Motivation for decomposition Functional decomposition

causal influences on west side highway traffic. weather , gw bridge traffic screen off other influences

intuitively, reduction in representation size achieved because each variable depends on subset of other variables. thus, variable




x

1




{\displaystyle x_{1}}

depends directly on variable




x

2




{\displaystyle x_{2}}

, rather depending on entire set of variables. variable




x

2




{\displaystyle x_{2}}

screens off variable




x

1




{\displaystyle x_{1}}

rest of world. practical examples of phenomenon surround us, discussed in philosophical considerations below, let s consider particular case of northbound traffic on west side highway. let assume variable (





x

1





{\displaystyle {x_{1}}}

) takes on 3 possible values of { moving slow , moving deadly slow , not moving @ }. let s variable





x

1





{\displaystyle {x_{1}}}

depends on 2 other variables, weather values of { sun , rain , snow }, , gw bridge traffic values { 10mph , 5mph , 1mph }. point here while there many secondary variables affect weather variable (e.g., low pressure system on canada, butterfly flapping in japan, etc.) , bridge traffic variable (e.g., accident on i-95, presidential motorcade, etc.) these other secondary variables not directly relevant west side highway traffic. need (hypothetically) in order predict west side highway traffic weather , gw bridge traffic, because these 2 variables screen off west side highway traffic other potential influences. is, other influences act through them.

outside of purely mathematical considerations, perhaps greatest value of functional decomposition insight provides structure of world. when functional decomposition can achieved, provides ontological information structures exist in world, , how can predicted , manipulated. example, in illustration above, if learned





x

1





{\displaystyle {x_{1}}}

depends directly on





x

2





{\displaystyle {x_{2}}}

, means purposes of prediction of





x

1





{\displaystyle {x_{1}}}

, suffices know





x

2





{\displaystyle {x_{2}}}

. moreover, interventions influence





x

1





{\displaystyle {x_{1}}}

can taken directly on





x

2





{\displaystyle {x_{2}}}

, , nothing additional can gained intervening on variables



{

x

3


,

x

4


,

x

5


}


{\displaystyle \{x_{3},x_{4},x_{5}\}}

, since these act through





x

2





{\displaystyle {x_{2}}}

in case.