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Example 27
As the sample size increases, the likelihood function becomes more and more
tightly concentrated about the ML estimate. In that case, a diffuse prior tends to be
nearly flat or constant over the region where the likelihood is high; the shape of the
posterior distribution is largely determined by the likelihood, that is by the data
themselves.
Under a uniform prior distribution for θ, p(θ) is completely flat, and the posterior
distribution is simply a re-normalized version of the likelihood. Even under a non-
uniform prior distribution, the influence of the prior distribution diminishes as the
sample size increases. Moreover, as the sample size increases, the joint posterior
distribution for θ comes to resemble a normal distribution. For this reason, Bayesian
and classical maximum likelihood analyses yield equivalent asymptotic results
(Jackman, 2000). In smaller samples, if you can supply sensible prior information to
the Bayesian procedure, the parameter estimates from a Bayesian analysis can be more
precise. (The other side of the coin is that a bad prior can do harm by introducing bias.)
Bayesian Analysis and Improper Solutions
One familiar problem in the fitting of latent variable models is the occurrence of
improper solutions (Chen, Bollen, Paxton, Curran, and Kirby, 2001). An improper
solution occurs, for example, when a variance estimate is negative. Such a solution is
called improper because it is impossible for a variance to be less than 0. An improper
solution may indicate that the sample is too small or that the model is wrong. Bayesian
estimation cannot help with a bad model, but it can be used to avoid improper solutions
that result from the use of small samples. Martin and McDonald (1975), discussing
Bayesian estimation for exploratory factor analysis, suggested that estimates can be
improved and improper solutions can be avoided by choosing a prior distribution that
assigns zero probability to improper solutions. The present example demonstrates
Martin and McDonald’s approach to avoiding improper solutions by a suitable choice
of prior distribution.
About the Data
Jamison and Scogin (1995) conducted an experimental study of the effectiveness of a
new treatment for depression in which participants were asked to read and complete
the homework exercises in Feeling Good: The New Mood Therapy (Burns, 1999).
Jamison and Scogin randomly assigned participants to a control condition or an