Remark/Question about independency

From: andre (andrevh_at_sci.kun.nl)
Date: 02/03/05 

Date: Thu, 3 Feb 2005 12:34:19 +0000 (UTC)

Suppose x=(x1,x2,...,xN) consists of N observations, and the hypothesis
is that they are i.i.d. following a pdf f(.;y), where y is some
parameter:

P(xj|y) = f(xj;y)

and

P(x|y) = f(x1;y)f(x2;y)...f(xN;y) .

What I find somehow surprising is that the independency of the variables
only exists when y is known. If it is not certain what y is, if P(y) has
a non-zero width, then the observations must not be considered
independent, since (notation: Int(g(y),y)=integral of g(y) )

P(x) = Int( f(x1;y)f(x2;y)...f(xN;y)P(y), y)

while

P(x1)P(x2)...P(xN)
= Int( f(x1;y)P(y), y) Int( f(x2;y)P(y), y) ... Int( f(xN;y)P(y), y) .

Or am I doing something wrong here?