Statistical Methods in Medical Research 4ed

566 Further Bayesian methods
probabilities can be found by applying the usual methods, i.e. the joint probability
of the data and model M1 is
p…y, M1† ˆ „ p…yju1, M1†p…u1jM1†p…M1† du1: …16:18†
There is a similar expression for p…y, M2†, so Bayes' theorem (§3.3) immediately
gives p…M1jy† ˆp…M1, y†=‰p…M1, y†‡p…M2, y†Š:
The usual way to express these quantities is through the ratio of the posterior
odds to the prior odds of model M1, a quantity known as the Bayes factor, BF
(§3.3); that is,
BF ˆ p…M1jy†=p…M2jy†
p…M1†=p…M2†
This kind of analysis can readily be extended to consider m competing models,
M1, ..., Mm. Bayes factors can then be used to make pairwise comparisons
between models. Alternatively, interest may be focused on the models with the
largest posterior probabilities.
There are some difficulties with the Bayes factor for comparing two models.
The first concerns the practicalities of computing BF via (16.18), as this involves
the specification of prior distributions for the parameters in each of the models.
If the priors have to be established using an elicitation procedure, then having to
accommodate several possible models can make the elicitation procedure much
more cumbersome and complicated. An approach which avoids the need to
specify prior distributions was developed by Schwarz (1978), who showed that
for samples of size n, with n large,
2 log BF 2 log LR …k2 k1† log n, …16:19†
where ki is the number of parameters in model Mi and LR is the usual (frequentist)
likelihood ratio between the models, i.e. LR ˆ p…yj ^ u1†=p…yj ^ u2†, where ^ ui is
the maximum likelihood estimator for model Mi. The quantity in (16.19) is
sometimes referred to as the Bayesian information criterion, (BIC).
It might be asked to what extent it is possible to approximate a quantity such
as the Bayes factor, which depends on prior distributions, using a quantity which
does not depend on any priors at all. However, the approximation in (16.19) is
valid only for large n, and for large n (and priors that are not too concentrated)
the likelihood plays a more important role in determining quantities such as
(16.18) than does the prior.
It might have been thought that an alternative to avoiding the specification of
the priors was to use non-informative priors. This would be possible if the noninformative
priors are proper but is not possible for improper priors. This is
because (16.18) does not define a proper distribution if the prior for the parameters
of the model is not proper. This occurs for exactly the same reason as was