It is quite often useful to consider the
effect of some specific propositions in the context of some general
background evidence that remains fixed, rather than in the complete
absence of information. The following questions ask you to prove more
general versions of the product rule and Bayes’ rule, with respect to
some background evidence $\textbf{e}$:

1. Prove the conditionalized version of the general product rule: $${\textbf{P}}(X,Y \textbf{e}) = {\textbf{P}}(XY,\textbf{e}) {\textbf{P}}(Y\textbf{e})\ .$$

2. Prove the conditionalized version of Bayes’ rule in Equation (conditional-bayes-equation).

1. Prove the conditionalized version of the general product rule: $${\textbf{P}}(X,Y \textbf{e}) = {\textbf{P}}(XY,\textbf{e}) {\textbf{P}}(Y\textbf{e})\ .$$

2. Prove the conditionalized version of Bayes’ rule in Equation (conditional-bayes-equation).

1. Prove the conditionalized version of the general product rule:
$${\textbf{P}}(X,Y \textbf{e}) = {\textbf{P}}(XY,\textbf{e}) {\textbf{P}}(Y\textbf{e})\ .$$

2. Prove the conditionalized version of Bayes’ rule in
Equation (conditional-bayes-equation).