MCMC (Markov Chain Monte Carlo)
MCMC is a Bayesian parameter estimation procedure that is most useful in MARK for estimating variance components. The variance components analysis in MARK was previously limited to a method of moments procedure. With the MCMC estimation procedure, the process variance-covariance of the beta parameters can be estimated. When you specify the hyperdistribution parameters, i.e., the means and standard deviations, you have specified the hyperdistributions of the beta. However, you may want to go further and estimate the covariance or correlation (correlation(1,2) = covariance(1,2)/(sigma1*sigma2) between the beta parameters. For example, to assess the process correlation between the survival (S) and recovery rates (f) of the Brownie et al. (1985) band recovery model, you must check the variance-covariance specification box.
When this box is checked, you are given the opportunity to specify the upper off-diagonal elements of the variance-covariance matrix. The default is all zero values, which is the same as if no variance-covariance matrix is specified. On the diagonal of the matrix are the sigma values (which you should not need to change if they were correctly specified in the specification of the hyperdistribution. You only have to specify the off-diagonal elements above the diagonal, as the matrix is symmetric and the lower off-diagonal elements should be blacked out. The following is an example for the default (no correlations) for 5 beta parameters in a single hyperdistribution.
sigma(1) 0 0 0 0
sigma(1) 0 0 0
sigma(1) 0 0
A useful matrix to estimate the autocorrelation of the beta parameters is the following, again for just 5 beta values.
sigma(1) rho(1) rho(1)**2 rho(1)**3 rho(1)**4
sigma(1) rho(1) rho(1)**2 rho(1)**3
sigma(1) rho(1) rho(1)**2
The above matrix models the correlation between consecutive parameters. If beta(1) and beta(2) are correlated as rho(1), and beta(2) and beta(3) are also correlated as rho(1), then beta(1) and beta(3) have to be correlated with rho(1)**2. This autocorrelation matrix can be obtained easily by right-clicking the VC matrix window and selecting the Autocorrelation matrix option for the list of options. Likewise, you can get back to the default matrix of zero off-diagonal elements with the No correlation menu choice.
Another, more conceptually complex example, is modeling the correlation between sets of parameters. As an example, consider the process correlation between the survival rate (S) and the recovery rate (f) of the Brownie et al. (1985) models. For 3 banding occasions, with a different hyperdistribution for the survival rates and the recovery rates, this matrix would look like the following.
sigma(1) 0 rho(1) 0 0
sigma(1) 0 rho(1) 0
sigma(2) 0 0
The first 2 beta values are the survival rates, modeled with the first hyperdistribution. The next 3 parameters are the recovery rates, modeled with the second hyperdistribution. The 2 rho(1) entries correspond to the correlations of S1 with f1 and S2 with f2. The design matrix command Copy Value Diagonal is useful for creating large matrices where many values of the rho parameter must be entered.
You also have the option to paste a VC matrix into the window, but you may want to specify the entire matrix (both above and below the diagonal) and paste it into the window, because trying to construct the values to paste with different numbers of elements per row can be painful. For the values below the diagonal, specify zeros, as these will be ignored. Note that only zeros or rho values are legal values in the upper off-diagonal portion of the matrix, and only sigmas are valid on the diagonal. So, technically, this matrix is a correlation matrix above the diagonal, and a standard deviation vector on the diagonal.