A.I. in health informatics lecture 3 clinical reasoning & probabilistic inference, II*

kevin small & byron wallace

*Slides  borrow  heavily  from  Andrew  Moore,  Weng-­‐Keen  Wong  and  Longin  Jan  Latecki  

today

•  probabilistic reasoning – Bayesian networks –  reasoning with uncertainty –  crucial building block for automated clinical

reasoning systems

•  review conditional independence and (a little) graph theory

introduction

•  diagnosing inhalational anthrax

•  observe the following symptoms –  patient has difficulty breathing –  patient has a cough –  patient has a fever –  patient has diarrhea –  patient has inflamed mediastinum

introduction

•  diagnoses often stated in probabilities (e.g. 30% chance of inhalational anthrax)

•  additional evidence should change your degree of belief in the diagnosis

•  how much evidence until absolutely certain?

•  Bayesian networks are a methodology for reasoning with uncertainty

review: random variables  

•  basic  element  of  probabilisAc  reasoning  

•  refers  to  an  event  drawn  from  a  distribuAon  modeling  the  uncertain  outcome  of  the  event  

Boolean random variables

•  takes the values true or false

•  can be thought of event occurred or event didn’t occur

•  examples notation –  patient has inhalational anthrax A –  patient has difficulty breathing B –  patient has a cough C –  patient has a fever F –  patient has diarrhea D –  patient has inflamed mediastinum M

joint probability distribution  

•  expresses  probability  between  arbitrary  number  of  variables  

•  for  each  combinaAon,  states  how  probable  said  combinaAon  is  

A   D   M   P(A,D,M)  

false   false   false   0.65  

false   false   true   0.03  

false   true   false   0.1  

false   true   true   0.04  

true   false   false   0.02  

true   false   true   0.06  

true   true   false   0.03  

true   true   true   0.07  

must  sum  to  1  

reasoning with the joint  

•  with  the  joint,  you  can  compute  anything  

•  may  need  need  marginalizaAon  and/or  Bayes’  rule  to  do  so    

A   D   M   P(A,D,M)  

false   false   false   0.65  

false   false   true   0.03  

false   true   false   0.1  

false   true   true   0.04  

true   false   false   0.02  

true   false   true   0.06  

true   true   false   0.03  

true   true   true   0.07  

€

p(D) = p(A,D,M) + p(A,D,¬M) + p(¬A,D,M) + p(¬A,D,¬M) = 0.15

€

p(A,M |D) =p(A,M,D)p(D)

= 0.467

€

p(A |M,D) =p(A,M,D)p(M,D)

= 0.636

problems with the joint  

•  not  a  compact  representaAon  –  requires  2n-­‐1  parameters  to  express  

–  requires  a  lot  of  data  to  accurately  esAmate  

•  (condiAonal)  independence  to  the  rescue!  

independence  

•  random  variables  A  and  B  are  independent  if  – p(A,B)  =  p(A)  p(B)  – p(A|B)  =  p(A)  – p(B|A)  =  p(B)  

knowledge  regarding  outcome  of  A  provides  no  addiAonal  informaAon  about  the  outcome  of  B  

independence  

•  independence  allows  compact  representaAon  

•  suppose  n  coin  flips  –  joint  requires  2n-­‐1  parameters  –  if  flips  independent,  requires  n  parameters  

conditional independence  

•  random  variables  A  and  B  are  condiAonally  independent  if  – p(A,B|C)  =  p(A|C)  p(B|C)  – p(A|B,C)  =  p(A|C)  – p(B|A,C)  =  p(B|C)  

knowledge  regarding  outcome  of  A  provides  no  addiAonal  informaAon  about  the  outcome  of  B  

Bayesian networks (finally!)  

•  a  Bayesian  network  G=(V,E)  is  composed  of  – a  directed  acyclic  graph  – a  set  of  condiAonal  probability  tables  (CPT)  

A  

B  

C   D  

B   D   P(D|B)  

false   false   0.02  

false   true   0.98  

true   false   0.05  

true   true   0.95  

A   B   P(B|A)  

false   false   0.01  

false   true   0.99  

true   false   0.7  

true   true   0.3  

B   C   P(C|B)  

false   false   0.4  

false   true   0.6  

true   false   0.9  

true   true   0.1  

A   P(A)  

false   0.6  

true   0.4  

semantics of structure  

A  

B  

C   D  

A   P(A)  

false   0.6  

true   0.4  

each  vertex  is  a  random  variable  

B  is  a  parent  of  D;  D  is  condiAoned  on  B  

B   D   P(D|B)  

false   false   0.02  

false   true   0.98  

true   false   0.05  

true   true   0.95  

each  vertex  has  CPT  p(Xi|Parents(Xi))  

•  a  Boolean  variable  with  n  parents  has  2n+1  entries  (2n  which  must  be  stored)  

•  note  what  must  sum  to  1  

conditional probability tables  

A  

B  

A  

B  

E  A   B   P(B|A)  

false   false   0.01  

false   true   0.99  

true   false   0.7  

true   true   0.3  

A   B   E   P(B|A,E)  

false   false   false   0.2  

false   false   true   0.1  

false   true   false   0.8  

false   true   true   0.9  

true   false   false   0.25  

true   false   true   0.98  

true   true   false   0.75  

true   true   true   0.02  

utility of Bayes nets  

•  two  important  properAes  – encodes  condiAonal  independence  relaAonships  between  random  variables  in  the  graph    

– compact  representaAon  of  the  joint  

X  

P1   P2  

C1   C2  

ND2  ND1  

given  parents  (P1,P2),  a  vertex  X  is  condiAonally  independent  of  its  non-­‐descendents  (ND1,ND2)  

calculating the joint  

•  can  compute  joint  using  Markov  condiAon  

€

p(A,B,¬C,D) = p(A)⋅ p(B | A)⋅ p(¬C |B)⋅ p(D |B)

€

p X1 = x1,…,Xn = xn( ) = p Xi = xi |Parents Xi( )( )i=1

n

∏

A  

B  

C   D  €

= 0.4⋅ 0.3⋅ 0.9⋅ 0.95 = 0.1026

inference  

•  compuAng  probabiliAes  specified  by  model  

•  generally  queries  of  the  form  

€

p(X | E)

A  

B  

C   D  

evidence  variable(s)  query  variable(s)  

inference  

•  compuAng  probabiliAes  specified  by  model  

•  let’s  try  

€

p(C | A)

A  

B  

C   D  

evidence  variable(s)  query  variable(s)  

to  the  board!  

bad news  

•  exact  inference  is  feasible  in  only  small  to  medium  sized  networks  

•  exact  inference  in  larger  networks  takes  a  long  Ame  

•  can  use  approximate  inference  

network structure  

•  use  domain  expert  knowledge  to  design  

•  learn  it  from  data  – not  trivial  

•  good  news  is  clinical  experAse  is  high  

A  

B  

C   D  

naïve Bayes  

•  another  opAon  is  to  make  strong  (condiAonal)  independence  assumpAons  

•  ogen  effecAve  for  classificaAon  models  

A  

B   C   D   F   M  

Bayes revisited  

•  posterior  =  (prior  *  likelihood)  /  evidence  

€

p(A |B,C,D,F,M) =p(A)⋅ p(B,C,D,F,M | A)

p(B,C,D,F,M)

A  

B   C   D   F   M  

conditional independence  

•  assume  input  variables  condiAonally  indendent  

A  

B   C   D   F   M  

€

p(A | X) =

p(A)⋅ p Xi | A( )i=1

n

∏p(X)

naïve Bayes classification  

•  since  p(X)  is  the  same  for  all  outcome  of  A  

€

ˆ a = argmaxa '∈A

p(A = a')⋅ p Xi | A = a'( )i=1

n

∏⎛

⎝ ⎜

⎞

⎠ ⎟

A  

B   C   D   F   M  

number of parameters  

•  joint  probability  distribuAon  

•  naïve  Bayes  

•  inference  runAme  

•  to  esAmate  parameters,  count  (and  smooth)  

€

O n A( )€

2n −1 = 63

€

A −1+ A Xi −1( ) =11i=1

n

∑

example  day   outlook   temperature   humidity   wind   tennis  

1   sunny   hot   high   weak   no  

2   sunny   hot   high   strong   no  

3   overcast   hot   high   weak   yes  

4   rain   mild   high   weak   yes  

5   rain   cool   normal   weak   yes  

6   rain   cool   normal   strong   no  

7   overcast   cool   normal   strong   yes  

8   sunny   mild   high   weak   no  

9   sunny   cool   normal   weak   yes  

10   rain   mild   normal   weak   yes  

11   sunny   mild   normal   strong   yes  

12   overcast   mild   high   strong   yes  

13   overcast   hot   normal   weak   yes  

14   rain   mild   high   strong   no  

Given  today  is  sunny,  cool  but  windy  with  high  humidity,  will  we  play  tennis?  

[Mitchell’s  Machine  Learning  Book]  

example  Given  today  is  sunny,  cool  but  windy  with  high  humidity,  will  we  play  tennis?  

€

p(T = no | X) ≈ p(T = no)p(O = sunny |T = no)

€

p(M = cool |T = no)p(H = high |T = no)

€

p(W = strong |T = no)

€

p(T = yes | X) ≈ p(T = yes)p(O = sunny |T = yes)

€

p(M = cool |T = yes)p(H = high |T = yes)

€

p(W = strong |T = yes)

€

≈914⎛

⎝ ⎜

⎞

⎠ ⎟ 29⎛

⎝ ⎜ ⎞

⎠ ⎟ 39⎛

⎝ ⎜ ⎞

⎠ ⎟ 39⎛

⎝ ⎜ ⎞

⎠ ⎟ 39⎛

⎝ ⎜ ⎞

⎠ ⎟ = 5.2e - 3

€

≈514⎛

⎝ ⎜

⎞

⎠ ⎟ 35⎛

⎝ ⎜ ⎞

⎠ ⎟ 15⎛

⎝ ⎜ ⎞

⎠ ⎟ 45⎛

⎝ ⎜ ⎞

⎠ ⎟ 35⎛

⎝ ⎜ ⎞

⎠ ⎟ = 2.1e - 2

PopulaAon-­‐wide  ANomaly  DetecAon  and  Assessment  (PANDA)  

•  a  detector  for  a  large-­‐scale  outdoor  release  of  inhalaAonal  anthrax  

•  massive  Bayes  net  

•  populaAon-­‐wide  means  each  person  has  their  own  subnetwork  in  the  model  

[Wong  et  al.,  KDD  2005]  

population-wide approach  

•  anthrax  is  non-­‐contagious  –  reflected  in  network  structure  

Time of Release

Person Model

Anthrax Release

Location of Release

Person Model Person Model

person model  Anthrax Release

Location of Release Time Of Release

Anthrax Infection

Home Zip

Respiratory from Anthrax

Other ED Disease

Gender Age Decile

Respiratory CC From Other

Respiratory CC

Respiratory CC When Admitted

ED Admit from Anthrax

ED Admit from Other

ED Admission

Anthrax Infection

Home Zip

Respiratory from Anthrax

Other ED Disease

Gender Age Decile

Respiratory CC From Other

Respiratory CC

Respiratory CC When Admitted

ED Admit from Anthrax

ED Admit from Other

ED Admission

… …

Yesterday never

False

15213

20-30 Female

Unknown

15146

50-60 Male

advanced topics  

•  learning  network  structure  – generally  a  search  procedure  

•  Markov  networks  – considers  undirected  edges  

•  influence  diagrams  – generalized  with  determinisAc  verAces  

•  more  inference  – variable  eliminaAon,  approximate  inference  

more?  

current  standard  bearer  

the  classic  

really  interesAng