# Perfect samplers for mixtures of distributions

## Summary (1 min read)

### 2.2. The slice sampler

- While, for large values of n, a handwaving argument could justify the switch to a continuous state space, there exists a rigorous argument which validates this continuous embedding.
- If the authors apply the slice sampler to the continuous state space chain, the monotonicity argument holds.
- In particular, by moving the lower and upper chains downwards and upwards, the authors simply retard the moment of coalescence but ensure that the chains of interest will have coalesced at that moment.
- (In fact, this is impossible for large sample sizes.).
- Any value below (or above) will be acceptable.

### 3. The general case

- There is very little of what has been said in Section 2 that does not apply to the general case.
- The problem with the general case is not in extending the method, which does not depend on k, intrinsically, even though Algorithm 1] must be adapted to select the proper number of uniform random variables, but rather with nding a maximum starting value 1.
- The implementation of the slice sampler also gets more di cult as k increases; the authors are then forced to settle for simple accept-reject methods which are correct but may be slow.
- The authors describe in Sections 3.1 and 3.2 the particular cases of exponential and normal mixtures to show that perfect sampling can also be achieved in such settings.
- Note that the treatment of the Poisson case also extends to the general case, even if it may imply one numerical maximisation.

### 4. Conclusion

- The authors have obtained what they believe to be the rst general iid sampling method for mixture posterior distributions.
- This is of direct practical interest since mixtures are heavily used in statistical modelling and the corresponding inference is delicate (Titterington et al., 1985 , Robert, 1996) .
- The authors have also illustrated that perfect sampling can be achieved for realistic statistical models and not only for toy problems.

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### Cites background from "Perfect samplers for mixtures of di..."

...Casella et al. (2002) introduced a perfect sampling scheme, which is not easily extended to nonexponential families....

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111 citations

##### References

3,464 citations

### "Perfect samplers for mixtures of di..." refers background in this paper

...When considering realistic statistical models like those involving finite mixtures of distributions (Titterington et al., 1985), with densities of the form k Pi f(x Ii), ki=1 k(1) k EPi = 1, i=l...

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...When considering realistic statistical models like those involving finite mixtures of distributions (Titterington et al., 1985), with densities of the form k∑ i=1 pi f.x | θi/; k∑ i=1 pi = 1; .1/ Address for correspondence: C. P. Robert, Ceremade, Université Paris-Dauphine, Place du Maréchal de…...

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2,018 citations

### "Perfect samplers for mixtures of di..." refers result in this paper

...The constraint on this closed form representation is obviously that the prior distributions must be conjugate, but this is often the case in the literature (see Diebolt and Robert (1994) or Richardson and Green (1997))....

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1,235 citations

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895 citations

### "Perfect samplers for mixtures of di..." refers background or result in this paper

...The constraint on this closed form representation is obviously that the prior distributions must be conjugate, but this is often the case in the literature (see Diebolt and Robert (1994) or Richardson and Green (1997))....

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...One of the key features of the solution of Hobert et al. (1999) is to exploit the duality principle established by Diebolt and Robert (1994) for latent variable models....

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