Theory of Statistics - George Mason University

4.5 Bayesian Testing 365
• Bayes factors are very general and do not require alternative models to be
nested.
• Several techniques are available for computing Bayes factors, including
asymptotic approximations that are easy to compute using the output
from standard packages that maximize likelihoods.
• In “nonstandard” statistical models that do not satisfy common regularity
conditions, it can be technically simpler to calculate Bayes factors than to
derive non-Bayesian significance tests.
• The Schwarz criterion (or BIC) gives a rough approximation to the logarithm
of the Bayes factor, which is easy to use and does not require
evaluation of prior distributions. The BIC is
BIC = −2 log(L(θm|x)) + k log n,
where θm is the value of the parameters that specify a given model, k is
the number of unknown or free elements in θm, and n is the sample size.
The relationship is
−BIC/2 − log(BF)
→ 0,
log(BF)
as n → ∞.
• When we are interested in estimation or prediction, Bayes factors may be
converted to weights to be attached to various models so that a composite
estimate or prediction may be obtained that takes account of structural
or model uncertainty.
• Algorithms have been proposed that allow model uncertainty to be taken
into account when the class of models initially considered is very large.
• Bayes factors are useful for guiding an evolutionary model-building process.
• It is important, and feasible, to assess the sensitivity of conclusions to the
prior distributions used.
***** stuff to add:
pseudo-Bayes factors
training sample
arithmetic intrinsic Bayes factor
geometric intrinsic Bayes factor
median intrinsic Bayes factor
The Bayes Risk Set
A risk set can be useful in analyzing Bayesian procedures when the parameter
space is finite. If
Θ = {θ1, . . ., θk}, (4.55)
the risk set for a procedure T is a set in IR k :
{(z1, ..., zk) : zi = R(θi, T)}. (4.56)
Theory of Statistics c○2000–2013 James E. Gentle