An opportunity is provided to apply a menu of Bayesian prior probability
distributions, and their means and variances, to model parameters.
You may have occasion to undertake analyses where you have objective
information, or subjective beliefs, outside of the current model
and data concerning a parameter's value, and which you wish to have
considered by the current analysis.
Typically this information enters the model as Bayesian prior distributions
for the estimated parameters.
For example, you may already have a mean and standard error estimate
for a survivorship parameter from a previous study of the same population.
SmartStats © allows you to incorporate that information into
your current analysis by imposing a prior probability distribution
on the values for that parameter.
For a survivorship parameter, a beta distribution described by
its mean and variance might be the appropriate choice.
When such prior probability distributions are imposed upon parameter
values in a likelihood function minimizing analysis, the resulting
point estimate represents the most probable parameter value in the
posterior probability distribution for that parameter.
These estimates and their covariance matrix can be good starting
points for defining the posterior distributions of the parameters.
The mean of posterior probability distribution is referred to as
a Bayes estimate of the parameter value, but like the most probable
estimate, it has little inference value since the concept of a posterior
distribution presumes parameters to be random variables.
The more that the prior distribution of a parameter differs from
information on that parameter contained in the data being analyzed,
the more a maximum likelihood estimate and Bayes estimate will differ.
However, as sample size increases and asymptotic conditions prevail,
Bayes estimates converge to maximum-likelihood estimates with multinormal
properties.
Standard errors of those estimates can be calculated in the usual
way except that numerical derivatives, not analytical derivatives,
must be used to calculate the covariance matrix.
You might correctly interpret from this text that Bayes estimates
and posterior distributions generally have most relevance for situations
where sample sizes are small and prior objective information or
belief is strong.
In such cases the value of the posterior distribution is its robust
depiction of the probability distribution for values of their associated
parameter.
These distributions are likely to differ significantly from Gaussian
in shape and thus expressions of uncertainty using only means and
SEs are somewhat uninformative.
Estimates of the means, standard errors and shapes of parameter
posterior distributions can be produced
in SmartStats ©.
SmartStats © provides you with the option to save the Bayesian
prior distribution associated with an analysis in text files with
the suffix *.pri.
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