Merge pull request #917 from stan-dev/fix-reference-model-jacobian

fix Jacobian documentation

Author: WardBrian

src/reference-manual/optimization.qmd

@@ -17,16 +17,19 @@ has the advantage of setting its own stepsize.
 
 ## General configuration
 
-All of the optimizers have the option of including the
-the log absolute Jacobian determinant of inverse parameter transforms
-in the log probability computation.
-Without the Jacobian adjustment, optimization
-returns the maximum likelihood estimate (MLE),
-$\mathrm{argmax}_{\theta}\ p(y | \theta)$,
-the value which maximizes the likelihood of the data given the parameters.
-Applying the Jacobian adjustment produces the maximum a posteriori estimate (MAP),
-that maximizes the value of the posterior density in the unconstrained space,
-$\mathrm{argmax}_{\theta}\ p(y | \theta)\,p(\theta)$.
+All of the optimizers have the option of including the the log
+absolute Jacobian determinant of inverse parameter transforms in the
+log probability computation. If the Jacobian adjustment is not
+included (the default), the optimization returns parameter values that
+correspond to a mode of the target in the constrained space (if such
+mode exists). Thus this option is useful for any optimization where we
+want to find the mode in the original constrained parameter space. If
+the Jacobian adjustment is included, the optimization returns
+parameter values that correspond to a mode in the unconstrained
+space. This is useful, for example, if we want to make a
+distributional approximation of the posterior at the mode (see,
+Laplace sampling, as then Jacobian adjustment needs to be included for
+correct results.
 
 All of the optimizers are iterative and allow the maximum number of
 iterations to be specified;  the default maximum number of iterations