property of full connectedness.)
is a clique which
can be added
a proper subset of another
to a maximal clique without

Factorisation using cliques
Potential function
Normalisation factor / partition function
Major limitation of MRFs, computing this for the entire graph
is usually hard.

Conditional
independence and factorisation

Strictly positive potential functions
Potential function: expressing which configurations of the local variables are
preferred to others.
Global configurations
probability: those with a good balance in satisfying the (possibly conflicting) influences of the clique potentials.
with relatively high

image denoising

corresponding corrupted pixel tend to be the
Neighbouring
pixels tend to be the same.
A pixel is more likely to be background.

Graphical model 2
Conditional independence
Relation to directed graphs
Inference in
graphical models (part 1)
and factorisation

Directed vs
undirected chain graphs
Directed vs undirected chain graphs have the same factors.
Partition function Z

Moralisation

Example taken from https://web.mit.edu/jmn/www/6.034/d-separation.pdf

Bayes Net and MRFs

Graphical model 2
Conditional independence
Relation to directed graphs
Inference in
graphical models (part 1)
and factorisation

Inference in graphical
other nodes.
General ideas:
in graphical
observed values, compute the
¡ñ Exploit graphical structure
models: with some of the nodes in
posterior distributions of one or
– efficiency, transparency
¡ñ Many algorithms expressed in local ¡°message propagation¡± around the graph
¡ñ Focusing on exact inference in this class
(see Chap 10 for approximate inference algorithms)
an essential step
in learning (c.f. EM algorithm)
a graph are clamped to
subsets of

on the simplest graphical model

on a chain graph
to compute p(xn), naive algorithm: O(KN-1)
removing xN
also (8.49)

Recursive evaluation of messages

Other marginals and conditional
¡ñ Computing all marginals
Repeat the above N
times O(N2K2)
Storing all intermediate messages: O(N2K2)
¡ñ Observed nodes
Clamp instead of sum over
¡ñ Computing joint probabilities
distributions

Tree-shaped graphical models
Undirected graph: there is one, and only one, path between any pair
Directed graphs: there is all other nodes have one
Moralisation will
a single node, parent.
not add links ¡ú convert between directed and
Inference can be done similarly ¡ú
called the root, which has
sum-product algorithm, stay tuned
undirected tree.
no parents, and

Markov random
Conditional independence
Relation to directed graphs
Inference in
graphical models (part 1) –
and factorisation
chains and trees

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