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Is the best estimate of power equal to the power of the best estimate

In an inverse problem, such as the determination of brain activity given magnetic field measurements outside the head, the main quantity of interest is often the power associated with a source. The `standard' way to determine this has been to find the best linear estimate of the source and calculate the power associated with this. This paper proposes an alternative method and then relationship to this previous method of estimation is explored both algebraically and by numerical simulation. 
In abstract terms the problem can be stated as follows. Let H be a Hilbert space with inner product . Let L be a linear map: H->R^n. Suppose that we are a given data vector b in R^n such that b=Lx+e where e is a vector of random variables with zero mean and given covariance matrix that represents measurement errors. The problem that is addressed in this paper is to estimate where X is an operator on H (e.g. the characteristic function of a region of interest). 
Keywords: Linear inverse problem, biomagnetic inverse problem, magnetoencephalography (MEG).