Population and Ecological Models
 
 
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  Setting Bayesian prior distributions  

 

 

 

     
   

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.