Data Assimilation in Cardiovascular Fluid–Structure Interaction Problems: An Introduction - Fluid-Structure Interaction and Biomedical Applications - page 402

The marginal density function of one of the random variables w j in the pool may
be obtained by the joint one after integration over the range of the other variables,

Z

p W j D p W 1 W 2 :::W n d w 1 :::d w j 1 d w j C 1 :::d w n

n 1

R

Conditional Probability

The conditional p.d.f. of the random vector w given the occurrence of the random
vector y is defined as

p W;Y . w ; y /
p Y . y /

p W j Y . w j y / :

Similarly, we have the definition

p W;Y . w ; y /
p W . w /

p Y j W . y j w / ;

from which we obtain the Bayes law

p Y j W . y j w /p W . w /
p Y . y /

p W j Y . w j y / D : (6.5)

The conditional expectation is defined consequently as

Z

E . w j y / D w p W j Y . w j y /d w :

n

R

From the previous relations, it follows that

Z Z Z

E . w / D w p W . w /d w D w p W;Y . w ; y /d y d w

n

0

n

1

n

R R R

Z Z

@ A p Y d y D E

w p W j Y d w . E . w j y //:

R

n

R

n

6.2.2 Minimum Variance and Other In-Out Estimators

Let us consider a first example of estimator w of the random vector w upon data z
in the “steady” case—Fig. 6.3 .Let e w w be the estimate error, and define

J. e / D e T E e ; (6.6)