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Reasoning Under Uncertainty
Artificial Intelligence
CMSC 25000
February 19, 2008
Agenda
• Motivation
– Reasoning with uncertainty
• Medical Informatics
• Probability and Bayes’ Rule
– Bayesian Networks
– Noisy-Or
• Decision Trees and Rationality
• Conclusions
Uncertainty
• Search and Planning Agents
– Assume fully observable, deterministic, static
• Real World:
– Probabilities capture “Ignorance & Laziness”
• Lack relevant facts, conditions
• Failure to enumerate all conditions, exceptions
– Partially observable, stochastic, extremely complex
– Can't be sure of success, agent will maximize
– Bayesian (subjective) probabilities relate to knowledge
Motivation
• Uncertainty in medical diagnosis
– Diseases produce symptoms
– In diagnosis, observed symptoms => disease ID
– Uncertainties
• Symptoms may not occur
• Symptoms may not be reported
• Diagnostic tests not perfect
– False positive, false negative
• How do we estimate confidence?
Motivation II
• Uncertainty in medical decision-making
– Physicians, patients must decide on treatments
– Treatments may not be successful
– Treatments may have unpleasant side effects
• Choosing treatments
– Weigh risks of adverse outcomes
• People are BAD at reasoning intuitively
about probabilities
– Provide systematic analysis
Probability Basics
• The sample space:
– A set Ω ={ω1, ω2, ω3,… ωn}
• E.g 6 possible rolls of die;
• ωi is a sample point/atomic event
• Probability space/model is a sample space
with an assignment P(ω) for every ω in Ω
s.t. 0<= P(ω)<=1; Σ ωP(ω) = 1
– E.g. P(die roll < 4)=1/6+1/6+1/6=1/2
Random Variables
• A random variable is a function from sample
points to a range (e.g. reals, bools)
• E.g. Odd(1) = true
• P induces a probability distribution for any r.v X:
– P(X=xi) = Σ{ω:X(ω)=xi}P(ω)
– E.g. P(Odd=true)=1/6+1/6+1/6=1/2
• Proposition is event (set of sample pts) s.t.
proposition is true: e.g. event a= A(ω)=true
Why probabilities?
• Definitions imply that logically related
events have related probabilities
• In AI applications, sample points are
defined by set of random variables
– Random vars: boolean, discrete, continuous
Prior Probabilities
• Prior probabilities: belief prior to evidence
– E.g. P(cavity=t)=0.2; P(weather=sunny)=0.6
– Distribution gives values for all assignments
• Joint distribution on set of r.v.s gives probability
on every atomic event of r.v.s
– E.g. P(weather,cavity)=4x2 matrix of values
• Every question about a domain can be answered
with joint b/c every event is a sum of sample pts
Conditional Probabilities
• Conditional (posterior) probabilities
– E.g. P(cavity|toothache) = 0.8, given only that
– P(cavity|toothache)=2 elt vector of 2 elt vectors
• Can add new evidence, possibly irrelevant
• P(a|b) = P(a^b)/P(b) where P(b) ≠0
• Also, P(a^b)=P(a|b)P(b)=P(b|a)P(a)
– Product rule generalizes to chaining
Inference By Enumeration
Inference by Enumeration
Inference by Enumeration
Independence
Conditional Independence
Conditional Independence II
Probabilities Model Uncertainty
• The World - Features
– Random variables   { X 1 , X 2 ,..., X n }
{xi1,xi2,...,
xiki }
– Feature values
• States of the world
– Assignments of values to variables
n
k
i
i 1
– Exponential in # of variables
– k i  2;2 n possible states
Probabilities of World States
• P( Si ): Joint probability of assignments
– States are distinct and exhaustive
i1 ki
 P( S j )
n
j 1
• Typically care about SUBSET of assignments
– aka “Circumstance”
P( X 2  t , X 4  f ) 
  P({ X
u{t , f } v{t , f }
– Exponential in # of don’t cares
1
 u, X 2  t , X 3  v, X 4  f })
A Simpler World
• 2^n world states = Maximum entropy
– Know nothing about the world
• Many variables independent
– P(strep,ebola) = P(strep)P(ebola)
• Conditionally independent
– Depend on same factors but not on each other
– P(fever,cough|flu) = P(fever|flu)P(cough|flu)
Probabilistic Diagnosis
• Question:
– How likely is a patient to have a disease if they have the
symptoms?
• Probabilistic Model: Bayes’ Rule
• P(D|S) = P(S|D)P(D)/P(S)
– Where
• P(S|D) : Probability of symptom given disease
• P(D): Prior probability of having disease
• P(S): Prior probability of having symptom
Diagnosis
• Consider Meningitis:
–
–
–
–
–
–
Disease: Meningitis: m
Symptom: Stiff neck: s
P(s|m) = 0.5
P(m) =0.0001
P(s) = 0.1
How likely is it that someone with a stiff neck
actually has meningitis?
Modeling (In)dependence
• Simple, graphical notation for conditional
independence; compact spec of joint
• Bayesian network
– Nodes = Variables
– Directed acyclic graph: link ~ directly influences
– Arcs = Child depends on parent(s)
• No arcs = independent (0 incoming: only a priori)
• Parents of X =  ( X )
• For each X need P ( X | ( X ))
Example I
Simple Bayesian Network
• MCBN1
A = only a priori
B depends on A
C depends on A
D depends on B,C
E depends on C
A
B
C
D
Need:
P(A)
P(B|A)
P(C|A)
P(D|B,C)
P(E|C)
E
Truth table
2
2*2
2*2
2*2*2
2*2
Simplifying with Noisy-OR
• How many computations?
– p = # parents; k = # values for variable
– (k-1)k^p
– Very expensive! 10 binary parents=2^10=1024
• Reduce computation by simplifying model
– Treat each parent as possible independent cause
– Only 11 computations
• 10 causal probabilities + “leak” probability
– “Some other cause”
Noisy-OR Example
A
B
Pn(b|a) = 1-(1-ca)(1-L)
Pn(b|a) = (1-ca)(1-L)
Pn(b|a) = 1-(1 -L) = L = 0.5
Pn(b|a) = (1-L)
P(B|A)
b
b
a
0.6 0.4
a
0.5
0.5
Pn(b|a) = 1-(1-ca)(1-L)=0.6
(1-ca)(1-L)=0.4
(1-ca) =0.4/(1-L)
=0.4/0.5=0.8
ca = 0.2
Noisy-OR Example II
A
B
C
Full model: P(c|ab)P(c|ab)P(c|ab)P(c|ab) & neg
Noisy-Or: ca, cb, L
Pn(c|ab) = 1-(1-ca)(1-cb)(1-L)
Pn(c|ab) = 1-(1-cb)(1-L)
Pn(c|ab) = 1-(1-ca)(1-L)
Pn(c|ab) = 1-(1-L)
= L = 0.3
Pn(c|b)=Pn(c|ab)P(a)+Pn(c|ab)P(a)
1-0.7=(1-ca)(1-cb)(1-L)0.1+(1-cb)(1-L)0.9
0.3=0.5(1-cb)0.07+(1-cb)0.7*0.9
=0.035(1-cb)+0.63(1-cb)=0.665(1-cb)
0.55=cb
Assume:
P(a)=0.1
P(b)=0.05
Pn(c|ab)=0.3
ca= 0.5
Pn(c|b) = 0.7
Graph Models
• Bipartite graphs
– E.g. medical reasoning
– Generally, diseases cause symptom (not reverse)
s1
d1
d2
s2
s3
s4
d3
d4
s5
s6
Topologies
• Generally more complex
– Polytree: One path between any two nodes
• General Bayes Nets
– Graphs with undirected cycles
• No directed cycles - can’t be own cause
• Issue: Automatic net acquisition
– Update probabilities by observing data
– Learn topology: use statistical evidence of indep,
heuristic search to find most probable structure
Holmes Example (Pearl)
Holmes is worried that his house will be burgled. For
the time period of interest, there is a 10^-4 a priori chance
of this happening, and Holmes has installed a burglar alarm
to try to forestall this event. The alarm is 95% reliable in
sounding when a burglary happens, but also has a false
positive rate of 1%. Holmes’ neighbor, Watson, is 90% sure
to call Holmes at his office if the alarm sounds, but he is also
a bit of a practical joker and, knowing Holmes’ concern,
might (30%) call even if the alarm is silent. Holmes’ other
neighbor Mrs. Gibbons is a well-known lush and often
befuddled, but Holmes believes that she is four times more
likely to call him if there is an alarm than not.
Holmes Example: Model
There a four binary random variables:
B: whether Holmes’ house has been burgled
A: whether his alarm sounded
W: whether Watson called
G: whether Gibbons called
W
B
A
G
Holmes Example: Tables
B = #t
B=#f
A
W=#t
W=#f
0.0001
0.9999
#t
#f
0.90
0.30
0.10
0.70
A
G=#t
#t
#f
0.40
0.10
B
#t
#f
A=#t
0.95
0.01
A=#f
0.05
0.99
G=#f
0.60
0.90
Decision Making
• Design model of rational decision making
– Maximize expected value among alternatives
• Uncertainty from
– Outcomes of actions
– Choices taken
• To maximize outcome
– Select maximum over choices
– Weighted average value of chance outcomes
Gangrene Example
Medicine
Amputate foot
Worse 0.25
Die 0.05
0
Medicine
Die 0.4
0
Live 0.6
995
Full Recovery 0.7
Live 0.99
1000
850
Amputate leg
Die 0.02
0
Live 0.98
700
Die 0.01
0
Decision Tree Issues
• Problem 1: Tree size
– k activities : 2^k orders
• Solution 1: Hill-climbing
– Choose best apparent choice after one step
• Use entropy reduction
• Problem 2: Utility values
– Difficult to estimate, Sensitivity, Duration
• Change value depending on phrasing of question
• Solution 2c: Model effect of outcome over lifetime
Conclusion
• Reasoning with uncertainty
– Many real systems uncertain - e.g. medical
diagnosis
• Bayes’ Nets
– Model (in)dependence relations in reasoning
– Noisy-OR simplifies model/computation
• Assumes causes independent
• Decision Trees
– Model rational decision making
• Maximize outcome: Max choice, average outcomes
Holmes Example (Pearl)
Holmes is worried that his house will be burgled. For
the time period of interest, there is a 10^-4 a priori chance
of this happening, and Holmes has installed a burglar alarm
to try to forestall this event. The alarm is 95% reliable in
sounding when a burglary happens, but also has a false
positive rate of 1%. Holmes’ neighbor, Watson, is 90% sure
to call Holmes at his office if the alarm sounds, but he is also
a bit of a practical joker and, knowing Holmes’ concern,
might (30%) call even if the alarm is silent. Holmes’ other
neighbor Mrs. Gibbons is a well-known lush and often
befuddled, but Holmes believes that she is four times more
likely to call him if there is an alarm than not.
Holmes Example: Model
There a four binary random variables:
B: whether Holmes’ house has been burgled
A: whether his alarm sounded
W: whether Watson called
G: whether Gibbons called
W
B
A
G
Holmes Example: Tables
B = #t
B=#f
A
W=#t
W=#f
0.0001
0.9999
#t
#f
0.90
0.30
0.10
0.70
A
G=#t
#t
#f
0.40
0.10
B
#t
#f
A=#t
0.95
0.01
A=#f
0.05
0.99
G=#f
0.60
0.90