Since our reply is rather long, we are posting it as a new blog post. Please see:

–LW

]]>Rather than conditioning on ((p(x),pi(x)), O(X,R,YR)) {p,px being known functions}

what prevents one from conditioning instead on ((p(x),pi(x),MN(Y.i,p.IS(y)),O(X,R)) ?

{MN(Y.i,p.IS(y)) being the “known” importance sampling distribution of Y.i given pi(x.i) is known}

That is, trying to match the use of the importance sampling distribution by the Horwitz-Thompson estimator

by conditioning on that (perhaps stretching the meaning of known.)

Given the conditions in the problem you can write the joint in the following functional way:

p(x,y,r)=F(y, r, theta(x), pi(x))

for some function F. Given this relationship for a known pi(x) what is a reasonable prior for theta(x)?

We’ll here’s on natural criterion: theta_1 will have a higher prior probability than theta_2 if the entropy of the resulting p(x,y,r) is higher for theta_1 than it is for theta_2.

This natural criterion together with the functional relation F will make the prior for theta depend on pi! I’ll also note that it is easy to calulate F() and the resulting entropy of p(x,y,r) in terms of y,r,theta, and pi. So you can see the dependancy explicitly.

]]>Do you agree that the random variables and can be dependent (since they are both functions of X)

while the prior can make them independent functions i.e.

?

Larry

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