Kurt Vedros

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Kurt Vedros
Bayesian Basics
CSNR Fellow, Idaho State University Undergraduate,
Nuclear Engineering Mentor Curtis L. Smith
August 8, 2006
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Summer Missions

• Learn as much as you can about Bayesian Belief
  Networks

with applications in risk analysis.

• Use SAPHIRE to model
• launch risk of MSL09

move to Idaho Falls with all of the kids
stuffed toys and wifes beanie babies...
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Probability Theory

• A mathematical reasoning of uncertainty
• Attempts to answer the question how likely is it
  that an event will occur?
• Applications in risk assessment and planning.
• Two main camps
• Frequentist
• Bayesian

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Frequentist

• Uses data and/or models to gain probability
  distributions
• Maximum Likelihood Estimates (MLEs), confidence
  intervals and p-values
• Uses data and the confidence placed in that data

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Frequentist Inference

• Infers one hypothesis as valid by statistically
  eliminating others
• A implies that x is unlikely (low frequency)
• x is true
• therefore A is not credible

http//faculty.washington.edu/eeholmes/Files/Ellis
on.pdf
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Bayesian

• Requires both a sampling model and a prior
  distribution on all unknown quantities in the
  model
• Prior and likelihood used to compute conditional
  distribution of unknowns
• Deals in degrees of belief or initial data
  tempered by ensuing data

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Bayesian Inference

• Infers one hypothesis as valid by comparing and
  deciding which is most probable among possible
  hypotheses
• A implies x occurs with frequency a
• B implies x occurs with frequency b
• x is true
• The odds of A being true to B being true are a to
  b
• Therefore if agtgtb then A is more credible than B

http//faculty.washington.edu/eeholmes/Files/Ellis
on.pdf
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Who are the Bayesians?

• Didnt Captain Kirk fight one in the old TV show?

Take that, you Overacting Frequentist!
Last Star Trek reference, I promise!
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Thomas Bayes

• 1702-1761
• Non-Conformist Minister
• Accomplished Mathematician
• Published posthumously Essay towards solving a
  problem in the doctrine of chances published in
  the Philosophical Transactions of the Royal
  Society of London

http//www-history.mcs.st-andrews.ac.uk/Biographie
s/Bayes.html
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Why Bayesian in Risk Analysis?

• Deals with uncertainty and delivers a level of
  threat
• Can utilize expert knowledge for priors with
  greater confidence
• Provides a dynamic system that relearns and
  updates based on recent data

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The Medical Diagnosis Problem

• 1 of women age forty who have a routine
  screening have breast cancer.
• 80 of women with breast cancer test positive.
• 9.6 of women without breast cancer test
  positive.
• If a woman tests positive, what is the
  probability that she actually has cancer?

http//yudkowsky.net/bayes/bayes.html
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Diagnosis?

• Correct answer is 7.8
• Medical Doctors tested on this problem averaged
  15 correct.
• Common mistakes
• Ignoring the original fraction of women with
  breast cancer.
• Ignoring the false positives.

Casscells, Schoenberger, and Grayboys 1978 Eddy
1982 Gigerenzer and Hoffrage 1995 http//yudkowsk
y.net/bayes/bayes.html
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Diagnosis Walkthrough

• Sample 10000 patients
• Before screening
• Group 1 100 women with breast cancer
• Group 2 9900 women without breast cancer
• After screening
• Group A 80 women with cancer and test
• Group B 20 women with cancer and test
• Group C 950 women without cancer and test
• Group D 8950 women without cancer and test

http//yudkowsky.net/bayes/bayes.html
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The Solution

• Our positives are Groups A and C, therefore
• Pr(cancerpositive)
  __A_ __80_ 7.8
• 
  AC 1030
• Stated another way
• Pr(cancerpositive)
  Pr(positivecancer)Pr(cancer)____________________
  
• Pr(positivecancer)
  Pr(cancer) Pr(positivecancer)Pr(cancer)
• ________.80.01_____
  __ 7.8
• (.80.01)((950/990
  0).99)

http//yudkowsky.net/bayes/bayes.html
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Bayes Theorem

• Derivation of Bayes Theorem from conditional
  probability
• also
• rearranged
• Bayes Theorem
• (Pr(B)?0)

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Bayes Theorem

• further
• Pr(B) Pr(BA)Pr(A) Pr(BA)Pr(A)
• Soooo, Bayes Theorem again is
• P(AB) ______Pr(BA) Pr(A)_______
• Pr(BA)Pr(A) Pr(BA)Pr(A)
• Which led to our correct diagnosis answer

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Bayesian Breakdown

• Bayes Theorem
• A The prior estimate (belief) or data
• B The posterior data, new evidence
• Pr(BA) Conditional probability of B given A is
  true
• P(B) Marginal probability of B S
  Pr(BAi)Pr(Ai)
• Mutually exclusive of assumptions of prior
  beliefs or datasets
• Pr(AB) Posterior probability
• Updated Conditional Probability based on new
  evidence

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Thats Niiiiice

• but, what are we going to do with it?

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Bayesian Belief Network (BBN)

• Variables are shown as nodes in a directed
  acyclic graph (DAG)
• Variables have mutually exclusive states
• Interrelations are shown by connecting arrows on
  the DAG

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Basic BBN
Tree is Sick
Tree is Dry
Tree Drops Leaves
Example based on Hugin tool tutorial, hugin.com
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BBN in Tables
Example based on Hugin tool tutorial, hugin.com
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BBN in Conditional Probability Tables



Example based on Hugin tool tutorial, hugin.com
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Static BBN

• A static BBN is concerned only with the current
  status

Sick Tree
Dry Tree
Drops Leaves
Example based on Hugin tool tutorial, hugin.com
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Dynamic BBN

• Dynamic BBNs include temporal information to
  predict the future based on the past information

Sick Tree
Dry Tree
Sick Tree at Harvest
Dry Tree at Harvest
Drops Leaves
Drops Leaves at Harvest
Example based on Hugin tool tutorial, hugin.com
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Dynamic BBN

• Addition of decision nodes can affect the end
  states

Cost
Treat
Sick Tree
Dry Tree
Sick Tree at Harvest
Dry Tree at Harvest
Drops Leaves
Drops Leaves at Harvest
Harvest
Example based on Hugin tool tutorial, hugin.com
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Dynamic BBN
Example based on Hugin tool tutorial, hugin.com
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Dynamic BBN
Example based on Hugin tool tutorial, hugin.com
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Dynamic BBN
Example based on Hugin tool tutorial, hugin.com
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Simplified NTR Testing BBN
Overall Risk
Transport
Material Diversion
Signal Features
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BBN Structure

• Signal Features (measurements, video footage,
  etc) are not affected by any other status
• Material Diversion has Signal Features as a
  parent
• Overall Risk status is
• affected by two parents
• Many factors involved as
• parents to Transport and Diversion

Overall Risk
Transport
Material Diversion
Signal Features
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Summary

• Bayes Theorem and Bayesian Inference allows a way
  to update probabilities of occurrence both
  forward and reverse.
• Bayesian Belief Networks can create useful
  interactive, real-time tools for decisions in
  Risk Assessment.

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References

• An Intuitive Explanation of Bayesian Reasoning,
  Eliezer Yudkowsky, http//yudkowsky.net/bayes/baye
  s.html
• Bayes and Empirical Bayes Methods for Data
  Analysis, Bradley P. Carlin, Thomas A. Lewis,
  2000, Chapman Hall
• Baysian Data Analysis, Gelman, Carlin, Stone,
  Rubin, 2004, Chapman Hall
• Hugin Expert A/S, http//www.hugin.com/
• In response to Ellison (2003) an alternate
  explanation of Frequentist versus Bayesian
  inference, E.E. Holmes, http//faculty.washington.
  edu/eeholmes/Files/Ellison.pdf
• Thomas Bayes, JOC/EFR, School of Mathematics and
  Statistics, University of St Andrews, Scotland,
  June 2004, http//www-history.mcs.st-andrews.ac.uk
  /Biographies/Bayes.html