  Classical Bayes' rule lays the foundation for the classical causal relation
between cause (input) and effect (output). This causal relation is believed to
be universally true for all physical processes. Here we show, on the contrary,
that it is inadequate to establish correct correspondence between cause and
effect in quantum mechanics. In fact, there are instances within the framework
of quantum mechanics where the use of classical Bayes' rule leads to
inconsistencies in quantum measurement inferences, such as Frauchiger-Renner's
paradox. Similar inconsistency also appears in the context of Hardy's setup
even after assuming quantum mechanics as a non-local theory. As a remedy, we
introduce an input-output causal relation based on quantum Bayes' rule. It
applies to general quantum processes even when a cause (or effect) is in
coherent superposition with other causes (or effects), involves nonlocal
correlations as allowed by quantum mechanics, and in the cases where causes
belonging to one system induce effects in some other system as it happens in
quantum measurement processes. This enables us to propose a resolution to the
contradictions that appear in the context of Frauchiger-Renner's and Hardy's
setups. Our results thereby affirm that quantum mechanics, equipped with
quantum Bayes' rule, can indeed consistently explain the use of itself.
