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CS 2001: Bayesian Belief Networks for Uncertainty in Computer Science, Lab Reports of Computer Science

University of Pittsburgh (Pitt) - Medical Center-Health SystemComputer Science

The lecture notes for cs 2001, a course on bayesian belief networks (bbns) taught by milos hauskrecht at carnegie mellon university. The notes cover the basics of bbns, their application in medical diagnosis, and the representation of uncertainty. The course focuses on milos' research interests in artificial intelligence, planning, reasoning, optimization, and machine learning.

Typology: Lab Reports

Pre 2010

Uploaded on 09/02/2009

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CS 2001 Bayesian belief networks
CS 2001 –Lecture 1
Milos Hauskrecht
[email protected]
5329 Sennott Square
X4-8845
Bayesian belief networks
CS 2001 Bayesian belief networks
Milos’ research interests
Artificial Intelligence
•Planning, reasoning and optimization in the presence of
uncertainty
•Machine learning
•Applications:
–Medicine
–Finance and investments
Main research focus:
•Models of high dimensional stochastic problems and their
efficient solutions
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CS 2001 Bayesian belief networks

CS 2001 – Lecture 1

Milos Hauskrecht [email protected] 5329 Sennott Square X4-

Bayesian belief networks

CS 2001 Bayesian belief networks

Milos’ research interests

Artificial Intelligence

  • Planning, reasoning and optimization in the presence of uncertainty
  • Machine learning
  • Applications:
    • Medicine
    • Finance and investments

Main research focus:

  • Models of high dimensional stochastic problems and their efficient solutions

CS 2001 Bayesian belief networks

KB for medical diagnosis.

We want to build a KB system for the diagnosis of pneumonia. Problem description:

  • Disease: pneumonia
  • Patient symptoms (findings, lab tests) :
    • Fever, Cough, Paleness, WBC (white blood cells) count, Chest pain, etc. Representation of a patient case:
  • Statements that hold (are true) for that patient. E.g:

Diagnostic task: we want to infer whether the patient suffers from the pneumonia or not given the symptoms

Fever = True Cough = False WBCcount= High

CS 2001 Bayesian belief networks

Uncertainty To make diagnostic inference possible we need to represent rules or axioms that relate symptoms and diagnosis Problem: disease/symptoms relation is not deterministic (things may vary from patient to patient)

  • Disease Symptoms uncertainty
    • A patient suffering from pneumonia may not have fever all the times, may or may not have a cough, white blood cell test can be in a normal range.
  • Symptoms Disease uncertainty
    • High fever is typical for many diseases (e.g. bacterial diseases) and does not point specifically to pneumonia
    • Fever, cough, paleness, high WBC count combined do not always point to pneumonia

CS 2001 Bayesian belief networks

Representing certainty factors

  • Facts (propositional statements about the world) are assigned some certainty number reflecting the belief in that the statement is satisfied:
  • Rules incorporate tests on the certainty values
  • Methods for combination of conclusions

CF ( Pneumonia = True )= 0. 7

( A in[ 0. 5 , 1 ])∧ ( B in[ 0. 7 , 1 ])→ C with CF= 0.

( A in[ 0. 5 , 1 ])∧ ( B in[ 0. 7 , 1 ])→ C with CF= 0. ( E in[ 0. 8 , 1 ])∧ ( D in[ 0. 9 , 1 ])→ C with CF= 0. CF ( C )=max[ 0. 9 ; 0. 8 ]= 0. 9 CF ( C )= 0. 9 * 0. 8 = 0. 72 CF ( C )= 0. 9 + 0. 8 − 0. 9 * 0. 8 = 0. 98

?

CS 2001 Bayesian belief networks

Probability theory

a well-defined coherent theory for representing uncertainty and for reasoning with it Representation: Proposition statements – assignment of values to random variables

Probabilities over statements model the degree of belief in these statements P ( Pneumonia = True )= 0. 001

P ( Pneumonia = True , Fever = True )= 0. 0009 P ( Pneumonia = False , WBCcount = normal , Cough = False )= 0. 97

P ( WBCcount = high )= 0. 005

Pneumonia = True WBCcount = high

CS 2001 Bayesian belief networks

Joint probability distribution Joint probability distribution (for a set variables)

  • Defines probabilities for all possible assignments to values of variables in the set

P ( WBCcount )

0. 005 0.^9930.^002

P ( pneumonia , WBCcount )

high normal^ low

Pneumonia True False

WBCcount

P ( Pneumonia )

Marginalization (summing of rows, or columns)

  • summing out variables

2 × 3 table

CS 2001 Bayesian belief networks

Conditional probability distribution

Conditional probability distribution:

  • Probability distribution of A given (fixed B)
  • Conditional probability is defined in terms of joint probabilities
  • Joint probabilities can be expressed in terms of conditional probabilities
  • Conditional probability – is useful for diagnostic reasoning
    • the effect of a symptoms (findings) on the disease P ( Pneumonia = True | Fever = True , WBCcount = high , Cough = True )

P B

P A B = P AB

P ( A , B )= P ( A | B ) P ( B )

CS 2001 Bayesian belief networks

Modeling uncertainty with probabilities

  • Knowledge based system era (70s – early 80’s)
    • Extensional non-probabilistic models
    • Probability techniques avoided because of space, time and acquisition bottlenecks in defining full joint distributions
    • Negative effect on the advancement of KB systems and AI in 80s in general
  • Breakthrough (late 80s, beginning of 90s)
    • Bayesian belief networks
      • Give solutions to the space, acquisition bottlenecks
      • Significant improvements in the time cost of inferences

CS 2001 Bayesian belief networks

Bayesian belief networks (BBNs)

Bayesian belief networks.

  • Represent the full joint distribution more compactly with smaller number of parameters.
  • Take advantage of conditional and marginal independences among components in the distribution
  • A and B are independent
  • A and B are conditionally independent given C

P ( A , B )= P ( A ) P ( B )

P ( A , B | C )= P ( A | C ) P ( B | C )

P ( A | C , B )= P ( A | C )

CS 2001 Bayesian belief networks

Alarm system example.

  • Assume your house has an alarm system against burglary. You live in the seismically active area and the alarm system can get occasionally set off by an earthquake. You have two neighbors, Mary and John , who do not know each other. If they hear the alarm they call you, but this is not guaranteed.
  • We want to represent the probability distribution of events:
    • Burglary, Earthquake, Alarm, Mary calls and John calls

Burglary

JohnCalls

Alarm

Earthquake

MaryCalls

Causal relations

CS 2001 Bayesian belief networks

Bayesian belief network.

Burglary Earthquake

JohnCalls MaryCalls

Alarm

P (B) (^) P (E)

P (A|B,E)

P (J|A) P (M|A)

  1. Graph reflecting direct (causal) dependencies between variables
  2. Local conditional distributions relating variables and their parents

CS 2001 Bayesian belief networks

Full joint distribution in BBNs

Full joint distribution is defined in terms of local conditional distributions (obtained via the chain rule):

1 ,..

=

i n

P X X Xn P Xi pa Xi

M

A

B

J

E

P ( B = T , E = T , A = T , J = T , M = F ) =

Example:

P ( B = T ) P ( E = T ) P ( A = T | B = T , E = T ) P ( J = T | A = T ) P ( M = F | A = T )

Then its probability is:

Assume the following assignment of values to random variables

B = T , E = T , A = T , J = T , M = F

CS 2001 Bayesian belief networks

Independences in BBNs

  • 3 basic independence structures
  1. JohnCalls is independent of Burglary given Alarm
  2. Burglary is independent of Earthquake (not knowing Alarm) Burglary and Earthquake become dependent given Alarm !!
  3. MaryCalls is independent of JohnCalls given Alarm

Burglary

JohnCalls

Alarm

Burglary

Alarm

Earthquake

JohnCalls

Alarm

MaryCalls

CS 2001 Bayesian belief networks

Independences in BBNs

  • Other dependences/independences in the network
  • Earthquake and Burglary are dependent given MaryCalls
  • Burglary and MaryCalls are dependent (not knowing Alarm)
  • Burglary and RadioReport are independent given Earthquake
  • Burglary and RadioReport are dependent given MaryCalls

Burglary

JohnCalls

Alarm

Earthquake

MaryCalls

RadioReport

CS 2001 Bayesian belief networks

Parameters: full joint:

BBN:

Parameter complexity problem

  • In the BBN the full joint distribution is expressed as a product of conditionals (of smaller) complexity

Burglary

JohnCalls

Alarm

Earthquake

MaryCalls

1 ,..

=

i n

P X X Xn P Xi pa Xi

Parameters to be defined: full joint:

BBN:

CS 2001 Bayesian belief networks

Diagnosis of car engine

  • Diagnose the engine start problem

CS 2001 Bayesian belief networks

Car insurance example

  • Predict claim costs (medical, liability) based on application data

CS 2001 Bayesian belief networks

(ICU) Alarm network

CS 2001 Bayesian belief networks

CPCS

  • C omputer-based P atient C ase S imulation system (CPCS-PM) developed by Parker and Miller (at University of Pittsburgh)
  • 422 nodes and 867 arcs