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Bayesian Estimation
Selecting Priors
A prior distribution quantifies the researcher’s belief concerning where the unknown
parameter may lie. Knowledge of how a variable is distributed in the population can
sometimes be used to help researchers select reasonable priors for parameters of
interest. Hox (2002) cites the example of a normed intelligence test with a mean of 100
units and a standard deviation of 15 units in the general population. If the test is given
to participants in a study who are fairly representative of the general population, then
it would be reasonable to center the prior distributions for the mean and standard
deviation of the test score at 100 and 15, respectively. Knowing that an observed
variable is bounded may help us to place bounds on the parameters. For instance, the
mean of a Likert-type survey item taking values 0, 1, …, 10 must lie between 0 and 10,
and its maximum variance is 25. Prior distributions for the mean and variance of this
item can be specified to enforce these bounds.
In many cases, one would like to specify prior distribution that introduces as little
information as possible, so that the data may be allowed to speak for themselves. A
prior distribution is said to be diffuse if it spreads its probability over a very wide range
of parameter values. By default, Amos applies a uniform distribution from
to to each parameter.
Diffuse prior distributions are often said to be non-informative, and we will use that
term as well. In a strict sense, however, no prior distribution is ever completely non-
informative, not even a uniform distribution over the entire range of allowable values,
because it would cease to be uniform if the parameter were transformed. (Suppose, for
example, that the variance of a variable is uniformly distributed from 0 to ; then the
standard deviation will not be uniformly distributed.) Every prior distribution carries
with it at least some information. As the size of a dataset grows, the evidence from the
data eventually swamps this information, and the influence of the prior distribution
diminishes. Unless your sample is unusually small or if your model and/or prior
distribution are strongly contradicted by the data, you will find that the answers from a
Bayesian analysis tend to change very little if the prior is changed. Amos makes it easy
for you to change the prior distribution for any parameter, so you can easily perform
this kind of sensitivity check.
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