1. 1 in data, and …uncertainty and complexity in models

2. 2 What do I mean by structure? The key idea is conditional independence: x and z are conditionally independent given y if p(x,z|y) = p(x|y)p(z|y) … implying, for example, that p(x|y,z) = p(x|y) CI turns out to be a remarkably powerful and pervasive idea in probability and statistics

3. 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. 4 Markov chains Graphical models Contingency tables Spatial statistics Sufficiency Regression Covariance selection Statistical physics Genetics AI

5. 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. 6 Basic DAG in general: for example: ab c d p(a,b,c,d)=p(a)p(b)p(c|a,b)p(d|c)

7. 7 Basic DAG ab c d p(a,b,c,d)=p(a)p(b)p(c|a,b)p(d|c)

8. 8 A natural DAG from genetics ABAO OO

9. 9 A natural DAG from genetics ABAO OO AOABAO

10. 10 DAG for a trivial Bayesian model  y

11. 11 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?

12. 12 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)

13. 13 DNA forensics in Hugin

14. 14 DNA forensics The data: 2 of 8 markers show more than 2 alleles at crime scene  mixture of 2 or more people

15. 15 DNA forensics Population gene frequencies for D7S820 (used as ‘prior’ on ‘founder’ nodes):

16. 16

17. 17 DNA forensics Results (suspect+victim vs. unknown+victim):

18. 18 How does it work? (1) Manipulate DAG to corresponding (undirected) conditional independence graph (draw an (undirected) edge between variables  and  if they are not conditionally independent given all other variables)   

19. 19 How does it work? (2) If necessary, add edges so it is triangulated (=decomposable)

20. 20 76 5 23 4 1 12 2672363456 2636 2 a cliqueanother cliquea separator For any 2 cliques C and D, C  D is a subset of every node between them in the junction tree (3) Construct junction tree

21. 21 How does it work? (4) any joint distribution with a triangulated graph can be factorised: until cliques separators

22. 22 How does it work? (5) ‘pass messages’ along junction tree: manipulate the terms of the expression until from which marginal probabilities can be read off

23. 23 Probabilistic expert systems : Hugin for ‘Asia’ example

24. 24 Limitations of message passing: –all variables discrete, or –CG distributions (both continuous and discrete variables, but discrete precede continuous, determining a multivariate normal distribution for them) of Hugin: –complexity seems forbidding for truly realistic medical expert systems

25. 25 Graphical modelling [2] Assuming structure to do probability calculations Inferring structure to make substantive conclusions Structure in model building Inference about latent variables

26. 26 Conditional independence graph draw an (undirected) edge between variables  and  if they are not conditionally independent given all other variables   

27. 27 Infant mortality example Data on infant mortality from 2 clinics, by level of ante-natal care (Bishop, Biometrics, 1969) :

28. 28 Infant mortality example Same data broken down also by clinic:

29. 29 Analysis of deviance Resid Resid Df Deviance Df Dev P(>|Chi|) 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 Clinic:Ante 1 193.65 3 17.83 5.068e-44 Clinic:Survival 1 17.75 2 0.08 2.524e-05 Ante:Survival 1 0.04 1 0.04 0.84 Clinic:Ante:Survival 1 0.04 0 1.007e-12 0.84

30. 30 Infant mortality example ante clinic survival survival and clinic are dependent and ante and clinic are dependent but survival and ante are CI given clinic

31. 31 Prognostic factors for coronary heart disease strenuous physical work? family history of CHD? strenuous mental work? blood pressure > 140? smoking? ratio of  and  lipoproteins >3? Analysis of a 2 6 contingency table (Edwards & Havranek, Biometrika, 1985)

32. 32 How does it work? Hypothesis testing approaches: Tests on deviances, possibly penalised (AIC/BIC, etc.), MDL, cross-validation... Problem is how to search model space when dimension is large

33. 33 How does it work? Bayesian approaches: Typically place prior on all graphs, and conjugate prior on parameters (hyper- Markov laws, Dawid & Lauritzen), then use MCMC (see later) to update both graphs and parameters to simulate posterior distribution

34. 34 For example, Giudici & Green (Biometrika, 2000) use junction tree representation for fast local updates to graph 76 5 23 4 1 12 2672363456 2636 2

35. 35 76 5 23 4 1 127 2672363456 2636 27 12 2

36. 36 Graphical modelling [3] Assuming structure to do probability calculations Inferring structure to make substantive conclusions Structure in model building Inference about latent variables

37. 37 Mixture modelling DAG for a mixture model k w y 

38. 38 Mixture modelling DAG for a mixture model k w z y 

39. 39 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

40. 40 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)

41. 41 Scottish lip cancer data (2) The data include 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. the observed and expected cases (expected numbers based on the population and its age and sex distribution in the county),

42. 42 Scottish lip cancer data (3) CountyObsExpxSMR Adjacent casescases(% in counties agric.) 191.416652.2 5,9,11,19 2398.716450.3 7,10.................. 5601.8100.0 18,24,30,33,45,55

43. 43 Model for lip cancer data (1) Graph observed counts random spatial effects covariate regression coefficient relative risks

44. 44 Model for lip cancer data Data: Link function: Random spatial effects: Priors: (2) Distributions

45. 45 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

46. 46 Bugs code for lip cancer data model { b[1:regions] ~ car.normal(adj[], weights[], num[], tau) b.mean <- mean(b[]) for (i in 1 : regions) { O[i] ~ dpois(mu[i]) log(mu[i]) <- log(E[i]) + alpha0 + alpha1 * x[i] / 10 + b[i] SMRhat[i] <- 100 * mu[i] / E[i] } alpha1 ~ dnorm(0.0, 1.0E-5) alpha0 ~ dflat() tau ~ dgamma(r, d) sigma <- 1 / sqrt(tau) } skip

47. 47 Bugs code for lip cancer data model { b[1:regions] ~ car.normal(adj[], weights[], num[], tau) b.mean <- mean(b[]) for (i in 1 : regions) { O[i] ~ dpois(mu[i]) log(mu[i]) <- log(E[i]) + alpha0 + alpha1 * x[i] / 10 + b[i] SMRhat[i] <- 100 * mu[i] / E[i] } alpha1 ~ dnorm(0.0, 1.0E-5) alpha0 ~ dflat() tau ~ dgamma(r, d) sigma <- 1 / sqrt(tau) }

48. 48 Bugs code for lip cancer data model { b[1:regions] ~ car.normal(adj[], weights[], num[], tau) b.mean <- mean(b[]) for (i in 1 : regions) { O[i] ~ dpois(mu[i]) log(mu[i]) <- log(E[i]) + alpha0 + alpha1 * x[i] / 10 + b[i] SMRhat[i] <- 100 * mu[i] / E[i] } alpha1 ~ dnorm(0.0, 1.0E-5) alpha0 ~ dflat() tau ~ dgamma(r, d) sigma <- 1 / sqrt(tau) }

49. 49 Bugs code for lip cancer data model { b[1:regions] ~ car.normal(adj[], weights[], num[], tau) b.mean <- mean(b[]) for (i in 1 : regions) { O[i] ~ dpois(mu[i]) log(mu[i]) <- log(E[i]) + alpha0 + alpha1 * x[i] / 10 + b[i] SMRhat[i] <- 100 * mu[i] / E[i] } alpha1 ~ dnorm(0.0, 1.0E-5) alpha0 ~ dflat() tau ~ dgamma(r, d) sigma <- 1 / sqrt(tau) }

50. 50 Bugs code for lip cancer data model { b[1:regions] ~ car.normal(adj[], weights[], num[], tau) b.mean <- mean(b[]) for (i in 1 : regions) { O[i] ~ dpois(mu[i]) log(mu[i]) <- log(E[i]) + alpha0 + alpha1 * x[i] / 10 + b[i] SMRhat[i] <- 100 * mu[i] / E[i] } alpha1 ~ dnorm(0.0, 1.0E-5) alpha0 ~ dflat() tau ~ dgamma(r, d) sigma <- 1 / sqrt(tau) }

51. 51 WinBugs for lip cancer data Dynamic traces for some parameters:

52. 52 WinBugs for lip cancer data Posterior densities for some parameters:

53. 53 How does it work? The simplest MCMC method is the Gibbs sampler: in each sweep, ‘visit’ each variable in turn, and replace its current value by a random draw from its full conditional distribution - i.e. its conditional distribution given all other variables including the data skip

54. 54 Full conditionals in a DAG Basic DAG factorisation Bayes’ theorem gives full conditionals involving only parents, children and spouses. Often this is a standard distribution, by conjugacy.

55. 55 Full conditionals for lip cancer for example:

56. 56 Beyond the Gibbs sampler Where the full conditional is not a standard distribution, other MCMC updates can be used: the Metropolis- Hastings methods use the full conditionals algebraically

57. 57 Limitations of MCMC You can’t beat errors Autocorrelation limits efficiency Possibly-undiagnosed failure to converge

58. 58 Graphical modelling [4] Assuming structure to do probability calculations Inferring structure to make substantive conclusions Structure in model building Inference about latent variables

59. 59 Latent variable problems variable unknownvariable known edges known value set known value set unknown edges unknown

60. 60 Hidden Markov models z0z0 z1z1 z2z2 z3z3 z4z4 y1y1 y2y2 y3y3 y4y4 e.g. Hidden Markov chain observed hidden

61. 61 relative risk parameters Hidden Markov models Richardson & Green (2000) used a hidden Markov random field model for disease mapping observed incidence expected incidence hidden MRF

62. 62 Larynx cancer in females in France SMRs

63. 63 Latent variable problems variable unknownvariable known edges known value set knownvalue set unknown edges unknown

64. 64 Ion channel model choice Hodgson and Green, Proc Roy Soc Lond A, 1999

65. 65 Example: hidden continuous time models O2O2 O1O1 C1C1 C2C2 O1O1 O2O2 C1C1 C2C2 C3C3

66. 66 Ion channel model DAG levels & variances model indicator transition rates hidden state data binary signal

67. 67 levels & variances model indicator transition rates hidden state data binary signal O1O1 O2O2 C1C1 C2C2 C3C3 * * * * * * * * * * *

68. 68 Posterior model probabilities O1O1 C1C1 O2O2 O1O1 C1C1 O2O2 O1O1 C1C1 C2C2 O1O1 C1C1 C2C2.41.12.36.10

69. 69 ‘Alarm’ network Learning a Bayesian network, for an ICU ventilator management system, from 10000 cases on 37 variables (Spirtes & Meek, 1995)

70. 70 Latent variable problems variable unknown variable known edges known value set knownvalue set unknown edges unknown

71. 71 Wisconsin students college plans 10,318 high school seniors (Sewell & Shah, 1968, and many authors since) 5 categorical variables: sex (2) socioeconomic status (4) IQ (4) parental encouragement (2) college plans (2) sessex pe iq cp

72. 72 sessex pe iq cp 5 categorical variables: sex (2) socioeconomic status (4) IQ (4) parental encouragement (2) college plans (2) (Vastly) most probable graph according to an exact Bayesian analysis by Heckerman (1999)

73. 73 h Heckerman’s most probable graph with one hidden variable sessex pe iq cp

74. 74 CSS book (Complex Stochastic Systems) Graphical models and Causality: S Lauritzen Hidden Markov models: H Künsch Monte Carlo and Genetics: E Thompson MCMC: P Green F den Hollander and G Reinert ed: O Barndorff-Nielsen, D Cox and C Klüppelberg, Chapman and Hall (2001)

75. 75 HSSS book (Highly Structured Stochastic Systems) Graphical models and causality –T Richardson/P Spirtes, S Lauritzen, P Dawid, R Dahlhaus/M Eichler Spatial statistics –S Richardson, A Penttinen, H Rue/M Hurn/O Husby MCMC –G Roberts, P Green, C Berzuini/W Gilks

76. 76 HSSS book (ctd) Biological applications –N Becker, S Heath, R Griffiths Beyond parametrics –N Hjort, A O’Hagan... with 30 discussants editors: N Hjort, S Richardson & P Green OUP (2002?), to appear

77. 77 Further reading J Whittaker, Graphical models in applied multivariate statistics, Wiley, 1990 D Edwards, Introduction to graphical modelling, Springer, 1995 D Cox and N Wermuth, Multivariate dependencies, Chapman and Hall, 1996 S Lauritzen, Graphical models, Oxford, 1996 M Jordan (ed), Learning in graphical models, MIT press, 1999