We use local likelihood estimates of gravitational shear and convergence from lensed cosmic microwave background observations to estimate the projected mass spectral density. Typically there is an additive bias when using a plugin estimate of the spectral density from a noisy estimate of the random field. We explore the possibility of adjusting this bias by subtracting an approximate power spectrum of the noise in the reconstruction using unlensed simulations. We demonstrate some empirical results that suggest the remaining biases complement those seen in the quadratic estimate developed by Hu and Okamoto (ApJ 557:L79- L83, 2001; ApJ 574:566-574, 2002; Phys Rev D 67:083002, 2003). We finish the paper with a discussion regarding the potential scientific applications and the challenges associated with estimating the noise spectrum from simulations.