Profile likelihood is sometimes used to get estimates for the confidence limits of parameters from an n-dimension parameter fit to a model. It can be used for example instead of Monte Carlo estimation. I don't understand the intuition of the algorithm itself. See section 4.4 of the paper "Parameter uncertainty in biochemical models described by ordinary differential equations", by Vanlier et al. (2013), Math Biosci.
Assume a model has been optimized and a minimum located. According to the algorithm, a parameter is selected and slowly changed. After each change, all the other unchanged parameters are re-optimized at the new value of the changed parameter. The chi-square at this new optimized point is recorded. This is repeated until a chi-square profile is obtained. This process can be applied to each parameter in turn and the change in chi-square can be used to define a confidence region for the particular parameter.
I'd like to understand the intuition as to why the other parameters must be optimized as we profile the selected parameter? Why for example couldn't we just change the parameter (leave the other parameters fixed) and observe how the chi-square changes away from the optimum? Wouldn't that tell us how the curvature changed and therefore give us information on how confident we are in the parameter?