IP$,$ we can formulate a maximum a posterior $($MAP$)$ problem of the joint probability of x$1,...,$ x$5$ in the following factorized form:

p $($x$1,...,$ x$5)=1$

Z

fa $($x$1)$ fb $($x$2)$ fc $($x$3)$ fd $($x$4)$ fe $($x$5)$ fA $($x$1,$ x$2,$ x$3)$ fB $($x$3,$ x$4)$ fC $($x$4,$ x$5),$

where Z is the normalization constant, fa $($x$1)=$ exp $($x$1),$ fb $($x$2)=$ exp $(2$x$2),$ fc $($x$3)=$ exp $(3$x$3),$ fd $($x$4)=$

exp $(2$x$4),$ fe $($x$5)=$ exp $(2$x$5)$

fA $($x$1,$ x$2,$ x$3)=$

$1$ if x$1+$ x$2+$ x$3<=1$

$0$ otherwise $,$

fB $($x$3,$ x$4)=$

$1$ if x$3+$ x$4<=1$

$0$ otherwise $,$

fC $($x$4,$ x$5)=$

$1$ if x$4+$ x$5<=1$

$0$ otherwise