Bayesian Inference

Artificial IntelligenceCMSC 25000

February 26, 2002

Agenda

• Motivation– Reasoning with uncertainty

• Medical Informatics

• Probability and Bayes’ Rule– Bayesian Networks– Noisy-Or

• Decision Trees and Rationality• Conclusions

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

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

},...,,{ 21 nXXX}...,,{ ,21 iikii xxx

n

iik

1

nik 2;2

Probabilities of World States

• : Joint probability of assignments– States are distinct and exhaustive

• Typically care about SUBSET of assignments– aka “Circumstance”

– Exponential in # of don’t cares

}),,,({),( 43},{ },{

2142 fXvXtXuXPfXtXPftu ftv

)( iSP

)(1

1

n

i ik

jjSP

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

Modeling (In)dependence

• Bayesian network– Nodes = Variables– Arcs = Child depends on parent(s)

• No arcs = independent (0 incoming: only a priori)• Parents of X = • For each X need

)(X))(|( XXP

Simple Bayesian Network

• MCBN1

A

B C

D E

A = only a prioriB depends on AC depends on AD depends on B,CE depends on C

Need:P(A)P(B|A)P(C|A)P(D|B,C)P(E|C)

Truth table22*22*22*2*22*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 = 0.5Pn(b|a) = (1-l)

P(B|A) b b

a

a

0.6 0.4

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 IIA B

C

Full model: P(c|ab)P(c|ab)P(c|ab)P(c|ab) & neg

Noisy-Or: ca, cb, lPn(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)

Assume:

P(a)=0.1

P(b)=0.05

P(c|ab)=0.3

ca= 0.5

P(c|b) = 0.7

= l = 0.3

Pn(c|b)=Pn(c|ab)Pn(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

Graph Models

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

d1

d2

d3

d4

s1

s2

s3

s4

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

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

Live 0.99 Die 0.01850 0

Die 0.05 0

Full Recovery 0.7 1000

Worse 0.25

Medicine Amputate leg

Die 0.4 0

Live 0.6 995

Die 0.02 0

Live 0.98 700

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. Forthe time period of interest, there is a 10^-4 a priori chanceof this happening, and Holmes has installed a burglar alarmto try to forestall this event. The alarm is 95% reliable insounding 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 alsoa bit of a practical joker and, knowing Holmes’ concern, might (30%) call even if the alarm is silent. Holmes’ otherneighbor Mrs. Gibbons is a well-known lush and often befuddled, but Holmes believes that she is four times morelikely 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 burgledA: whether his alarm soundedW: whether Watson calledG: whether Gibbons called

B A

W

G

Holmes Example: Tables

B = #t B=#f

0.0001 0.9999

A=#t A=#fB

#t#f

0.95 0.05 0.01 0.99

W=#t W=#fA

#t#f

0.90 0.10 0.30 0.70

G=#t G=#fA

#t#f

0.40 0.60 0.10 0.90