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I have to integrate a 2 dimensional function between a range which has two peaks. I am trying to combine two Gaussian functions to get a distribution which is close to the function so as to use importance sampling to make a better estimate with lower variance. Do i have to draw from both the samples equally? My second question is how do i combine the samples drawn from both these distributions?

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If, when targeting the density $g(x)$ you use an importance density function of the form $$\alpha f_1(x)+(1-\alpha) f_2(x)$$ which is called a mixture distribution you can either

  1. simulate directly from the mixture, which amounts to simulate $$N_1\sim\mathcal{B}(n,\alpha)\quad x_1,\ldots,x_{N_1}\sim f_1\quad y_1,\ldots,y_{n-N_1}\sim f_2$$and use the importance weight$$\dfrac{g(x)}{\alpha f_1(x)+(1-\alpha) f_2(x)}$$for both the $x_i$'s and the $y_i$'s
  2. set $1\le n_1\le n-1$ in an arbitrary manner and simulate $$x_1,\ldots,x_{n_1}\sim f_1\quad y_1,\ldots,y_{n-n_1}\sim f_2$$and use the importance weight$$\dfrac{n g(x)}{n_1 f_1(x)+(n-n_1) f_2(x)}$$for both the $x_i$'s and the $y_i$'s

since both approaches are correct. The second approach is described in Owen and Zhou (2000, JASA) and extended in our adaptive multiple mixture paper.

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  • $\begingroup$ What if the we only know $f_1$ and $f_2$ up to a constant where these two constants are different? $\endgroup$
    – TrungDung
    Sep 12, 2023 at 14:00
  • $\begingroup$ If unbiasedness is not relevant, one can use bridge sampling to estimate the constants at no additional computational cost. $\endgroup$
    – Xi'an
    Sep 12, 2023 at 16:01
  • $\begingroup$ @Xian: Given the Bayes theorem, of the denominator, p(y), is intractable, so for a complicated model, estimate that constant p(y) is not straightforward and could be time consuming. Or maybe I do not understand you? $\endgroup$
    – TrungDung
    Sep 13, 2023 at 9:58
  • $\begingroup$ See e.g. arxiv.org/abs/0801.3887, projecteuclid.org/journalArticle/…, conservancy.umn.edu/bitstream/handle/11299/199589/Technical Report 568 Reweighting Monte Carlo Mixtures.pdf $\endgroup$
    – Xi'an
    Sep 14, 2023 at 12:19

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