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I'd like to sample a pair of continuous parameters which has the constraint that one has to be smaller than the other one. I understand one approach is by rejection sampling by rejecting the samples that are violating this constraint. Another way would be propose a pair of parameters using a 2D Gaussian distribution that are centered at the current sample, and assign the bigger value to the first parameter and the smaller one to the second. However, I'm not sure if this still keeps detailed-balance property. Should I compute the forward and the backward probabilities? Any ideas would be appreciated.

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  • $\begingroup$ Might post a full answer later if i have time. I think the key is to sample, say, the big one first, then to consider the conditional distribution of the second one given the first. $\endgroup$ – John Madden May 18 '17 at 18:00
  • $\begingroup$ Thanks for the reply. I guess by doing this way, we need to make a clever choice of the conditional distribution of the second sample. This conditional distribution might not be symmetric anymore (since when we generate the second sample, it has to be always smaller than the first sample) $\endgroup$ – eva22hay Dec 16 '17 at 4:20
  • $\begingroup$ Yes, a distribution truncated on one side will almost surely lose any symmetry it had. This isn't important if you can write down the full conditional (Gibbs sampling has no symmetry requirements). If you can't you will need to do a MH step instead of a Metropolis step, still not a big deal. Depending on your computing environment, sampling from a truncated standard distribution can be easy and efficient. $\endgroup$ – John Madden Dec 16 '17 at 15:53

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