10 cv mil_graphical_models

Computer vision: models,
learning and inference
Chapter 10
Graphical Models

Please send errata to s.prince@cs.ucl.ac.uk

Independence
• Two variables x1 and x2 are independent if their
joint probability distribution factorizes as
Pr(x1, x2)=Pr(x1) Pr(x2)

Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 2

Conditional independence
• The variable x1 is said to be conditionally
independent of x3 given x2 when x1 and x3 are
independent for fixed x2.

• When this is true the joint density factorizes in a
certain way and is hence redundant.



• Consider joint pdf of three discrete variables x1, x2, x3



• The three marginal distributions show that no pair of variables is independent



• The three marginal distributions show that no pair of variables is independent
• But x1 is independent of x2 given x3


Graphical models
• A graphical model is a graph based
representation that makes both factorization
and conditional independence relations easy
to establish

• Two important types:
– Directed graphical model or Bayesian network
– Undirected graphical model or Markov network


Directed graphical models
• Directed graphical model represents probability
distribution that factorizes as a product of
conditional probability distributions

where pa[n] denotes the parents of node n


Directed graphical models
• To visualize graphical model from factorization
– add one node per random variable and draw arrow to each
variable from each of its parents.

• To extract factorization from graphical model
– Add one term per node in the graph Pr(xn| xpa[n])
– If no parents then just add Pr(xn)


Example 1


Example 1

= Markov Blanket of variable x8 – Parents, children
and parents of children

Example 1

If there is no route between to variables and they
share no ancestors they are independent.

Example 1

A variable is conditionally independent of all others
given its Markov Blanket

Example 1

General rule:


Example 2

This joint pdf of this graphical model factorizes as:


Example 2

This joint pdf of this graphical model factorizes as:

It describes the original example:


Example 2

Here the arrows meet head to tail at x2 and so x1 is
conditionally independent of x3 given x2.

General rule:


Example 2

Algebraic proof:

No dependence on x3 implies that x1 is conditionally
independent of x3 given x2.

Redundancy
Conditional independence can be
thought of as redundancy in the full
distribution

4 + 3x4 + 2x3

4 x 3 x 2 = 24 entries = 22 entries

Redundancy here only very small, but with larger models
can be very significant.

Example 3

Mixture of t-distribution Factor analyzer
Gaussians
Blue boxes = Plates. Interpretation: repeat contents of box
number of times in bottom right corner.
Bullet = variables which are not treated as uncertain

Undirected graphical models
Probability distribution factorizes as:

Partition Product over Potential function
function C functions (returns non-
(normalization negative number)
constant)

Probability distribution factorizes as:

Partition
function
(normalization For large systems, intractable to compute
constant)

Alternative form

Can be written as Gibbs Distribution:

where Cost function
(positive or negative)

Cliques
Better to write undirected model as

Product over Clique
cliques Subset of variables


• To visualize graphical model from factorization
– Draw one node per random variable
– For every clique draw connection from every node to
every other

• To extract factorization from graphical model
– Add one term to factorization per maximal clique (fully
connected subset of nodes where it is not possible to add
another node and remain fully connected)


• Much simpler than for directed models:

One set of nodes is conditionally independent of another
given a third if the third set separates them (i.e.
Blocks any path from the first node to the second)


Example 1

Represents factorization:


Example 1

By inspection of graphical model:

x1 is conditionally independent of x3 given x2 as the
route from x1 to x3 is blocked by x2.


Example 1

Algebraically:

No dependence on x3 implies that x1 is conditionally
independent of x3 given x2.

Example 2

• Variables x1 and x2 form a clique (both connected to
each other)
• But not a maximal clique as we can add x3 and it is
connected to both

Example 2

Graphical model implies factorization:


Example 2

Or could be....

... but this is less general

Comparing directed and undirected models

Executive summary:

• Some conditional independence patterns can be
represented by both directed and undirected
• Some can be represented only by directed
• Some can be represented only by undirected
• Some can be represented by neither



These models represents same independence / There is no undirected model that
conditional independence relations can describe these relations



There is no directed model that Closest example,
can describe these relations but not the same

Graphical models in computer vision

Chain model Interpreting sign
(hidden Markov model) language sequences



Tree model Parsing the human
body



Grid model Semantic
Markov random field segmentation
(blue nodes)


Chain model Tracking contours
Kalman filter


Inference in models with many
unknowns
• Ideally we would compute full posterior
distribution Pr(w1...N|x1...N).
• But for most models this is a very large
discrete distribution – intractable to compute
• Other solutions:
– Find MAP solution
– Find marginal posterior distributions
– Maximum marginals
– Sampling posterior


Finding MAP solution

• Still difficult to compute – must search
through very large number of states to find
the best one.


Marginal posterior distributions

• Compute one distribution for each variable wn.
• Obviously cannot be computed by computing
full distribution and explicitly marginalizing.


Maximum marginals

• Maximum of marginal posterior distribution for
each variable wn.
• May have probability zero; the states can be
individually probably but never co-occur.


Sampling the posterior
• Draw samples from posterior Pr(w1...N|x1...N).
– use samples as representation of distribution
– select sample with highest prob. as point sample
– compute empirical max-marginals
• Look at marginal statistics of samples


Drawing samples - directed

To sample from directed model, use ancestral sampling

• work through graphical model, sampling one variable at
a time.
• Always sample parents before sampling variable
• Condition on previously sampled values


Ancestral sampling example


Ancestral sampling example

To generate one sample:
1. Sample x1* from Pr(x1) 4. Sample x3* from Pr(x3| x2*,x4*)
2. Sample x2* from Pr(x2| x1*) 5. Sample x5* from Pr(x5| x3*)
3. Sample x4* from Pr(x4| x1*, x2*)

Drawing samples - undirected
• Can’t use ancestral sampling as no sense of
parents / children and don’t have conditional
probability distributions
• Instead us Markov chain Monte Carlo method
– Generate series of samples (chain)
– Each depends on previous sample (Markov)
– Generation stochastic (Monte Carlo)
• Example MCMC method = Gibbs sampling


Gibbs sampling
To generate new sample x in the chain
– Sample each dimension in any order
– To update nth dimension xn
• Fix other N-1 dimensions
• Draw from conditional distribution Pr(xn| x1...Nn)
Get samples by selecting from chain
– Needs burn in period
– Choose samples spaced apart so not correlated


Gibbs sampling example: bi-variate
normal distribution


Learning in directed models

Use standard ML formulation

where xi,n is the nth dimension of the ith training example.

Learning in undirected models
Write in form of Gibbs distribution

Maximum likelihood formulation


Learning in undirected models

PROBLEM: To compute first term, we must sum over
all possible states. This is intractable


Contrastive divergence
Some algebraic manipulation


Now approximate:

Where xj* is one of J samples from the distribution.
Can be computed using Gibbs sampling. In practice, it is
possible to run MCMC for just 1 iteration and still OK.

Conclusions
Can characterize joint distributions as
– Graphical models
– Sets of conditional independence relations
– Factorizations
Two types of graphical model, represent
different but overlapping subsets of possible
conditional independence relations
– Directed (learning easy, sampling easy)
– Undirected (learning hard, sampling hard)


10 cv mil_graphical_models

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