- #1
kasraa
- 16
- 0
Hi,
Suppose I have N iid samples from a distribution q, and I want to estimate another distributin, p, using those samples (Importance Sampling).
By "standard importance sampling", I mean the case where samples (prior samples. i.e. samples from q) have equal weights ([tex] w_i = 1/N [/tex]).
In the case of "standard importance sampling", I should perform these steps:
1) compute (unnormalized) weights for those sample according to [tex]p(s_i)/q(s_i)[/tex] ([tex]s_{i}[/tex] is the i'th sample from q)
2) normalize those weights
3) then an estimate of p would be this:
[tex] \hat{p} = \sum_{i=1}^N w_{i} \delta(i) [/tex]
(w_i are normalized weights computed at step 2. delta(i) is the Dirac delta function at s_i)
Now consider the case where samples (prior samples, i.e. samples from q) are weighted (differnt weights, and normalized. for example [tex] u_i [/tex]).
Is it enough (justified) to change the (unnormalized) weights (computed at step 1) to [tex]p(s_i)u_{i}/q(s_i)[/tex]?
(multiplying prior weights and "standard importance sampling" weights together?)
Thanks in advance.
Suppose I have N iid samples from a distribution q, and I want to estimate another distributin, p, using those samples (Importance Sampling).
By "standard importance sampling", I mean the case where samples (prior samples. i.e. samples from q) have equal weights ([tex] w_i = 1/N [/tex]).
In the case of "standard importance sampling", I should perform these steps:
1) compute (unnormalized) weights for those sample according to [tex]p(s_i)/q(s_i)[/tex] ([tex]s_{i}[/tex] is the i'th sample from q)
2) normalize those weights
3) then an estimate of p would be this:
[tex] \hat{p} = \sum_{i=1}^N w_{i} \delta(i) [/tex]
(w_i are normalized weights computed at step 2. delta(i) is the Dirac delta function at s_i)
Now consider the case where samples (prior samples, i.e. samples from q) are weighted (differnt weights, and normalized. for example [tex] u_i [/tex]).
Is it enough (justified) to change the (unnormalized) weights (computed at step 1) to [tex]p(s_i)u_{i}/q(s_i)[/tex]?
(multiplying prior weights and "standard importance sampling" weights together?)
Thanks in advance.