Chapter 14 (14.1, 14.2, 14.3, 14.4)
• Capturing uncertain knowledge
• Probabilistic inference
Copyright, 1996 © Dale Carnegie & Associates, Inc.

Joint probability distribution

 can answer any question about the domain can become intractably large as #RV grows
 can be difficult to specify P for atomic events
Conditional independence can simplify probabilistic assignment
A data structure -
or
that represents the dependence between variables and gives a concise specification of the joint.
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A Bayesian network is a graph:


A set of random variables
A set of directed links connects pairs of nodes


Each node has a conditional P table that quantifies the effects that the parents have on the node
The graph has no directed cycles (DAG)
It is usually much easier for an expert to decide conditional dependence relationships than specifying probabilities

Sometimes, experts can have very different opinions 
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Once the network is specified, we need only specify conditional probabilities for the nodes that participate in direct dependencies, and use those to compute any other probabilities.
A simple Bayesian network (Fig 14.1)
An example of burglary-alarm-call (Fig 14.2)
The topology of the network can be thought of as the general structure of the causal process.
Many details (Mary listening to loud music, or phone ringing and confusing John) are summarized in the uncertainty associated with the links from Alarm to
JohnCalls and MaryCalls.
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The probabilities actually summarize a potentially infinite set of possible circumstances
Specifying the CPT for each node (Fig 14.2)

A conditioning case - a possible combination of values for the parent nodes (2 n )


Each row in a CPT must sum to 1
A node with no parents has only one row (priors)
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Two equivalent views of a Bayesian network


Representing the JPD - helpful in understanding how to construct networks
Representing conditional independence relations helpful in designing inference procedures
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A Bayesian network provides a complete description of the domain. Every entry in the
JPD can be calculated from the info in the network.
A generic entry in the joint is the probability of a conjunction of particular assignments to each variable.
1
n

i
i
(14.1)
What’s the probability of the event of
J^M^A^!B^!E?


=P(j|a)P(m|a)P(a|!b^!e)P(!b)P(!e)
Find the values in Figure 14.2 and done
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Eq 14.1 defines what a given BN means but implies certain conditional independence relationships that can be used to guide the construction.
P(x
1
,…,x n
)=P(x n
|x n-1
,…,x
1
)P(x n-1
,…,x
1
)
 continue for P(x n-1
,…,x
1
), we get (14.2) below
P(X i
|X i-1
,…,X
1
)=P(X i
|Parents(X i
)) (14.2)
The BN is a correct representation of the domain only if each node is C-independent of its predecessors in the node ordering, given its parents.

E.g., P(M|J,A,E,B)=P(M|A)
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


Pick a var and add a node to the network
Set its parents to some minimal set of nodes already in the net to satisfy Eq.14.2
Define the CPT for the var.
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A Bayesian network can often be far more compact than the full joint.
In a locally structured system, each subcomponent interacts directly with only
of other components.
A local structure is usually associated with linear rather than exponential growth in complexity.
With 30 ( by 5 (
& joint?
) nodes, if a node is directly influenced
) nodes, what’s the difference between BN

30*2^5 vs. 2^30, or n*2^k vs. 2^n
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The correct order to add nodes is to add the “root causes” first, then the variables they influence, and so on until we reach the leaves that have no direct causal influence on the other variables.

Domain knowledge helps!
What if we happen to choose the wrong order?
Fig 14.3 shows an example.
If we stick to a
having to specify causal model, we end up
, and the numbers will often be easier to come up
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Designing inference algorithms, we need to know if more general conditional independences hold.
Given a network, can we know if a set of nodes X is independent of another set Y, given a set of evidence nodes E? It boils down to the concept of non-descendants.

As in Fig 14.2, JohnCalls is indept of Burglary and
Earthquake, given Alarm.
A node is cond independent of all other nodes in the network, given its parents, children, and children’s parents (its Markov blanket).

Burglary is indept of JohnCalls and MaryCalls, given
Alarm and Earthquake
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Given canonical distributions, the complete table can be specified by naming the distribution with some parameters.
A deterministic node has its value specified exactly by the values of its parents.
Uncertain relationships can often be characterized by
“noisy” logical relationships.

Noisy-OR (page 500)
An example for determine cond probabilities starting with P(!fever) on page 501 given the individual inhibition probabilities given cold, flu, malaria as

P(!fever|c,!f,!m) = 0.6, P(!fever|!c,f,!m) = 0.2, and
P(!fever|!c,!f,m) = 0.1
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Exact inference


Inference by enumeration
The variable elimination algorithm


The complexity of exact inference
Clustering algorithms
Approximate inference

Direct sampling methods



Rejection sampling
Likelihood weighting
Inference by Markov chain simulation
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Decide what to talk about
Decide on a vocabulary of random variables
Encode general knowledge about the dependence
Encode a description of the specific problem instance
Pose queries to the inference procedure and get answers
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Different generations of expert systems


Strict logic reasoning (ignore uncertainty)
Probabilistic techniques using the full Joint




Default reasoning - believed until a better reason is found to believe something else
Rules with certainty factors
Handling ignorance - Dempster-Shafer theory
Vagueness - something is sort of true (fuzzy logic)
Probability makes the same ontological commitment as logic: the event is true or false
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The four-wheel car conclusion is reached by default.
New evidence can cause the conclusion retracted, while FOL is strictly monotonic.
Representatives are default logic, nonmonotonic logic, circumscription
There are problematic issues

Details in Chapter 10
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Logical reasoning systems have properties like:

Monotonicity



Locality
Detachment
Truth-functionality
These properties are good for obvious computational advantages; bad as they’re inappropriate for uncertain reasoning.
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Reasoning properly


In FOL, it means conclusions follow from premises
In probability, it means having beliefs that allow an agent to act rationally
Conditional independence info is vital
A Bayesian network is a complete representation for the JPD, but exponentially smaller in size
Bayesian networks can reason causally, diagnostically, intercausally, or combining two or more of the three.
For polytrees (singly connected networks), the computational time is linear in network size.
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