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Question on Importance Sampling (Monte Carlo method)

  1. Sep 8, 2010 #1

    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.
  2. jcsd
  3. Sep 9, 2010 #2
    Possibly not, because any Monte Carlo simulation (including importance sampling) is essentially based on approximating the (cumulative) distribution by the empirical distribution, i.e.

    [tex] P(x) = Prob[X\le x] = E_P[I[X\le x]] \approx \frac{1}{N}\sum_{i=1}^N I[X_i\le x] [/tex]

    where I is the Boolean indicator function and the [tex]X_i[/tex] are taken from distribution P. To change the weights from (1/N) to other numbers you'd need to change the sampling method to ensure that the "weighted" empirical distribution remains a good approximation to the CDF.

    However if you do find a way to overcome that, the new importance sampling formula would easily follow from the change of measure formula, with

    [tex] P(x) = E_P[I[X\le x]] = E_Q\left[I[X\le x]\frac{dP}{dQ}\right] = E_Q\left[I[X\le x]\frac{p(X)}{q(X)}\right] \approx \sum_{i=1}^N w_i I[X_i\le x]\frac{p(X_i)}{q(X_i)} [/tex]

    where the [tex]X_i[/tex] are samples such that

    [tex]\sum_{i=1}^N w_i I[X_i\le x] [/tex]

    closely approximates Q(x).
  4. Sep 10, 2010 #3
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