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1 Structure and Uncertainty Peter Green, University of Bristol, 10 July 2003

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2 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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3 If your experiment needs statistics, you ought to have done a better experiment Statistics and science Ernest Rutherford (1871-1937)

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4 Organic chemist! said Tilley expressively. Probably knows no statistics whatever. Statistics and science Nigel Balchin (1908-1970) The Small Back Room

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5 Graphical models Modelling Inference Mathematics Algorithms

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6 Markov chains Graphical models Contingency tables Spatial statistics Sufficiency Regression Covariance selection Statistical physics Genetics AI

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

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8 1. Modelling Modelling Inference Mathematics Algorithms

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9 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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10 Structured systems A framework for building models, especially probabilistic models, for empirical data

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11 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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12 Mendelian inheritance - a natural structured model A O AB A O AO OO Mendel

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13 Ion channel model levels & variances model indicator transition rates hidden state data binary signal Hodgson and Green, Proc Roy Soc Lond A, 1999

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14 levels & variances model indicator transition rates hidden state data binary signal O1O1 O2O2 C1C1 C2C2 C3C3 * * * * * * * * * * *

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

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16 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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17 Mapping of rare diseases Larynx cancer in females in France, 1986-1993 (standardised ratios)

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18 Mapping of rare diseases observed counts random spatial effects covariate regression coefficient relative risks

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19 Mapping of rare diseases Estimated posterior probability of excess risk, using Hidden Markov model, G & Richardson, 2002

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20 Mapping of rare diseases using Hidden Markov model G & Richardson, 2002 Larynx cancer in females in France, 1986-1993 (standardised ratios) Posterior probability of excess risk

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22 Probabilistic expert systems

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23 2. Mathematics Modelling Inference Mathematics Algorithms

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24 Graphical models Use ideas from graph theory to represent structure of a joint probability distribution by encoding conditional independencies D EB C A F

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25 Genetics –pedigree (family connections) Lattice systems –interaction graph (e.g. nearest neighbours) Gaussian case –graph determined by non-zeroes in inverse variance matrix Where does the graph come from?

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26 ABCD A B C D ABCDABCD Inverse of (co)variance matrix: independent case

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27 ABCD non-zero A B C D ABCDABCD Inverse of (co)variance matrix: dependent case Few links implies few parameters - Occams razor

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28 Few links implies few parameters –stable estimation Parsimony –Occams razor Few links implies few parameters - Occams razor

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29 Conditional independence X and Z are conditionally independent given Y if, knowing Y, discovering Z tells you nothing more about X : p(X|Y,Z) = p(X|Y) X Z Y XYZ

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30 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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31 ante clinic survival survival and clinic are dependent and ante and clinic are dependent but survival and ante are CI given clinic Conditional independence

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32 D EB C A F Conditional independence provides a mathematical basis for splitting up a large system into smaller components

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33 B C E D A B F D E

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34 3. Inference Modelling Inference Mathematics Algorithms

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35 Bayesian or non-

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36 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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37 Repeated measures: seizure counts in a randomised trial of anti-convulsant therapy in epilepsy

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38 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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39 (thanks to J Mortera)

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41 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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42 4. Algorithms Modelling Inference Mathematics Algorithms

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43 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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44 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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45 Markov chain Monte Carlo ? Updating ? - need only look at neighbours

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46 Probability propagation 76 5 23 4 1 12 2672363456 2636 2 form junction tree

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47 root Message passing in junction tree

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48 root Message passing in junction tree

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49 Emission tomography Industry standard reconstruction, using Radon transform

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50 Emission tomography, continued Bayesian Reconstruction (G, 1994)

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51 Learning structure Learning a Bayesian network, for an ICU ventilator management system, from 10000 cases on 37 variables (Spirtes & Meek, 1995)

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52 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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53 http://www.stats.bris.ac.uk/~peter P.J.Green@bristol.ac.uk …thanks to many

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54 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?

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