Comments on Hierarchical models, and the need for Bayes

1. IWSM, Chania, July 2002 Comments on Hierarchical models, and the need for Bayes Peter Green, University of Bristol, UK P.J.Green@bristol.ac.uk

2. Complex data structures • Multiple sources of variability • >1 strata • Measurement error, indirect observation • Random effects, latent variables • Hierarchical population structure (multi-level models) • Experimental regimes, missing data

3. Complex data structures, ctd. • …. all features prevalent in complex biomedical data, especially •  need for Hierarchical Models • e.g. many talks here at IWSM17 • generalised linear models are just not enough (and that’s not because of linearity or exponential families)

4. Inference in hierarchical models • it is important to • plug-in estimates generally lead to under-estimating variability of quantities of interest - AVOID! • we need a coherent calculus of uncertainty propagate all sources of variability

5. Inference in hierarchical models, ctd. • a coherent calculus of uncertainty? • we have one - it’s called Probability! •  full probability modelling of all variables • reported inference: joint distribution of unknowns of interest, given observed data • how? Bayes’ theorem

6. Costs and benefits • costs • more modelling work • computational issues (?) • benefits • valid analysis • avoiding ad-hoc decisions • counts all data once and once only!

7. Costs and benefits, ctd. • by-product • simultaneous, coherent inference about multiple targets • and the old question: what about sensitivity to prior assumptions? • if sensitivity analysis reveals strong dependence on prior among reasonable prior choices, how can you trust the non-Bayesian analysis?

8. A simple prediction problem (an example that plugging in is wrong) • We make 10 observations; their mean is 15 and standard deviation 2. • What is the chance that the next observation will be more than 19?

9. … prediction, continued • Can’t do much without assumptions - let’s suppose the data are normal…….. • …. 19 is 2 s.d.’s more than the mean, and the normal distribution probability of that is 2.3%

10. … prediction, continued • But this supposes that 15 and 2 are the population mean and s.d. • We ought to allow for our uncertainty in these numbers - they are only estimates • This is awkward to do for a non-Bayesian

11. … prediction, continued The Bayesian answer - (1) if the mean  and s.d.  were known, the answer would be 1-((19-)/) (2) we should average this quantity over the posterior distribution of (,) - I did this and got 4.5% - twice the ‘plug-in’ answer!

12. Summary (1) • Bayes inference is completely sound mathematically - ‘coherent’ • All your conclusions are self-consistent • Handles prediction properly • Allows sequential updating • No logical somersaults (confidence intervals, hypothesis tests) • Bayes estimators are often more accurate

13. Summary (2) • But it does require more input than just the data • Sensitivity to priors should be checked • Computation is an issue except in very simple problems - that’s true for non-Bayes too

14. Reading • Migon, H.S. and Gamerman, D. Statistical Inference: an integrated approach. Arnold, 1999. • Box, G.E.P. and Tiao, G.C. Bayesian inference in Statistical Analysis. Addison-Wesley, 1973. • Carlin, B.P. and Louis, T.A. Bayes and empirical Bayes methods for data analysis. Chapman and Hall, 1996. • Gelman, A., Carlin, J.B., Stern, H.S. and Rubin, D.B. Bayesian data analysis. Chapman and Hall, 1995.

15. Professor Peter Green Department of Mathematics University of Bristol Bristol BS8 1TW, UK tel: +44 117 928 7967 fax: 7999 P.J.Green@bristol.ac.uk