Profile likelihood intervals - beta vs real parameters

[murray.efford]

I'm trying to get my head around profile likelihood for another application and naturally turned to MARK for insight... My problem arises when there is not a 1:1 relation between beta and real parameters. The procedure is clearcut when one wants to obtain an interval for a beta parameter, but what about real parameters? Fixing a real parameter (as required for PLI calculation) then implicitly fixes several betas at once and puts you outside likelihood theory. As I read the MARK help (in my old download), MARK takes an ad hoc approach to 'drag' the ('real') interval near where it should be:
"The method used in MARK to compute the profile likelihood lower bound for parameter i is to minimize the following function:
[-2 log likelihood of current parameter value - (-2 log likelihood of maximum likelihood estimates + c-hat*3.84) ]**2 - (maximum likelihood estimate of parameter i - current parameter value of i).
The first portion of this expression finds the value of the deviance that is c-hat*3.84 units larger than the deviance for the maximum likelihood parameter estimates. The second portion of the expression maximizes the difference between the maximum likelihood parameter estimate and the lower bound." [I'm guessing 'maximizes' is a typo, or maybe there's something I'm missing]

Perhaps it's best to restrict PLI to beta parameters (or, equivalently, real parameters with a 1:1 relation to beta parameters)? I'd appreciate any insights on this, or maybe references to the literature.

[murray.efford]

I've since discovered, with some help from David Fletcher, that the answer lies in Lagrange multipliers. Profile likelihood intervals can be got for a function of 'beta' parameters, which of course includes the 'real' parameters. This is included in the coming release of 'secr' (which ironically has little need of PLI because the asymptotic intervals have good coverage). For a biometrical example (not capture-recapture) see Fletcher & Faddy 2007 JABES 12:315-324. I'm still not clear how MARK is able to form PLI of real parameters without Lagrange multipliers... (maybe they're in there somewhere).

Murray