  We begin with a study of operations and the effects they measure. We define
the probability that an effect $a$ occurs when the system is in a state $\rho$
by $P_\rho (a)= tr(\rho a)$. If $P_\rho (a)\ne 0$ and $\mathcal{I}$ is an
operation that measures $a$, we define the conditional probability of an effect
$b$ given $a$ relative to $\mathcal{I}$ by \begin{equation*} P_\rho (b\mid a) =
tr[\mathcal{I} (\rho )b] /P_\rho (a) \end{equation*} We characterize when
Bayes' quantum second rule \begin{equation*} P_\rho (b\mid a)=\frac{P_\rho
(b)}{P_\rho (a)}\,P_\rho (a\mid b) \end{equation*} holds. We then consider
L\"uders and Holevo operations. We next discuss instruments and the observables
they measure. If $A$ and $B$ are observables and an instrument $\mathcal{I}$
measures $A$, we define the observable $B$ conditioned on $A$ relative to
$\mathcal{I}$ and denote it by $(B\mid A)$. Using these concepts, we introduce
Bayes' quantum first rule. We observe that this is the same as the classical
Bayes' first rule, except it depends on the instrument used to measure $A$. We
then extend this to Bayes' quantum first rule for expectations. We show that
two observables $B$ and $C$ are jointly commuting if and only if there exists
an atomic observable $A$ such that $B=(B\mid A)$ and $C=(C\mid A)$. We next
obtain a general uncertainty principle for conditioned observables. Finally, we
discuss observable conditioned quantum entropies. The theory is illustrated
with many examples.
