  We develop a weighted local likelihood estimate for the parameters that
govern the local spatial dependency of a locally stationary random field. The
advantage of this local likelihood estimate is that it smoothly downweights the
influence of far away observations, works for irregular sampling locations, and
when designed appropriately, can trade bias and variance for reducing
estimation error. This paper starts with an exposition of our technique on the
problem of estimating an unknown positive function when multiplied by a
stationary random field. This example gives concrete evidence of the benefits
of our local likelihood as compared to na\"ive local likelihoods where the
stationary model is assumed throughout a neighborhood. We then discuss the
difficult problem of estimating a bandwidth parameter that controls the amount
of influence from distant observations. Finally we present a simulation
experiment for estimating the local smoothness of a local Mat\'ern random field
when observing the field at random sampling locations in $[0,1]^2$.
