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Suppose i have a bivariate random vector which i can simulate from, taking values in a given domain, and for the sake of simplicity let's suppose that it takes values in whole $R^2$.

Suppose now that the density of my random vector in a given subdomain (e.g [0,1]^2) is very small.

To simulate the random values conditionally on being in this given subdomain, the easy technique "simulate unconditionally and discard if it's not in the subdomain" wont be very efficient.

Is there a generic way to simulate conditionally on being in a sub-domain that would be more efficient that this easy trick ?

I have access to a random number generator from my bivariate law, but i don't have access to the law itself (no expression of density, cdf or whatever).

Maybe it's not the right place to post this ?

Robert Dodier
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lrnv
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  • If you only can simulate from R^2 and don't have any extra information, I don't see what else you can do other than simulating and discarding. It would seem that if we draw, say, (15.2, 32.3), then you'd like to map it into [0,1]^2, but there is no way to do that without extra assumptions/information. So, other than the random number generator, is there anything else we can use? – Julius Vainora Jan 06 '19 at 17:55
  • Take a look at rejection methods. In that approach, you sample from a distribution, say q, which is easy to sample over the subdomain, then you keep or reject the sample x according to the ratio of p(x)/q(x) where p is the distribution you really want. If I remember correctly the only requirement is that p(x) < q(x) everywhere in the subdomain. – Robert Dodier Jan 07 '19 at 18:27
  • @RobertDodier, the OP doesn't seem to know p. – Julius Vainora Jan 07 '19 at 20:29
  • @JuliusVainora yes, you're right, my mistake. – Robert Dodier Jan 08 '19 at 01:39
  • Yeah, i dont. I think that my only option is to simulate and reject if not in the domain. Thanks anyway – lrnv Jan 09 '19 at 13:42
  • I would use importance sampling. – Michael Baudin Aug 19 '23 at 13:45

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