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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.

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# Question on Importance Sampling (Monte Carlo method)

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