Comments on Hierarchical models, and the need for Bayes Peter Green, University of Bristol, UK IWSM, Chania, July 2002

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

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)

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

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 Inference in hierarchical models, ctd.

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

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?

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?

… 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%

… 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

… 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!

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

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

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

Professor Peter Green Department of Mathematics University of Bristol Bristol BS8 1TW, UK tel: fax: 7999