Comments on Hierarchical models, and the need for Bayes

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Comments on Hierarchical models, and the need
for Bayes
IWSM, Chania, July 2002

• Peter Green,
• University of Bristol, UK
• P.J.Green_at_bristol.ac.uk

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Complex data structures

• Multiple sources of variability
• gt1 strata
• Measurement error, indirect observation
• Random effects, latent variables
• Hierarchical population structure (multi-level
  models)
• Experimental regimes, missing data

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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 thats not because of linearity or
  exponential families)

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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
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Inference in hierarchical models, ctd.

• a coherent calculus of uncertainty?
• we have one - its called Probability!
• ? full probability modelling of all variables
• reported inference joint distribution of
  unknowns of interest, given observed data
• how? Bayes theorem

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Costs and benefits

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

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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?

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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?

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prediction, continued

• Cant do much without assumptions - lets 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

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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

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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!

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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

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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 - thats true for non-Bayes too

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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.

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• Professor Peter Green
• Department of Mathematics
• University of Bristol
• Bristol BS8 1TW, UK
• tel [phone] fax 7999
• P.J.Green_at_bristol.ac.uk