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MCMC Prior Distributions

The Markov Chain Monte Carlo method requires that a prior distribution is specified for each parameter in the estimation procedure.  Two types of parameters are estimated with the MCMC procedure.

Prior Distribution on Parameters specifying the Hyperdistribution.

Hyperdistributions in MARK are assumed to be normal distributions.  Thus, the mean and standard deviation are the 2 parameters to be estimated.  The prior distribution on the mean is specified as another normal, with the default being a mean of zero and a large standard deviation of 100.  This makes the prior distribution very flat over the range of the possible values of the mean.  For the standard deviation of a hyperdistribution, a gamma distribution prior is used with parameters alpha and beta.  The mean of a gamma distribution is alpha times beta, and the variance is alpha times beta^2.  Values of alpha = 1.0001 and beta = 0.0001 result in a reasonably flat prior distribution.   For rho parameters specified in the VC matrix, the prior is a uniform distribution over the range specified by the lower bound in the third edit box to the upper bound in the fourth edit box.  The default values for rho are -1 to 1, but to force the correlation to be positive, you might use 0 to 1, or forced to be negative, -1 to 1.

Note that for the beta parameters being modeled in the hyperdistribution, the prior is just the normal distribution defined by the hyperdistribution parameters.

Prior Distribution on Parameters not in Hyperdistributions.

Although the most likely use of the MCMC estimation procedure is to estimate the mean and variance of hyperdistributions, most models in MARK include other nuisance parameters in the models.  These parameters also require a prior distribution.  Three options are provided.  The first is to ignore the prior distribution, and never use it to decide whether a new value in the Markov Chain is accepted or rejected.  The second option is to specify a default prior distribution, consisting of a normal distribution with the mean and variance provided.  All parameters not included in a hyperdistribution will use this normal prior.  The third option is to specify the prior distribution for each parameter individually.  However, only normal priors are allowed, so you can only specify a mean and standard deviation appropriate for each of the non-hyperdistribution parameters.

Care must be taken in what prior is used for these parameters.  A mean of zero is appropriate for most parameters.  However, a very large standard deviation will result in a back-transformed logit value with a U-shaped distribution, i.e., the real parameter value is much more likely to take on the values close to zero or close to 1.  A standard deviation of 1.5 results in a back-transformed distribution that as about 95% of the probability between 0.05 and 0.95, so is actually results in a pretty reasonable prior distribution on the real scale.

Further consideration must be given to how the prior distributions of a function of the beta parameters will interact.  For example, the intercept and slope of a trend model might not be appropriately specified as normal(0, 1.5) priors, depending on how the beta parameters are expected to change over the range of the data.