  We propose a framework for general Bayesian inference. We argue that a valid
update of a prior belief distribution to a posterior can be made for parameters
which are connected to observations through a loss function rather than the
traditional likelihood function, which is recovered under the special case of
using self information loss. Modern application areas make it is increasingly
challenging for Bayesians to attempt to model the true data generating
mechanism. Moreover, when the object of interest is low dimensional, such as a
mean or median, it is cumbersome to have to achieve this via a complete model
for the whole data distribution. More importantly, there are settings where the
parameter of interest does not directly index a family of density functions and
thus the Bayesian approach to learning about such parameters is currently
regarded as problematic. Our proposed framework uses loss-functions to connect
information in the data to functionals of interest. The updating of beliefs
then follows from a decision theoretic approach involving cumulative loss
functions. Importantly, the procedure coincides with Bayesian updating when a
true likelihood is known, yet provides coherent subjective inference in much
more general settings. Connections to other inference frameworks are
highlighted.
