  Multiplicative noise arises in inverse problems when, for example,
uncertainty on measurements is proportional to the size of the measurement
itself. The likelihood that arises is hence more complicated than that from
additive noise. We consider two multiplicative noise models: purely
multiplicative noise, and a mixture of multiplicative noise and additive noise.
Adopting a Bayesian approach, we provide conditions for the resulting posterior
distributions on Banach space to be continuous with respect to perturbations in
the data; the inclusion of additive noise in addition to multiplicative noise
acts as a form of regularization, allowing for milder conditions on the forward
map. Additionally, we show that MAP estimators exist for both the purely
multiplicative and mixed noise models when a Gaussian prior is employed, and
for the latter prove that they are consistent in the small noise limit when all
noise is Gaussian.
