An extended maximum likelihood principle is described by which inverse solutions for problems with uncertainties in known model parameters can be treated. The method introduces the concept of an equivalent experimental noise which differs significantly from the measurement noise when the system response is sensitive to the uncertainties in the known parameters. When the equivalent noise varies smoothly and significantly over the range of uncertainty, the inverse solution tends to be independent of the uncertainties, By minimizing the equivalent noise through appropriate choice of a measurement protocol, an optimal experiment can be defined. Examples are given of designing an experiment for estimating conductivity and contact resistance when surface convective coefficients are uncertain,