We present and axiomatize several update rules for probabilities (and preferences) where there is no unique additive prior. In the context of non-additive probabilities we define and axiomatize Bayesian update rules; in the context of multiple (additive) priors we define maximum likelihood rules. It turns out that for decision makers which can be described by both theories, the two approaches coincide. Thus, we suggest an axiomatically based ambiguous beliefs update rule, which is needed for applications in many economic theory models. Journal of Economic Literature classification numbers: D80, D81, C11, C71.