$*$Multiplexer CPDs$.$ What is the form of the independence that is implied by the multiplexer CPD and that we used in our derivation of the posterior over the parameters of the simple Bayesian Network x$->$yx$->$y$?($i$.$e$.$ the factorization of P$(\backslash$theta x$,\backslash$theta Y$$x$1,\backslash$theta Y$$x$0$D$)$P$(\backslash$theta x$,\backslash$theta Y$$x$1,\backslash$theta Y$$x$0$D$)).$ Recall that a CPD is defined as a multiplexer if it has the structure P$($Y$$A$,$Z$1,...,$Zk$)=$I$\{$Y$=$Za$\}$P$($Y$$A$,$Z$1,...,$Zk$)=$I$\{$Y$=$Za$\}$ where the values of AA are the natural numbers $1$ through kk$.$ Also note that the answer is specific to multiplexer CPDs and is not implied by the graph structure alone.