in data

1
in data
structure
structure
in models
and

• uncertainty and complexity

2
What do I mean by structure?

• The key idea is conditional independence
• x and z are conditionally independent given y
  if p(x,zy) p(xy)p(zy)
• implying, for example, that
  p(xy,z) p(xy)
• CI turns out to be a remarkably powerful and
  pervasive idea in probability and statistics

3
How to represent this structure?

• The idea of graphical modelling we draw graphs
  in which nodes represent variables, connected by
  lines and arrows representing relationships
• We separate logical (the graph) and quantitative
  (the assumed distributions) aspects of the model

4
Contingency tables
Spatial statistics
Regression
Genetics
Graphical models
Markov chains
AI
Statistical physics
Sufficiency
Covariance selection
5
Graphical modelling 1

• Assuming structure to do probability calculations
• Inferring structure to make substantive
  conclusions
• Structure in model building
• Inference about latent variables

6
Basic DAG
in general
for example
7
A natural DAG from genetics
AB
AO
AO
OO
OO
8
A natural DAG from genetics
AB
AO
AO
OO
OO
9
DNA forensics example(thanks to Julia Mortera)

• A blood stain is found at a crime scene
• A body is found somewhere else!
• There is a suspect
• DNA profiles on all three - crime scene sample is
  a mixed trace is it a mix of the victim and
  the suspect?

10
DNA forensics in Hugin

• Disaggregate problem in terms of paternal and
  maternal genes of both victim and suspect.
• Assume Hardy-Weinberg equilibrium
• We have profiles on 8 STR markers - treated as
  independent (linkage equilibrium)

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DNA forensics

• The data
• 2 of 8 markers show more than 2 alleles at crime
  scene ?mixture of 2 or more people

12
DNA forensics in Hugin
13
DNA forensics

• Population gene frequencies for D7S820 (used as
  prior on founder nodes)

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(No Transcript)
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DNA forensics

• Results (suspectvictim vs. unknownvictim)

16
Graphical modelling 2

• Assuming structure to do probability calculations
• Inferring structure to make substantive
  conclusions
• Structure in model building
• Inference about latent variables

17
Conditional independence graph

• draw an (undirected) edge between variables ? and
  ? if they are not conditionally independent given
  all other variables

?
?
?
18
Infant mortality example

• Data on infant mortality from 2 clinics, by level
  of ante-natal care (Bishop, Biometrics, 1969)

19
Infant mortality example

• Same data broken down also by clinic

20
Analysis of deviance

• Resid Resid
• Df Deviance Df Dev
  P(gtChi)
• NULL 7 1066.43
  
• Clinic 1 80.06 6 986.36
  3.625e-19
• Ante 1 7.06 5 979.30
  0.01
• Survival 1 767.82 4 211.48
  5.355e-169
• ClinicAnte 1 193.65 3 17.83
  5.068e-44
• ClinicSurvival 1 17.75 2 0.08
  2.524e-05
• AnteSurvival 1 0.04 1 0.04
  0.84
• ClinicAnteSurvival 1 0.04 0 1.007e-12
  0.84

21
Infant mortality example
survival
ante
clinic
survival and clinic are dependent
and ante and clinic are dependent
but survival and ante are CI given clinic
22
Prognostic factors for coronary heart disease
Analysis of a 26 contingency table (Edwards
Havranek, Biometrika, 1985)
strenuous physical work?
smoking?
family history of CHD?
blood pressure gt 140?
strenuous mental work?
ratio of ? and ? lipoproteins gt3?
23
Graphical modelling 3

• Assuming structure to do probability calculations
• Inferring structure to make substantive
  conclusions
• Structure in model building
• Inference about latent variables

24
Modelling with undirected graphs

• Directed acyclic graphs are a natural
  representation of the way we usually specify a
  statistical model - directionally
• disease ? symptom
• past ? future
• parameters ? data ..
• However, sometimes (e.g. spatial models) there is
  no natural direction

25
Scottish lip cancer data

• The rates of lip cancer in 56 counties in
  Scotland have been analysed by Clayton and Kaldor
  (1987) and Breslow and Clayton (1993)
• (the analysis here is based on the example in the
  WinBugs manual)

26
Scottish lip cancer data (2)

• The data include

• the observed and expected cases (expected
  numbers based on the population and its age and
  sex distribution in the county),

• a covariate measuring the percentage of the
  population engaged in agriculture, fishing, or
  forestry, and

• the "position'' of each county expressed as a
  list of adjacent counties.

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Scottish lip cancer data (3)

• County Obs Exp x SMR Adjacent
• cases cases ( in counties
• agric.)
• 1 9 1.4 16 652.2 5,9,11,19
• 2 39 8.7 16 450.3 7,10
• ... ... ... ... ... ...
• 56 0 1.8 10 0.0 18,24,30,33,45,55

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Model for lip cancer data
(1) Graph
regression coefficient
covariate
random spatial effects
relative risks
observed counts
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Model for lip cancer data
(2) Distributions

• Data
• Link function
• Random spatial effects
• Priors

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WinBugs for lip cancer data

• Bugs and WinBugs are systems for estimating the
  posterior distribution in a Bayesian model by
  simulation, using MCMC
• Data analytic techniques can be used to summarise
  (marginal) posteriors for parameters of interest

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WinBugs for lip cancer data
Dynamic traces for some parameters
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WinBugs for lip cancer data
Posterior densities for some parameters
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Graphical modelling 4

• Assuming structure to do probability calculations
• Inferring structure to make substantive
  conclusions
• Structure in model building
• Inference about latent variables

34
Latent variable problems
variable unknown
variable known
edges known
edges unknown
value set unknown
value set known
35
Hidden Markov models
e.g. Hidden Markov chain
z0
z1
z2
z3
z4
hidden
y1
y2
y3
y4
observed
36
Hidden Markov models

• Richardson Green (2000) used a hidden Markov
  random field model for disease mapping

observed incidence
relative risk parameters
expected incidence
hidden MRF
37
Larynx cancer in females in France
SMRs
38
Latent variable problems
variable unknown
variable known
edges unknown
edges known
value set known
value set unknown
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Ion channel model choice
Hodgson and Green, Proc Roy Soc Lond A, 1999
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Example hidden continuous time models
O2
O1
C1
C2
C1
C2
C3
O1
O2
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Ion channelmodel DAG
model indicator
transition rates
hidden state
binary signal
levels variances
data
42
model indicator
C1
C2
C3
O1
O2
transition rates
hidden state
binary signal
levels variances
data











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Posterior model probabilities
.41
O1
C1
.12
O2
O1
C1
.36
O1
C1
C2
O2
O1
C1
C2
.10
44
Alarm network
Learning a Bayesian network, for an
ICU ventilator management system, from 10000
cases on 37 variables (Spirtes Meek, 1995)
45
Latent variable problems
variable unknown
variable known
edges known
edges unknown
value set known
value set unknown
46
Wisconsin students college plans
10,318 high school seniors (Sewell Shah, 1968,
and many authors since)
ses
sex
5 categorical variables sex (2) socioeconomic
status (4) IQ (4) parental encouragement
(2) college plans (2)
pe
iq
cp
47
(Vastly) most probable graph according to an
exact Bayesian analysis by Heckerman (1999)
ses
sex
5 categorical variables sex (2) socioeconomic
status (4) IQ (4) parental encouragement
(2) college plans (2)
pe
iq
cp
48
h
ses
sex
pe
iq
Heckermans most probable graph with one hidden
variable
decompos
cp