Structure and Uncertainty

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Structure and Uncertainty

• Peter Green, University of Bristol, 10 July 2003

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Statistician a term that is more or less
equivalent to that of Statesman. Galton,
Francis Memories of My Life Chapter XXI
We are not concerned with the very poor. They
are unthinkable, and only to be appreciated by
the statistician or the poet. Forster, E.M.
Howards End
Organic chemist! said Tilley expressively.
Probably knows no statistics whatever.
Balchin, Nigel The Small Back Room
Before the curse of statistics fell upon mankind
we lived a happy, innocent life, full of
merriment and go, and informed by fairly good
judgment. Belloc, Hilaire The Silence of the Sea
Like dreams, statistics are a form of
wish fulfillment. Baudrillard, Jean Cool
Memories
It is commonly believed that anyone who
tabulates numbers is a statistician. This is
like believing that anyone who owns a scalpel
is a surgeon. Hooke, R. How to Tell the Liars
from the Statisticians
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Statistics and science

• If your experiment needs statistics, you ought
  to have done a better experiment

Ernest Rutherford (1871-1937)
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Statistics and science

• Organic chemist! said Tilley expressively.
  Probably knows no statistics whatever.

Nigel Balchin (1908-1970) The Small Back Room
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Graphical models
Modelling
Mathematics
Inference
Algorithms
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Contingency tables
Graphical models
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Modular structure

• Basis for
• understanding the real system
• capturing important characteristics statistically
• defining appropriate methods
• computation
• inference and interpretation

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1. Modelling
Modelling
Mathematics
Inference
Algorithms
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Structured systems

• A framework for building models, especially
  probabilistic models, for empirical data
• Key idea -
• understand complex system
• through global model
• built from small pieces
• comprehensible
• each with only a few variables
• modular

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Structured systems

• A framework for building models, especially
  probabilistic models, for empirical data

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Structured systems

• Key idea -
• understand complex system
• through global model
• built from small pieces
• comprehensible
• each with only a few variables
• modular

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Mendelian inheritance - a natural structured model
A
AB
A
O
O
Mendel
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Ion channelmodel
model indicator
transition rates
hidden state
Hodgson and Green, Proc Roy Soc Lond A, 1999
binary signal
levels variances
data
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model indicator
C1
C2
C3
O1
O2
transition rates
hidden state
binary signal
levels variances
data











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Gene expression using Affymetrix chips
Zoom Image of Hybridised Array
Hybridised Spot
Single stranded, labeled RNA sample
Oligonucleotide element
20µm
Millions of copies of a specific oligonucleotide
sequence element
Expressed genes
Approx. ½ million different complementary
oligonucleotides
Non-expressed genes
1.28cm
Slide courtesy of Affymetrix
Image of Hybridised Array
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Gene expression is a hierarchical process

• Substantive question
• Experimental design
• Sample preparation
• Array design manufacture
• Gene expression matrix
• Probe level data
• Image level data

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Mapping of rare diseases
Larynx cancer in females in France, 1986-1993
(standardised ratios)
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Mapping of rare diseases
regression coefficient
covariate
random spatial effects
relative risks
observed counts
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Mapping of rare diseases
Estimated posterior probability of excess risk,
using Hidden Markov model, G Richardson, 2002
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Mapping of rare diseases using Hidden Markov model
Larynx cancer in females in France, 1986-1993
(standardised ratios)
Posterior probability of excess risk
G Richardson, 2002
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(No Transcript)
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Probabilistic expert systems
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2. Mathematics
Modelling
Mathematics
Inference
Algorithms
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Graphical models

• Use ideas from graph theory to
• represent structure of a joint probability
  distribution
• by encoding conditional independencies

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Where does the graph come from?

• Genetics
• pedigree (family connections)
• Lattice systems
• interaction graph (e.g. nearest neighbours)
• Gaussian case
• graph determined by non-zeroes in inverse
  variance matrix

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A B C D
A B C D
Inverse of (co)variance matrix independent case
A
B
C
D
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A B C D
A B C D
Inverse of (co)variance matrix dependent case
non-zero ? non-zero
A
B
C
D
Few links implies few parameters - Occams razor
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• Few links implies few parameters
• stable estimation
• Parsimony
• Occams razor

Few links implies few parameters - Occams razor
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Conditional independence

• X and Z are conditionally independent given Y if,
  knowing Y, discovering Z tells you nothing more
  about X p(XY,Z) p(XY)
• X ? Z ? Y

X
Y
Z
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Conditional independence

• as seen in data on perinatal mortality vs.
  ante-natal care.

Does survival depend on ante-natal care?
.... what if you know the clinic?
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Conditional independence
ante
survival
clinic
survival and clinic are dependent
and ante and clinic are dependent
but survival and ante are CI given clinic
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Conditional independence provides a mathematical
basis for splitting up a large system into
smaller components
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C
D
D
F
B
E
B
E
A
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3. Inference
Modelling
Mathematics
Inference
Algorithms
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or non-
Bayesian
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Bayesian paradigm in structured modelling

• borrowing strength
• automatically integrates out all sources of
  uncertainty
• properly accounting for variability at all levels
• including, in principle, uncertainty in model
  itself
• avoids over-optimistic claims of certainty

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Repeated measures seizure counts in a randomised
trial of anti-convulsant therapy in epilepsy
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Bayesian structured modelling

• borrowing strength
• automatically integrates out all sources of
  uncertainty
• for example in forensic statistics with DNA
  probe data..

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(thanks to J Mortera)
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(No Transcript)
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Bayesian structured modelling

• borrowing strength
• automatically integrates out all sources of
  uncertainty
• for example in modelling complex biomedical
  systems like ion channels..

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4. Algorithms
Modelling
Mathematics
Inference
Algorithms
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Algorithms for probability and likelihood
calculations

• Exploiting graphical structure
• Markov chain Monte Carlo
• Probability propagation (Bayes nets)
• Expectation-Maximisation
• Variational methods

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Markov chain Monte Carlo

• Subgroups of one or more variables updated
  randomly,
• maintaining detailed balance with respect to
  target distribution
• Ensemble converges to equilibrium target
  distribution ( Bayesian posterior, e.g.)

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Markov chain Monte Carlo
?
?
Updating
- need only look at neighbours
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Probability propagation
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5
4
2
3
1
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Message passing in junction tree
root
root
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Message passing in junction tree
root
root
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Emission tomography

Industry standard reconstruction, using Radon
transform
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Emission tomography, continued

Bayesian Reconstruction (G, 1994)
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Learningstructure
Learning a Bayesian network, for an
ICU ventilator management system, from 10000
cases on 37 variables (Spirtes Meek, 1995)
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Structured systems success stories include...

• Genomics bioinformatics
• DNA protein sequencing,
  gene mapping, evolutionary genetics
• Spatial statistics
• image analysis, environmetrics,
  geographical epidemiology, ecology
• Temporal problems
• longitudinal data, financial time series, signal
  processing

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thanks to many
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The role of statistical modelling

• Discipline in creation of methodology
• Framework
• for study of foundations
• for expressing principles
• for provision of computational tools
• Use more to communicate ideas
• break down barriers between theory and practice?