The Mathematics of Changing one's Mind, via Jeffrey's or via Pearl's
  update rule
  Evidence in probabilistic reasoning may be 'hard' or 'soft', that is, it may
be of yes/no form, or it may involve a strength of belief, in the unit interval
[0, 1]. Reasoning with soft, [0, 1]-valued evidence is important in many
situations but may lead to different, confusing interpretations. This paper
intends to bring more mathematical and conceptual clarity to the field by
shifting the existing focus from specification of soft evidence to accomodation
of soft evidence. There are two main approaches, known as Jeffrey's rule and
Pearl's method; they give different outcomes on soft evidence. This paper
argues that they can be understood as correction and as improvement. It
describes these two approaches as different ways of updating with soft
evidence, highlighting their differences, similarities and applications. This
account is based on a novel channel-based approach to Bayesian probability.
Proper understanding of these two update mechanisms is highly relevant for
inference, decision tools and probabilistic programming languages.
