Monte Carlo Newton-Raphson

If one uses the formulas (1.35) and (1.36) or (1.37) and (1.38) in the special case , there is considerable simplification. In this case, all the importance weights are . Hence, for example, (1.37) and (1.38) become

and

and one can use these approximations to do a Newton-Raphson update of the Monte Carlo approximation of the MLE.

Denote the current MCNR iterate by , then the Newton-Raphson update

gives the next iterate. There are, however, three problems with this approach, compared to the MCL approach.

• MCNR must iterate indefinitely, `until convergence' which is ill-defined. MCL requires only one sample if is close enough to so that the importance weights behave.
• Geyer (1994) gives an asymptotic approximation for the Monte Carlo error using MCL. No analogous formulas exist for MCNR.
• When is not close to neither approach works, but MCL can be fixed up by using a trust region in the maximization of the (Geyer and Thompson, 1992). No analogous fix-up exists for MCNR.

Thus MCNR has no benefits over MCL except that the calculations of the Monte Carlo approximation are slightly simpler. MCNR only makes sense as an approximation to MCL, which is the right thing.