The product rule:

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Reviewing Bayes rule
The product rule
The Bayes rule
P(a n b) P(ab) P(b)
?
P(a n b) P(ba) P(a)
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Bayes rule and its use
The product rule
The Bayes rule
P(a n b) P(ab) P(b)
?
P(a n b) P(ba) P(a)
Why is Bayes rule useful in practice ?
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Bayes rule and its use
The product rule
The Bayes rule
P(a n b) P(ab) P(b)
?
P(a n b) P(ba) P(a)
Why is Bayes rule useful in practice ?
Because there are many cases where we have good
probability estimates for three of the four
probabilities involved, and therefore can compute
the fourth one.
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Bayes rule and its use An Excercise
A doctor knows that the disease meningitis causes
the patient to have a stiff neck 50 of the time.
The doctor also knows that the probability that a
patient has meningitis is 1/50,000, and the
probability that any patient has a stiff neck is
1/20. Find the probability that a patient with a
stiff neck has meningitis.
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Bayes rule and its use An Excercise
A doctor knows that the disease meningitis causes
the patient to have a stiff neck 50 of the time.
The doctor also knows that the probability that a
patient has meningitis is 1/50,000, and the
probability that any patient has a stiff neck is
1/20. Find the probability that a patient with a
stiff neck has meningitis.
P(M S) P(S M) P(M) / P(S) 0.5
0.00002 / 0.05 0.0002
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Definition

• A Bayesian network is a directed acyclic graph
  which consists of
• A set of random variables which makes up the
  nodes of the network.
• A set of directed links (arrows) connecting pairs
  of nodes. If there is an arrow from node X to
  node Y, X is said to be a parent of Y.
• Each node Xi has a conditional probability
  distribution P(XiParents(Xi)) that quantifies
  the effect of the parents on the node.

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Definition

• Intuitions
• A Bayesian network models our incomplete
  understanding of the causal relationships from an
  application domain.
• A node represents some state of affairs or event.
• A link from X to Y means that X has a direct
  influence on Y.

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The probabilities associated with the nodes
reflect our representation of the causal
relationships.
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A Bayesian network provides a complete
description of the domain in the sense one can
compute the probability of any state of the world
(represented as a particular assignment to each
variable).
Example What is the probability that the alarm
has sounded, but neither burglary nor an
earthquake has occurred, and both John and Mary
call?
P(j, m, a, b, e) ???
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A Bayesian network provides a complete
description of the domain in the sense one can
compute the probability of any state of the world
(represented as a particular assignment to each
variable).
Example What is the probability that the alarm
has sounded, but neither burglary nor an
earthquake has occurred, and both John and Mary
call?
P(j, m, a, b, e) P(ja) P(ma) P(a, b, e)
P(b) P(e) 0.900.700.0010.9990.998
0.00062
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A Bayesian network provides a complete
description of the domain in the sense one can
compute the probability of any state of the world
(represented as a particular assignment to each
variable).
Example What is the probability that the alarm
has sounded, but neither burglary nor an
earthquake has occurred, and both John and Mary
call?
P(j, m, a, b, e) P(ja) P(ma) P(a, b, e)
P(b) P(e) 0.900.700.0010.9990.998
0.00062
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A Bayesian network provides a complete
description of the domain in the sense one can
compute the probability of any state of the world
(represented as a particular assignment to each
variable).
Example What is the probability that the alarm
has sounded, but neither burglary nor an
earthquake has occurred, and both John and Mary
call?
P(j, m, a, b, e) P(ja) P(ma) P(a, b, e)
P(b) P(e) 0.900.700.0010.9990.998
0.00062
In general
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Inference in Bayesian networks
The basic inference task for a Bayesian network
is to compute the posterior probability
distribution for a set of query variables, given
some observed event (i.e. some assignment of
values to a set of evidence variables).
Example Assume the event in which John and Mary
called. What is the probability that a burglary
has occurred?
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Inference in Bayesian networks
Let us use the following notations X denotes the
query variable e denotes the set of evidence
variables E1, , En y denotes the set of
nonevidence (hidden) variables Y1, , Yn
A typical query asks for the posterior
probability distribution P(Xe) ltP(Xtruee)
P(Xfalsee)gt
According to the probability theory
with a chosen such that ap1 ap2 1
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Inference in Bayesian networks Illustration
Example Assume the event in which John and Mary
called. What is the probability that a burglary
has occurred?
P(BurglaryJohnCallstrue, MaryCallstrue)
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Inference in Bayesian networks Illustration
P(Bj, m) a lt P(bj, m), P(bj, m) gt a
lt0.00059224, 0.0014919gt lt0.284, 0.716gt
Where a 1/(0.00059224 0.0014919gt
What is the probability that a burglary has
occurred, assuming that both John and Mary
called? 28.4