
26 Statistical Computing in Nuclear Imaging
1.8.1 CHAIN RULE AND MARGINALIZATION
T
he chain rule allows expressing the joint probability (e.g., probability distri-
bution of vector QoI) and is the generalization of Eq uation (1.6),
p( f
1
|{z}
, f
2
, f
3
, . . . , f
I
|
{z }
) = p(f
1
|f
2
, f
3
, . . . , f
I
)p(f
2
, f
3
, . . . , f
I
) (1.12)
where underbraces indicate two probability distributio ns: probability distri-
bution of f
1
and joint probability distribution f
2
, . . . , f
I
. Applying the above
I − 1 additional times the original joint distribution p(f
1
, . . . , f
I
) can be ex-
pressed as a product:
p(f
1
, f
2
, f
3
, . . . , f
I
) = p(f
1
|f
2
, f
3
, . . . , f
I
)p(f
2
|f
3
, . . .