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MCMC Hyperdistributions


MCMC is a Bayesian procedure useful for estimating the variance components of the beta parameters.  In MARK, only normally distributed hyperdistributions on the beta parameters are available.  

Selection of Beta Parameters

If you specify 1 or more hyperdistributions to be modeled in the initial MCMC dialog window, then the list of beta parameters is presented.  From this list, you can select all of the beta parameters that will be included in any of the hyperdistributions.  Two ways are available to select the beta parameters.

In the edit box at the top of the window, you can specify ranges of parameter numbers separated by commas, e.g., specifying "1 to 15, 31 to 45" without the quotation marks would select parameters 1, 2, ..., 15, and 31, 32, ..., 45.

Specification of the Hyperdistribution Structure

Once beta parameters have been selected, you must specify the appropriate hyperdistribution mean (mu) and standard deviation (sigma) to associate with each of the beta parameters.  Note that if only one hyperdistribution is specified, then each of the beta parameters selected will be modeled with a mean [mu(1)] and standard deviation [sigma(1)].  However, if >1 hyperdistributions are requested, then you must specify which hyper parameters are associated with each of the beta parameters.  First, you are given the list of beta parameters with a drop down box next to it to select which mean [mu(1), mu(2), ...] to use to model the beta parameter.  These selections can be made rapidly with the tab key and the down arrow to scroll through the list of mu values in the drop down boxes.  Once a mean has been selected for each beta parameter, the process is repeated for standard deviations.

If you checked the Variance-Covariance checkbox in the initial MCMC dialog window, then you will also be offered the chance to fill in the upper-triangular portion of the variance-covariance (VC) matrix for the hyperdistributions.  Each of the upper triangular cells of the VC matrix can have an entry consisting of the correlation of the corresponding beta values.   The correlation parameter is specified as rho(x), where x takes on the value 1, 2, ... to correspond to the number of hyperdistribution parameters.  The notation corresponds to the means (mu(1), mu(2), ...) and standard deviations (sigma(1), sigma(2), ...) of the hyperdistributions.  In addition, you can specify powers of the rho parmeters, e.g., rho(1)**2, rho(1)**3, etc., to create an autocorrelation matrix.  See the MCMC VC matrix help page for more details.  Process correlation parameters can only be specified in the VC matrix.

Note that you do not have to relate means and standard deviations exactly.  For example, you might want to have a hyperstructure with 3 different mean values, but a common standard deviation.  So, you would specify 3 different mu values for the 3 sets of beta parameters, but only 1 standard deviation for the entire list of beta parameters.  Likewise, you can specify a common mean, but different standard deviations for a set of beta parameters.  This capability can also be implemented through the design matrix capability.

Specification of the Hyperparameter Priors and Estimation Inputs

Based on the lists of means and standard deviations selected, a new dialog window appears that is requesting information about how to estimate these hyperparameters.  Four inputs are requested for each hyperparameter.  

The first edit box is to specify the step size to be used to generate new parameter values with which to sample the posterior distribution.  As with the default step size, the goal is to tune the estimation so that approximately 45% of the steps are accepted.

The second edit box is to specify an initial value to use to start the Markov Chain.  The default is "compute", which tells MARK to compute an estimate from the initial values of the beta parameters.  Note that generally you should run the model that you want to use for MCMC estimation as a typical MARK analysis, so that you can provide initial estimates to start the MCMC estimation.

The third and fourth edit boxes specify the parameters for the prior distribution to be used with this hyperdistribution.  For mean parameters, a normally distributed prior is used.  The third edit box specifies the mean, and the fourth edit box specifies the standard deviation.  The defaults are mean = 0 and standard deviation = 100, giving a very flat and uninformative prior.  For sigma parameters, a gamma prior is used to model the sigma in a transformation: 1/sigma**2 is assumed to be distributed as a gamma distribution with parameters alpha (third edit box) and beta (fourth edit box).  Again, the defaults of alpha = 1.00001 and beta = 0.00001 result in a very flat, uninformative prior.  For rho parameters, 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.

Once these values are acceptable, you click the OK button to proceed.  Presumably you also checked the provide initial estimates check box, so that you will now be requested to supply these initial estimates.  The Retrieve button is the most efficient method to retrieve estimates from the model stored in the MARK Results Browser Window.