# Question on Importance Sampling (Monte Carlo method)

1. Sep 8, 2010

### kasraa

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 ($$w_i = 1/N$$).

In the case of "standard importance sampling", I should perform these steps:

1) compute (unnormalized) weights for those sample according to $$p(s_i)/q(s_i)$$ ($$s_{i}$$ is the i'th sample from q)
2) normalize those weights
3) then an estimate of p would be this:
$$\hat{p} = \sum_{i=1}^N w_{i} \delta(i)$$

(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 $$u_i$$).

Is it enough (justified) to change the (unnormalized) weights (computed at step 1) to $$p(s_i)u_{i}/q(s_i)$$?
(multiplying prior weights and "standard importance sampling" weights together?)

2. Sep 9, 2010

### bpet

Possibly not, because any Monte Carlo simulation (including importance sampling) is essentially based on approximating the (cumulative) distribution by the empirical distribution, i.e.

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

where I is the Boolean indicator function and the $$X_i$$ 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

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

where the $$X_i$$ are samples such that

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

closely approximates Q(x).

3. Sep 10, 2010

Thanks.