mode-compute-6.md

+mode_compute=6 algorithm
+------------------------
+
+In some situations the posterior mode (that will be used to initialize
+the Metropolis Hastings (MH) and to define the jumping distribution)
+is hard to obtain with standard (gradient based or not) optimization
+routines. Option `mode_compute = 6`, triggers a
+Monte-Carlo based optimization routine. Actually, it is not a true
+optimization routine, the goal is only to identify an interesting
+region to start the Metropolis Hastings algorithm and an initial
+estimate of the posterior covariance matrix for the estimated
+parameters. The MH algorithm does not need to start from the posterior
+mode to converge to the posterior distribution. It is only required to 
+start from a point (in parameters space) with a high posterior density
+value and to use an estimate of the covariance matrix for the jumping
+distribution.
+
+In practice Dynare minimizes the opposite of the logged posterior
+kernel. Very often the default optimization algorithm fails to find a
+minimum with a positive definite definite hessian matrix. Consequently
+Dynare cannot run the MH, since a positive hessian matrix is required
+to approximate the posterior covariance matrix (the inverse of the
+Hessian matrix provides an accurate approximation of the covariance
+matrix if the posterior distribution is not too far from a Gaussian
+distribution).
+
+The `mode_compute=6` algorithm uses a Metropolis Hastings algorithm
+with a diagonal covariance matrix (prior variances or a covariance
+matrix proportional to unity) and continuously updates the posterior
+covariance matrix and the posterior mode estimates through the MH
+draws. After each MH draw $`\theta_t`$ in the posterior distribution the
+posterior mean, the posterior covariance and the posterior mode are
+updated as follows:
+```math
+\mu_t = \mu_{t-1} + \frac{1}{t}\left(\theta_t-\mu_{t-1}\right)
+```
+```math
+\Sigma_t = \Sigma_{t-1} + \mu_{t-1}\mu_{t-1}'-\mu_{t}\mu_{t}'+
+            \frac{1}{t}\left(\theta_t\theta_t'-\Sigma_{t-1}-\mu_{t-1}\mu_{t-1}'\right)
+```
+and
+```math
+\mathrm{mode}_t = 
+\begin{cases}
+\theta_t, & \text{if } p(\theta_t|\mathcal Y) > p(\mathrm{mode}_{t-1}|\mathcal Y) \\
+\mathrm{mode}_{t-1}, & \text{otherwise.}
+\end{cases}
+```
+where $`p(\bullet, \mathcal Y)`$ is the posterior density or kernel. Following options are available:
+ - `options_.gmhmaxlik.iterations` = [integer] to call repeatedly the optimization routine. Improves the estimates of the posterior covariance matrix and of the posterior mode. Default is 3.
+ - `options_.gmhmaxlik.number` = [integer] sets the number of simulations used in the estimation of the covariance matrix. Default is 20000.
+ - `options_.gmhmaxlik.nscale` = [integer] sets the number of simulations used when tuning the scale parameter. Default is 200000.
+ - `options_.gmhmaxlik.nclimb` = [integer] sets the number of simulations used when climbing the hill (last step). Default is 200000.
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