Download Learning with Bayesian Networks

Author:
David Heckerman
Presented By:
Yan Zhang - 2006
Jeremy Gould – 2013
1
Outline
 Bayesian Approach
 Bayesian vs. classical probability methods
 Examples
 Bayesian Network
 Structure
 Inference
 Learning Probabilities
 Learning the Network Structure
 Two coin toss – an example
 Conclusions
 Exam Questions
2
Bayesian vs. the Classical Approach
 The Bayesian probability of an event x, represents the
person’s degree of belief or confidence in that event’s
occurrence based on prior and observed facts.
 Classical probability refers to the true or actual
probability of the event and is not concerned with
observed behavior.
3
Example – Is this Man a Martian
Spy?
4
Example
We start with two concepts:
1. Hypothesis (H) – He either is or is not a Martian
spy.
2. Data (D) – Some set of information about the
subject. Perhaps financial data, phone records,
maybe we bugged his office…
5
Example
Frequentist Says
Given a hypothesis (He IS a
Martian) there is a
probability P of seeing this
data:
P( D | H )
(Considers absolute ground
truth, the uncertainty/noise
is in the data.)
Bayesian Says
Given this data there is a
probability P of this
hypothesis being true:
P( H | D )
(This probability indicates
our level of belief in the
hypothesis.)
6
Bayesian vs. the Classical Approach
 Bayesian approach restricts its prediction to the next
(N+1) occurrence of an event given the observed
previous (N) events.
 Classical approach is to predict likelihood of any given
event regardless of the number of occurrences.
NOTE: The Bayesian approach can be
updated as new data is observed.
7
Bayes Theorem
where
𝑃 𝐵 =
𝑃 𝐵 =
𝑃 𝐵 𝐴 𝑃(𝐴)
𝑃 𝐴𝐵 =
𝑃(𝐵)
𝑃 𝐵 𝐴 𝑃 𝐴 𝑑𝐴
𝑖 𝑃 𝐵 𝐴𝑖 𝑃(𝐴𝑖 )
Continuous Realm
Discrete Realm
For the
continuous case
imagine an
infinite number
of infinitesimally
small partitions.
8
Example – Coin Toss

I want to toss a coin n = 100 times. Let’s denote the
random variable X as the outcome of one flip:



p(X=head) = θ
p(X=tail) =1- θ
Before doing this experiment we have some belief in
our mind: the prior probability ξ. Let’s assume
that this event will have a Beta distribution (a
common assumption):
Sample Beta
Distributions:
Example – Coin Toss
𝑃 𝜃|𝜉 = 𝐵𝑒𝑡𝑎 𝜃|𝛼, 𝛽 =
𝛼
𝐸𝜃 =
𝛼+𝛽
Γ 𝛼+𝛽
𝜃 𝛼−1 1 − 𝜃
Γ 𝛼 Γ 𝛽
𝑉𝐴𝑅 𝜃 =
𝛽−1
𝛼𝛽
𝛼+𝛽 2+ 𝛼+𝛽+1
If we assume a 50-50 coin we can use
α = β = 5 which gives:
𝛼
5
𝐸𝜃 =
=
= 0.5
𝛼+𝛽
10
(Hopefully, what you
were expecting!)
Example – Coin Toss
Now I can run my experiment. As I go I can update my beliefs based on
the observed heads (h) and tails (t) by applying Bayes Law to the Beta
Distribution:
𝑃 𝜃|𝜉 𝑃 𝐷|𝜃, 𝜉
𝑃 𝜃|𝜉 𝜃 ℎ 1 − 𝜃
𝑃 𝜃|𝐷, 𝜉 =
=
𝑃 𝐷|𝜉
𝑃 𝐷|𝜉
𝑡
11
Example – Coin Toss
Since we’re assuming a Beta distribution this becomes:
𝑃 𝜃|𝐷, 𝜉 = 𝐵𝑒𝑡𝑎 𝜃|𝛼 + ℎ, 𝛽 + 𝑡 =
Γ 𝛼 + 𝛽 + ℎ + 𝑡 𝛼+ℎ−1
𝜃
1−𝜃
Γ 𝛼+ℎ Γ 𝛽+𝑡
𝛽+𝑡−1
…our posterior probability. Supposing that we observed
h = 65, t = 35, we would get:
𝛼+ℎ
5 + 65
𝐸 𝜃|𝐷, 𝜉 =
=
= 0.64
𝛼+ℎ+𝛽+𝑡
6 + 65 + 5 + 35
12
Example – Coin Toss
13
Integration
To find the probability that Xn+1= heads, we
could also integrate over all possible values of
θ to find the average value of θ which yields:
𝑃 𝑋𝑛+1 = ℎ𝑒𝑎𝑑𝑠 | 𝐷, 𝜉 =
=
𝑃 𝑋𝑛+1 = ℎ𝑒𝑎𝑑𝑠 | 𝐷, 𝜉 𝑃 𝜃 𝐷, 𝜉 𝑑𝜃
𝜃𝑃 𝜃|𝐷, 𝜉 𝑑𝜃 = 𝐸 𝜃|𝐷, 𝜉
This might be necessary if we were working with a
distribution with a less obvious Expected Value.
14
More than Two Outcomes
 In the previous example, we used a Beta distribution to encode the
states of the random variable. This was possible because there were
only 2 states/outcomes of the variable X.
 In general, if the observed variable X is discrete, having r possible states
{1,…,r}, the likelihood function is given by:
𝑃 𝑋 = 𝑥𝑘 | 𝜃, 𝜉 = 𝜃𝑘 𝑤ℎ𝑒𝑟𝑒:
𝑘 = 1, 2, … . , 𝑟
𝜃 ∈ 𝜃1 , … . , 𝜃𝑟 and 𝜃𝑘 = 1
 In this general case we can use a Dirichlet distribution instead:
Γ 𝛼
𝛼𝑘 −1
𝑟
𝜃
𝑘=1 𝑘
Prior
𝑃 𝜃|𝜉 = Dir 𝜃| 𝛼1 , 𝛼2 , … , 𝛼𝑟 =
Posterior
𝑃 𝜃|𝐷, 𝜉 = Dir 𝜃| 𝛼1 + 𝑛1 , 𝛼2 + 𝑛2 , … , 𝛼𝑟 + 𝑛𝑟
𝑟
𝑘=1 Γ
𝛼𝑘
15
Vocabulary Review
 Prior Probability, P( θ | ξ ): Prior Probability of a
particular value of θ given no observed data (our
previous “belief”)
 Posterior Probability, P(θ | D, ξ): Probability of a
particular value of θ given that D has been observed
(our final value of θ).
 Observed Probability or “Likelihood”, P(D|θ, ξ):
Likelihood of sequence of coin tosses D being
observed given that θ is a particular value.
 P(D|ξ): Raw probability of D
16
Outline
 Bayesian Approach
 Bayes Therom
 Bayesian vs. classical probability methods
 coin toss – an example
 Bayesian Network
 Structure
 Inference
 Learning Probabilities
 Learning the Network Structure
 Two coin toss – an example
 Conclusions
 Exam Questions
17
OK, But So What?
That’s great but this is Data Mining not Philosophy of
Mathematics.
Why should we care about all of this ugly math?
18
Bayesian Advantages
It turns out that the Bayesian technique permits us to do
some very useful things from a mining perspective!
1. We can use the Chain Rule with Bayesian Probabilities:
𝑛
𝑛
𝑃
𝐴𝑘 =
𝑘=1
Ex.
𝑘−1
𝑃 𝐴𝑘 |
𝑘=1
𝑗=1
𝐴𝑗
𝑃 𝐴, 𝐵, 𝐶 = 𝑃 𝐴 𝐵, 𝐶 𝑃 𝐵|𝐶 𝑃 𝐶
This isn’t
something we
can’t easily do
with classical
probability!
2. As we’ve already seen using the Bayesian model
permits us to update our beliefs based on new data.
19
Example Network
To create a Bayesian network we will ultimately
need 3 things:
 A set of Variables X={X1,…, Xn}
 A Network Structure
 Conditional Probability Table (CPT)
Note that when we start we may not have any of
these things or a given element may be
incomplete!
20
Let’s start with a simple case where we are given all three things: a credit
fraud network designed to determine the probability of credit fraud.
21
Set of Variables
Each node
represents a
random variable.
(Let’s assume
discrete for now.)
22
Network Structure
Each edge
represents a
conditional
dependence
between variables.
23
Conditional Probability Table
Each rule
represents the
quantification of a
conditional
dependency.
24
Since we’ve been given the
network structure we can
easily see the conditional
dependencies:
P(A|F,A,S,G) = P(A)
P(S|F,A,S,G) = P(S)
P(G|F,A,S,G) = P(G|F)
P(J|F,A,S,G) = P(J|F,A,S)
25
Note that the absence of an edge
indicates conditional independence:
P(A|G) = P(A)
26
Important Note:
The presence of a of cycle
will render one or more of
the relationships
intractable!
27
Inference
Now suppose we want to calculate (infer) our
confidence level in a hypothesis on the fraud
variable f given some knowledge about the other
variables. This can be directly calculated via:
𝑃 𝑓, 𝑎, 𝑠, 𝑔, 𝑗
𝑃 𝑓|𝑎, 𝑠, 𝑔, 𝑗 =
=
𝑃 𝑎, 𝑠, 𝑔, 𝑗
𝑃 𝑓, 𝑎, 𝑠, 𝑔, 𝑗
′ , 𝑎, 𝑠, 𝑔, 𝑗
𝑃
𝑓
𝑓′
(Kind of messy…)
28
Inference
Fortunately, we can use the Chain Rule to simplify!
𝑃 𝑓|𝑎, 𝑠, 𝑔, 𝑗 =
𝑃 𝑓 𝑃 𝑎 𝑃 𝑠 𝑃 𝑔|𝑓 𝑃 𝑗|𝑓,𝑎,𝑠
𝑓′ 𝑃 𝑓′ 𝑃 𝑎 𝑃 𝑠 𝑃 𝑔|𝑓′ 𝑃 𝑗|𝑓′,𝑎,𝑠
=
𝑃 𝑓 𝑃 𝑔|𝑓 𝑃 𝑗|𝑓,𝑎,𝑠
𝑓′ 𝑃 𝑓′ 𝑃 𝑔|𝑓′ 𝑃 𝑗|𝑓′,𝑎,𝑠
This Simplification is especially powerful when the network is
sparse which is frequently the case in real world problems.
This shows how we can use a Bayesian Network to
infer
a probability not stored directly in the model.
29
Now for the Data Mining!
So far we haven’t added much value to the data. So let’s take advantage
of the Bayesian model’s ability to update our beliefs and learn from
new data.
First we’ll rewrite our joint probability distribution in a more compact
form:
𝑛
𝑃 𝑥|𝜃𝑠 , 𝑆 ℎ =
𝑃 𝑥𝑖 |𝑝𝑎𝑖, 𝜃𝑖 , 𝑆 ℎ
𝑖=1
𝑆 ℎ 𝑖𝑠 𝑡ℎ𝑒 𝑐𝑢𝑟𝑟𝑒𝑛𝑡 ℎ𝑦𝑝𝑜𝑡ℎ𝑒𝑠𝑖𝑠
(𝐵𝑎𝑠𝑖𝑐𝑎𝑙𝑙𝑦 𝑙𝑖𝑘𝑒 𝜉 𝑓𝑟𝑜𝑚 𝑏𝑒𝑓𝑜𝑟𝑒)
𝜃𝑖 𝑖𝑠 𝑡ℎ𝑒 𝑣𝑒𝑐𝑡𝑜𝑟 𝑜𝑓 𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟𝑠 𝑓𝑜𝑟 𝑃 𝑥𝑖 |𝑝𝑎𝑖, 𝜃𝑖 , 𝑆 ℎ
𝑥 ∈ 𝑥1 , 𝑥2 , … , 𝑥𝑛
𝜃𝑠 = 𝜃1 , 𝜃2 , … , 𝜃𝑛
𝑝𝑎𝑖 𝑖𝑠 𝑡ℎ𝑒 𝑐𝑜𝑛𝑓𝑖𝑔𝑢𝑟𝑎𝑡𝑖𝑜𝑛 𝑣𝑒𝑐𝑡𝑜𝑟:
𝑗
𝑃 𝑥𝑖𝑘 |𝑝𝛼𝑖 , 𝜃𝑖 , 𝑆 ℎ = 𝜃𝑖𝑗𝑘 > 0
𝑭𝒓𝒐𝒎 𝒉𝒆𝒓𝒆 𝒘𝒆 𝒄𝒂𝒏 𝒇𝒊𝒏𝒅 𝑷 𝜽𝒔 |𝑺𝒉
30
Learning Probabilities in a Bayesian Network
 First we need to make two assumptions:
There is no missing data (i.e. the data
accurately describes the distribution)
 The parameter vectors are
independent (generally a good
assumption, at least locally).

31
Learning Probabilities in a Bayesian Network
If these assumptions hold we can express the probabilities as:
Prior 𝑃 𝜃𝑠 |𝑆 ℎ =
𝑛
𝑖=1
Posterior 𝑃 𝜃𝑠 |𝐷, 𝑆 ℎ =
𝑞𝑖
𝑗=1 𝑃
𝑛
𝑖=1
𝜃𝑖𝑗 |𝑆 ℎ
𝑞𝑖
𝑗=1 𝑃
𝜃𝑖𝑗 |𝐷, 𝑆 ℎ
This means we can update each vector of parameters θij
independently, just as one-variable case!
•
If each vector θij has the prior distribution Dir(θij |aij1,…,
aijri)…
•
…Then the posterior distribution is:
𝑃 𝜃𝑠 |𝐷, 𝑆 ℎ = 𝐷𝑖𝑟 𝜃𝑖𝑗 |𝛼𝑖𝑗1 + 𝑛𝑖𝑗1 , … , 𝛼𝑖𝑗𝑟𝑖 + 𝑛𝑖𝑗𝑟𝑖
Where nijk is the number of cases in D in which Xi=xik and Pai=paij
32
Dealing with Unknowns
Whew! Now we know how to use our network to infer
conditional relationships and how to update our
network with new data. But what if we aren’t given a well
defined network? We could start with missing or
incomplete:
1. Set of Variables
2. Conditional Relationship Data
3. Network Structure
33
Unknown Variable Set
Our goal when choosing variables is to:
“Organize…into variables having mutually exclusive and
collectively exhaustive states.”
This is a problem shared by all data mining algorithms: What
should we measure and why? There is not and probably
cannot be an algorithmic solution to this problem as arriving
at any solution requires intelligent and creative thought.
34
Unknown Conditional Relationships
This can be easy.
So long as we can generate a plausible initial belief about
a conditional relationship we can simply start with our
assumption and let our data refine our model via the
mechanism shown in the Learning Probabilities in a
Bayesian Network slide.
35
Unknown Conditional Relationships
However, when our ignorance becomes serious
enough that we no longer even know what is
dependent on what we segue into the Unknown
Structure scenario.
36
Learning the Network Structure
Sometimes the conditional relationships are
not obvious. In this case we are uncertain with
the network structure: we don’t know where
the edges should be.
37
Learning the Network Structure
Theoretically, we can use a Bayesian approach to get the
posterior distribution of the network structure:
𝑃 𝑆 ℎ |𝐷 =
𝑃 𝐷|𝑆 ℎ 𝑃 𝑆 ℎ
𝑚
ℎ
ℎ𝑖 𝑃 𝑆ℎ𝑖
𝑖=1 𝑃 𝐷|𝑆 = 𝑆
Unfortunately, the number of possible network
structure increase exponentially with n – the number of
nodes. We’re basically asking ourselves to consider every
possible graph with n nodes!
38
Learning the Network Structure
Heckerman describes two main methods for shortening the
search for a network model:
 Model Selection


To select a “good” model (i.e. the network structure) from all
possible models, and use it as if it were the correct model.
Selective Model Averaging

To select a manageable number of good models from among all
possible models and pretend that these models are exhaustive.
The math behind both techniques is quite involved so I’m afraid
we’ll have to content ourselves with a toy example today.
39
Two Coin Toss Example
S
h
1
X1
X2
p(H)=p(T)=0.5




S
h
2
X1
p(H)=p(T)=0.5
X2
P(X2|X1)
p(H|H)
p(T|H)
p(H|T)
p(T|T)
=
=
=
=
0.1
0.9
0.9
0.1
Experiment: flip two coins and observe the outcome
Propose two network structures: Sh1 or Sh2
Assume P(Sh1)=P(Sh2)=0.5
After observing some data, which model is more
accurate for this collection of data?
40
Two Coin Toss Example
X1
X2
𝑃 𝐷|𝑆 ℎ 𝑃 𝑆 ℎ
ℎ
𝑃 𝑆 |𝐷 =
1
T
T
2
T
H
3
H
T
4
H
T
2
𝑖=1 𝑃
5
T
H
10
6
H
T
7
T
H
8
T
H
9
H
T
10
H
T
𝑃 𝑆 ℎ |𝐷 =
𝑚
𝑖=1 𝑃
𝐷|𝑆ℎ = 𝑆𝑖ℎ 𝑃 𝑆𝑖ℎ
𝑃 𝐷|𝑆 ℎ 𝑃 𝑆 ℎ
𝐷|𝑆ℎ = 𝑆𝑖ℎ 𝑃 𝑆𝑖ℎ
2
𝑃 𝐷|𝑆 ℎ =
𝑃 𝑋𝑑𝑖 |𝑃𝛼𝑖 , 𝑆 ℎ
𝑑=1 𝑖=1
41
Two Coin Toss Example
X1
X2
1
T
T
2
T
H
3
H
T
4
H
T
5
T
H
6
H
T
7
T
H
8
T
H
9
H
T
10
H
T
𝐹𝑜𝑟 𝑆1ℎ :
10
2
𝑃 𝐷|𝑆 ℎ =
𝑃 𝑋𝑑𝑖 |𝑃𝛼𝑖 , 𝑆 ℎ
𝑑=1 𝑖=1
= 𝑃 𝑋1 = 𝑇 𝑃 𝑋2 = 𝑇
=
0.5 0.5
= 0.52
𝑃 𝑋1 = 𝑇 𝑃 𝑋2 = 𝐻 ….
0.5 0.5 ….
10
42
Two Coin Toss Example
10
𝐹𝑜𝑟 𝑆2ℎ :
2
𝑃 𝐷|𝑆 ℎ =
𝑃 𝑋𝑑𝑖 |𝑃𝛼𝑖 , 𝑆 ℎ
𝑑=1 𝑖=1
X1 X2
1
T
T
𝑃 𝑋1 = 𝑇 𝑃 𝑋2 = 𝑇|𝑋1 = 𝑇
= 0.5 0.1
2
T
H
𝑃 𝑋1 = 𝑇 𝑃 𝑋2 = 𝐻|𝑋1 = 𝑇
= 0.5 0.9
3
H
T
𝑃 𝑋1 = 𝐻 𝑃 𝑋2 = 𝑇|𝑋1 = 𝐻
= 0.5 0.9
4
H
T
5
T
H
6
H
T
7
T
H
8
T
H
9
H
T
10
H
T
.
.
.
10
2
𝑃 𝐷|𝑆 ℎ =
𝑃 𝑋𝑑𝑖 |𝑃𝛼𝑖 , 𝑆 ℎ = 0.5
10
0.1
1
0.9
9
𝑑=1 𝑖=1
43
Two Coin Toss Example
𝑃 𝑆1ℎ |𝐷 =
𝑃 𝑆1ℎ |𝐷 =
𝑃 𝐷|𝑆1ℎ 𝑃 𝑆1ℎ
𝑃 𝐷|𝑆1ℎ 𝑃 𝑆1ℎ + 𝑃 𝐷|𝑆2ℎ 𝑃 𝑆2ℎ
0.52
10
0.52 10 0.5
0.5 + 0.5 10 0.1
10
0.5
𝑃 𝑆1ℎ |𝐷 =
0.5 10 + 0.1 1 0.9
1
0.9
9
0.5
9
𝑃 𝑆1ℎ |𝐷 ≈ 2.5%
44
Two Coin Toss Example
𝑃 𝑆2ℎ |𝐷 =
𝑃 𝑆2ℎ |𝐷 =
𝑃 𝐷|𝑆2ℎ 𝑃 𝑆2ℎ
𝑃 𝐷|𝑆1ℎ 𝑃 𝑆1ℎ + 𝑃 𝐷|𝑆2ℎ 𝑃 𝑆2ℎ
0.52
10
0.5 10 0.1 1 0.9 9 0.5
0.5 + 0.5 10 0.1 1 0.9
1
9
0.1
0.9
𝑃 𝑆2ℎ |𝐷 =
0.5 10 + 0.1 1 0.9
9
0.5
9
𝑃 𝑆2ℎ |𝐷 ≈ 97.5%
45
Two Coin Toss Example
𝑃 𝑆2ℎ |𝐷 ≈ 97.5% > 2.5% ≈ 𝑃 𝑆1ℎ |𝐷
𝑇ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒 𝑤𝑒 𝑠ℎ𝑜𝑢𝑙𝑑 𝑝𝑟𝑒𝑓𝑒𝑟 𝑆2ℎ 𝑏𝑦 𝑠𝑐𝑜𝑟𝑒.
46
Outline
 Bayesian Approach
 Bayes Therom
 Bayesian vs. classical probability methods
 coin toss – an example
 Bayesian Network
 Structure
 Inference
 Learning Probabilities
 Learning the Network Structure
 Two coin toss – an example
 Conclusions
 Exam Questions
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Conclusions
 Bayesian method
 Bayesian network
 Structure
 Inference
 Learn parameters and structure
 Advantages
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Question1: What is Bayesian Probability?
 A person’s degree of belief in a certain event
 Your own degree of certainty that a tossed coin will
land “heads”
 A degree of confidence in an outcome given some data.
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Question 2: Compare the Bayesian and classical
approaches to probability (any one point).
Bayesian Approach:
Classical Probability:
 +Reflects an expert’s
 +Objective and unbiased
 - Generally not available
 It takes a long time to
measure the object’s physical
characteristics
 Wants P( D | H )
knowledge
 +The belief is kept updating
when new data item arrives
 - Arbitrary (More subjective)
 Wants P( H | D )
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Question 3: Mention at least 1 Advantage of
Bayesian analysis
 Handle incomplete data sets
 Learning about causal relationships
 Combine domain knowledge and data
 Avoid over fitting
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The End
 Any Questions?
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