Reasoning Under Uncertainty

1
Reasoning Under Uncertainty

• Artificial Intelligence
• CMSC 25000
• February 19, 2008

2
Agenda

• Motivation
• Reasoning with uncertainty
• Medical Informatics
• Probability and Bayes Rule
• Bayesian Networks
• Noisy-Or
• Decision Trees and Rationality
• Conclusions

3
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

4
Motivation

• Uncertainty in medical diagnosis
• Diseases produce symptoms
• In diagnosis, observed symptoms gt 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?

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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

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Probability Basics

• The sample space
• A set O ?1, ?2, ?3, ?n
• E.g 6 possible rolls of die
• ?i is a sample point/atomic event
• Probability space/model is a sample space with an
  assignment P(?) for every ? in O s.t. 0lt
  P(?)lt1 S ?P(?) 1
• E.g. P(die roll lt 4)1/61/61/61/2

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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(Xxi) S?X(?)xiP(?)
• E.g. P(Oddtrue)1/61/61/61/2
• Proposition is event (set of sample pts) s.t.
  proposition is true e.g. event a A(?)true

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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

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Prior Probabilities

• Prior probabilities belief prior to evidence
• E.g. P(cavityt)0.2 P(weathersunny)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

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Conditional Probabilities

• Conditional (posterior) probabilities
• E.g. P(cavitytoothache) 0.8, given only that
• P(cavitytoothache)2 elt vector of 2 elt vectors
• Can add new evidence, possibly irrelevant
• P(ab) P(ab)/P(b) where P(b) ?0
• Also, P(ab)P(ab)P(b)P(ba)P(a)
• Product rule generalizes to chaining

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Inference By Enumeration
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Inference by Enumeration
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Inference by Enumeration
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Independence
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Conditional Independence
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Conditional Independence II
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Probabilities Model Uncertainty

• The World - Features
• Random variables
• Feature values
• States of the world
• Assignments of values to variables
• Exponential in of variables
• possible states

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Probabilities of World States

• Joint probability of assignments
• States are distinct and exhaustive
• Typically care about SUBSET of assignments
• aka Circumstance
• Exponential in of dont cares

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A Simpler World

• 2n 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,coughflu) P(feverflu)P(coughflu)

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Probabilistic Diagnosis

• Question
• How likely is a patient to have a disease if they
  have the symptoms?
• Probabilistic Model Bayes Rule
• P(DS) P(SD)P(D)/P(S)
• Where
• P(SD) Probability of symptom given disease
• P(D) Prior probability of having disease
• P(S) Prior probability of having symptom

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Diagnosis

• Consider Meningitis
• Disease Meningitis m
• Symptom Stiff neck s
• P(sm) 0.5
• P(m) 0.0001
• P(s) 0.1
• How likely is it that someone with a stiff neck
  actually has meningitis?

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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
• For each X need

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Example I
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Simple Bayesian Network

• MCBN1

Need P(A) P(BA) P(CA) P(DB,C) P(EC)
Truth table 2 22 22 222 22
A only a priori B depends on A C depends on A D
depends on B,C E depends on C
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Simplifying with Noisy-OR

• How many computations?
• p parents k values for variable
• (k-1)kp
• Very expensive! 10 binary parents2101024
• Reduce computation by simplifying model
• Treat each parent as possible independent cause
• Only 11 computations
• 10 causal probabilities leak probability
• Some other cause

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Noisy-OR Example
Pn(ba) 1-(1-ca)(1-L) Pn(ba)
(1-ca)(1-L) Pn(ba) 1-(1 -L) L 0.5 Pn(ba)
(1-L)
P(BA)
b b
Pn(ba) 1-(1-ca)(1-L)0.6
(1-ca)(1-L)0.4 (1-ca)
0.4/(1-L) 0.4/0.50.8
ca 0.2
a a
0.6 0.4 0.5 0.5
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Noisy-OR Example II
Full model P(cab)P(cab)P(cab)P(cab) neg
Assume P(a)0.1 P(b)0.05 Pn(cab)0.3 ca
0.5 Pn(cb) 0.7
Noisy-Or ca, cb, L
Pn(cab) 1-(1-ca)(1-cb)(1-L) Pn(cab)
1-(1-cb)(1-L) Pn(cab) 1-(1-ca)(1-L) Pn(cab)
1-(1-L)
L 0.3
Pn(cb)Pn(cab)P(a)Pn(cab)P(a)
1-0.7(1-ca)(1-cb)(1-L)0.1(1-cb)(1-L)0.9
0.30.5(1-cb)0.07(1-cb)0.70.9
0.035(1-cb)0.63(1-cb)0.665(1-cb) 0.55cb
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Graph Models

• Bipartite graphs
• E.g. medical reasoning
• Generally, diseases cause symptom (not reverse)

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Topologies

• Generally more complex
• Polytree One path between any two nodes
• General Bayes Nets
• Graphs with undirected cycles
• No directed cycles - cant 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

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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.
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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
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Holmes Example Tables
B t Bf
Wt Wf
A t f
0.0001 0.9999
0.90 0.10 0.30 0.70
At Af
B t f
Gt Gf
A t f
0.95 0.05 0.01 0.99
0.40 0.60 0.10 0.90
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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

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Gangrene Example
Medicine
Amputate foot
Worse 0.25
Full Recovery 0.7 1000
Die 0.05 0
Die 0.01
Live 0.99
850
0
Medicine
Amputate leg
Live 0.6 995
Live 0.98 700
Die 0.4 0
Die 0.02 0
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Decision Tree Issues

• Problem 1 Tree size
• k activities 2k 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

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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

37
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.
38
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
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Holmes Example Tables
B t Bf
Wt Wf
A t f
0.0001 0.9999
0.90 0.10 0.30 0.70
At Af
B t f
Gt Gf
A t f
0.95 0.05 0.01 0.99
0.40 0.60 0.10 0.90